Given a rational function H(x), the problem of rational function decomposition is to determine two rational functions J(x) and F(x), where J'<>0, F'<>0, such that their composition satisfies J o F = H. The FunctionDecomposition command handles the following related problem:

We have a system of i equations of the form (J[i] o F)(x) = H[i](x), where all the J[i] and H[i] are known while the unknown F (F' <> 0) is always the same. The input system is expected to have at least two equations of this form.

The J[i] and H[i] may depend on one unknown parameter, C, such that a solution F exists for only one value of C, which is to be determined. When this variable C to be solved for is indicated, the input system must contain a minimum of three equations involving F, x, and C.

All of {J[i], H[i], F, C} may also depend in a rational way on an arbitrary number, say k, of symbolic parameters a[k]. FunctionDecomposition is designed to efficiently solve this problem, which appears when solving differential equations using an equivalence approach. FunctionDecomposition is used by the Maple ODE solver dsolve. This type of function decomposition problem however appears in various areas and so the command has applications beyond differential equations.

When a solution {C, F(x)} decomposing such an overdetermined rational system exists, FunctionDecomposition returns a sequence, with the value of the parameter C such that F(x) exists - only one value of C is computed - followed by a set with all the solutions F(x) found. When no C parameter is specified, only the set with solutions for F(x) - when it exists - is returned. When the input system (the first argument) contains extra parameters a[k], only solutions {C, F(x)} depending on the a[k] in a rational way are computed.

$\mathrm{with}\left(\mathrm{DEtools}\&comma;\mathrm{FunctionDecomposition}\right)\&colon;$

$\mathrm{sys}≔\left[\frac{\frac{1}{4}{\left(9C-15F\left(x\right)C+15C^2{F\left(x\right)}^3+2-4C{F\left(x\right)}^2+2C^2{F\left(x\right)}^4\right)}^3}{C^3{\left(1-F\left(x\right)+C{F\left(x\right)}^3\right)}^5}=\frac{\frac{9}{4}{x}^3{\left(4+30x-12x^2-45x^3+36x^4\right)}^3}{{\left(-3x^2+2+3x^3\right)}^5}\&comma;\frac{C\left(4C^3{F\left(x\right)}^7-4F\left(x\right)+135C+60C^3{F\left(x\right)}^6+315C^2{F\left(x\right)}^3-315F\left(x\right)C-60C^2{F\left(x\right)}^4-12C^2{F\left(x\right)}^5-60C{F\left(x\right)}^2+60+12C{F\left(x\right)}^3\right)\left(1-F\left(x\right)+C{F\left(x\right)}^3\right)}{{\left(9C-15F\left(x\right)C+15C^2{F\left(x\right)}^3+2-4C{F\left(x\right)}^2+2C^2{F\left(x\right)}^4\right)}^2}=\frac{\frac{1}{3}\left(-3x^2+2+3x^3\right)\left(16-72x^2+1998x^4+240x-360x^3-540x^5-2889x^6+2025x^7\right)}{x^2{\left(4+30x-12x^2-45x^3+36x^4\right)}^2}\&comma;\frac{\frac{1}{4}\left(4C^3{F\left(x\right)}^7-4F\left(x\right)+135C+60C^3{F\left(x\right)}^6+315C^2{F\left(x\right)}^3-315F\left(x\right)C-60C^2{F\left(x\right)}^4-12C^2{F\left(x\right)}^5-60C{F\left(x\right)}^2+60+12C{F\left(x\right)}^3\right)\left(9C-15F\left(x\right)C+15C^2{F\left(x\right)}^3+2-4C{F\left(x\right)}^2+2C^2{F\left(x\right)}^4\right)}{C^2{\left(1-F\left(x\right)+C{F\left(x\right)}^3\right)}^4}=\frac{\frac{3}{4}\left(16-72x^2+1998x^4+240x-360x^3-540x^5-2889x^6+2025x^7\right)x\left(4+30x-12x^2-45x^3+36x^4\right)}{{\left(-3x^2+2+3x^3\right)}^4}\right]$

$\mathrm{sys}≔\left[\frac{{\left(9C-15F\left(x\right)C+15C^2{F\left(x\right)}^3+2-4C{F\left(x\right)}^2+2C^2{F\left(x\right)}^4\right)}^3}{4C^3{\left(1-F\left(x\right)+C{F\left(x\right)}^3\right)}^5}=\frac{9x^3{\left(36x^4-45x^3-12x^2+30x+4\right)}^3}{4{\left(3x^3-3x^2+2\right)}^5}\&comma;\frac{C\left(4C^3{F\left(x\right)}^7-4F\left(x\right)+135C+60C^3{F\left(x\right)}^6+315C^2{F\left(x\right)}^3-315F\left(x\right)C-60C^2{F\left(x\right)}^4-12C^2{F\left(x\right)}^5-60C{F\left(x\right)}^2+60+12C{F\left(x\right)}^3\right)\left(1-F\left(x\right)+C{F\left(x\right)}^3\right)}{{\left(9C-15F\left(x\right)C+15C^2{F\left(x\right)}^3+2-4C{F\left(x\right)}^2+2C^2{F\left(x\right)}^4\right)}^2}=\frac{\left(3x^3-3x^2+2\right)\left(2025x^7-2889x^6-540x^5+1998x^4-360x^3-72x^2+240x+16\right)}{3x^2{\left(36x^4-45x^3-12x^2+30x+4\right)}^2}\&comma;\frac{\left(4C^3{F\left(x\right)}^7-4F\left(x\right)+135C+60C^3{F\left(x\right)}^6+315C^2{F\left(x\right)}^3-315F\left(x\right)C-60C^2{F\left(x\right)}^4-12C^2{F\left(x\right)}^5-60C{F\left(x\right)}^2+60+12C{F\left(x\right)}^3\right)\left(9C-15F\left(x\right)C+15C^2{F\left(x\right)}^3+2-4C{F\left(x\right)}^2+2C^2{F\left(x\right)}^4\right)}{4C^2{\left(1-F\left(x\right)+C{F\left(x\right)}^3\right)}^4}=\frac{3\left(2025x^7-2889x^6-540x^5+1998x^4-360x^3-72x^2+240x+16\right)x\left(36x^4-45x^3-12x^2+30x+4\right)}{4{\left(3x^3-3x^2+2\right)}^4}\right]$

$\mathrm{map}\left(u\mapsto \mathrm{map}\left(\mathrm{degree}\&comma;\left[\left(\mathrm{numer}\&comma;\mathrm{denom}\right)\left(\mathrm{subs}\left(F\left(x\right)=F\&comma;\left(\mathrm{lhs}-\mathrm{rhs}\right)\left(u\right)\right)\right)\right]\&comma;x\right)\&comma;\mathrm{sys}\right)$

$\left[\left[15\&comma;15\right]\&comma;\left[10\&comma;10\right]\&comma;\left[12\&comma;12\right]\right]$

$\mathrm{FunctionDecomposition}\left(\mathrm{sys}\&comma;F\left(x\right)\&comma;C\right)$

$C=\frac{2}{3},\left\{F\left(x\right)=\frac{1}{x}\right\}$

$\mathrm{sys}≔\left[-\frac{729{\left(-72C{F\left(x\right)}^6-270C{F\left(x\right)}^4+54{F\left(x\right)}^4+36{F\left(x\right)}^2+135-15C{F\left(x\right)}^8+15{F\left(x\right)}^6+2{F\left(x\right)}^8+27C^2{F\left(x\right)}^8\right)}^3}{{F\left(x\right)}^4{\left(-2{F\left(x\right)}^4+9C{F\left(x\right)}^4-9{F\left(x\right)}^2-27\right)}^5}=\frac{729{\left(243-45a+13608b^3{x}^3+6804b^2{x}^2+1944bx-810a{x}^4+135a^2{x}^8+17010b^4{x}^4-1296a{x}^{3}b-3240a{x}^4{b}^2-4320a{x}^5{b}^3-3240a{x}^6{b}^4-1296a{x}^7{b}^5-216a{x}^8{b}^6-3240a{x}^{5}b-4860a{x}^6{b}^2-3240a{x}^7{b}^3-810a{x}^8{b}^4+54a^2{x}^4+216a^2{x}^{5}b+324a^2{x}^6{b}^2+216a^2{x}^7{b}^3+54a^2{x}^8{b}^4+36a^2{x}^6+72a^2{x}^{7}b+36a^2{x}^8{b}^2-1260a{b}^2{x}^2-2520a{b}^3{x}^3-3150a{b}^4{x}^4-2520a{b}^5{x}^5-1260a{b}^6{x}^6-360a{b}^7{x}^7-45a{b}^8{x}^8+90a^2{x}^{3}b+225a^2{x}^4{b}^2+300a^2{x}^5{b}^3+225a^2{x}^6{b}^4+90a^2{x}^7{b}^5+15a^2{x}^8{b}^6+56a^2{b}^2{x}^2+112a^2{b}^3{x}^3+140a^2{b}^4{x}^4+112a^2{b}^5{x}^5+56a^2{b}^6{x}^6+16a^2{b}^7{x}^7+2a^2{b}^8{x}^8-216a{x}^2+13608b^5{x}^5+6804b^6{x}^6+1944b^7{x}^7+243b^8{x}^8-360abx+15a^2{x}^2+16a^{2}bx+2a^2\right)}^3}{a{\left(1+bx\right)}^4{\left(2a+8abx+12a{b}^2{x}^2+8a{b}^3{x}^3+2a{b}^4{x}^4-27-108bx-162b^2{x}^2-108b^3{x}^3-27b^4{x}^4+9a{x}^2+18a{x}^{3}b+9a{x}^4{b}^2+27a{x}^4\right)}^5}\&comma;\frac{\frac{1}{3}\left(-2835+243{F\left(x\right)}^2-2349{F\left(x\right)}^4-927{F\left(x\right)}^6+14175C{F\left(x\right)}^4-495{F\left(x\right)}^8-105{F\left(x\right)}^{10}-10{F\left(x\right)}^{12}+4806C{F\left(x\right)}^6+4131C{F\left(x\right)}^8+900C{F\left(x\right)}^{10}+105C{F\left(x\right)}^{12}-8505C^2{F\left(x\right)}^8-1917C^2{F\left(x\right)}^{10}-360C^2{F\left(x\right)}^{12}+405C^3{F\left(x\right)}^{12}\right)\left(-2{F\left(x\right)}^4+9C{F\left(x\right)}^4-9{F\left(x\right)}^2-27\right)}{{\left(-72C{F\left(x\right)}^6-270C{F\left(x\right)}^4+54{F\left(x\right)}^4+36{F\left(x\right)}^2+135-15C{F\left(x\right)}^8+15{F\left(x\right)}^6+2{F\left(x\right)}^8+27C^2{F\left(x\right)}^8\right)}^2}=\frac{\frac{1}{3}\left(-10935+3240a-2405700b^3{x}^3-721710b^2{x}^2-131220bx+76545a{x}^4-42525a^2{x}^8-5412825b^4{x}^4+172530a{x}^{3}b+776385a{x}^4{b}^2+2070360a{x}^5{b}^3+3623130a{x}^6{b}^4+4347756a{x}^7{b}^5+3623130a{x}^8{b}^6+612360a{x}^{5}b+2143260a{x}^6{b}^2+4286520a{x}^7{b}^3+5358150a{x}^8{b}^4-12393a^2{x}^4-99144a^2{x}^{5}b-347004a^2{x}^6{b}^2-694008a^2{x}^7{b}^3-867510a^2{x}^8{b}^4-14418a^2{x}^6-86508a^2{x}^{7}b-216270a^2{x}^8{b}^2+213840a{b}^2{x}^2+712800a{b}^3{x}^3+1603800a{b}^4{x}^4+2566080a{b}^5{x}^5+2993760a{b}^6{x}^6+2566080a{b}^7{x}^7+1603800a{b}^8{x}^8-27000a^2{x}^{3}b-121500a^2{x}^4{b}^2-324000a^2{x}^5{b}^3-567000a^2{x}^6{b}^4-680400a^2{x}^7{b}^5-567000a^2{x}^8{b}^6-20790a^2{b}^2{x}^2-69300a^2{b}^3{x}^3-155925a^2{b}^4{x}^4-249480a^2{b}^5{x}^5-291060a^2{b}^6{x}^6-249480a^2{b}^7{x}^7-155925a^2{b}^8{x}^8+17253a{x}^2-8660520b^5{x}^5-10103940b^6{x}^6-8660520b^7{x}^7-5412825b^8{x}^8+38880abx-2700a^2{x}^2-3780a^{2}bx-315a^2+2835a^3{x}^{12}-243a^3{x}^{10}-486a^3{x}^{11}b-243a^3{x}^{12}{b}^2+2349a^3{x}^8+9396a^3{x}^{9}b+14094a^3{x}^{10}{b}^2+9396a^3{x}^{11}{b}^3+2349a^3{x}^{12}{b}^4+5562a^3{x}^{7}b+13905a^3{x}^8{b}^2+18540a^3{x}^9{b}^3+13905a^3{x}^{10}{b}^4+5562a^3{x}^{11}{b}^5+927a^3{x}^{12}{b}^6-170100a^2{x}^{9}b-255150a^2{x}^{10}{b}^2-170100a^2{x}^{11}{b}^3-42525a^2{x}^{12}{b}^4+3960a^3{x}^{5}b+13860a^3{x}^6{b}^2+27720a^3{x}^7{b}^3+34650a^3{x}^8{b}^4+27720a^3{x}^9{b}^5+13860a^3{x}^{10}{b}^6+3960a^3{x}^{11}{b}^7+495a^3{x}^{12}{b}^8+1050a^3{x}^{3}b+4725a^3{x}^4{b}^2+12600a^3{x}^5{b}^3+22050a^3{x}^6{b}^4+26460a^3{x}^7{b}^5+22050a^3{x}^8{b}^6+12600a^3{x}^9{b}^7+4725a^3{x}^{10}{b}^8+1050a^3{x}^{11}{b}^9+105a^3{x}^{12}{b}^{10}+927a^3{x}^6+495a^3{x}^4-2405700b^9{x}^9-721710b^{10}{x}^{10}+105a^3{x}^2-131220b^{11}{x}^{11}+10a^3+120a^{3}bx+660a^3{b}^2{x}^2+2200a^3{b}^3{x}^3+4950a^3{b}^4{x}^4+7920a^3{b}^5{x}^5+9240a^3{b}^6{x}^6+7920a^3{b}^7{x}^7+4950a^3{b}^8{x}^8+2200a^3{b}^9{x}^9+660a^3{b}^{10}{x}^{10}+120a^3{b}^{11}{x}^{11}+10a^3{b}^{12}{x}^{12}-288360a^2{x}^9{b}^3-216270a^2{x}^{10}{b}^4-86508a^2{x}^{11}{b}^5-14418a^2{x}^{12}{b}^6-694008a^2{x}^9{b}^5-347004a^2{x}^{10}{b}^6-99144a^2{x}^{11}{b}^7-12393a^2{x}^{12}{b}^8-324000a^2{x}^9{b}^7-121500a^2{x}^{10}{b}^8-27000a^2{x}^{11}{b}^9-2700a^2{x}^{12}{b}^{10}-69300a^2{b}^9{x}^9-20790a^2{b}^{10}{x}^{10}-3780a^2{b}^{11}{x}^{11}-315a^2{b}^{12}{x}^{12}+4286520a{x}^9{b}^5+2143260a{x}^{10}{b}^6+612360a{x}^{11}{b}^7+76545a{x}^{12}{b}^8+2070360a{x}^9{b}^7-10935b^{12}{x}^{12}+776385a{x}^{10}{b}^8+172530a{x}^{11}{b}^9+17253a{x}^{12}{b}^{10}+712800a{b}^9{x}^9+213840a{b}^{10}{x}^{10}+38880a{b}^{11}{x}^{11}+3240a{b}^{12}{x}^{12}\right)\left(2a+8abx+12a{b}^2{x}^2+8a{b}^3{x}^3+2a{b}^4{x}^4-27-108bx-162b^2{x}^2-108b^3{x}^3-27b^4{x}^4+9a{x}^2+18a{x}^{3}b+9a{x}^4{b}^2+27a{x}^4\right)}{{\left(243-45a+13608b^3{x}^3+6804b^2{x}^2+1944bx-810a{x}^4+135a^2{x}^8+17010b^4{x}^4-1296a{x}^{3}b-3240a{x}^4{b}^2-4320a{x}^5{b}^3-3240a{x}^6{b}^4-1296a{x}^7{b}^5-216a{x}^8{b}^6-3240a{x}^{5}b-4860a{x}^6{b}^2-3240a{x}^7{b}^3-810a{x}^8{b}^4+54a^2{x}^4+216a^2{x}^{5}b+324a^2{x}^6{b}^2+216a^2{x}^7{b}^3+54a^2{x}^8{b}^4+36a^2{x}^6+72a^2{x}^{7}b+36a^2{x}^8{b}^2-1260a{b}^2{x}^2-2520a{b}^3{x}^3-3150a{b}^4{x}^4-2520a{b}^5{x}^5-1260a{b}^6{x}^6-360a{b}^7{x}^7-45a{b}^8{x}^8+90a^2{x}^{3}b+225a^2{x}^4{b}^2+300a^2{x}^5{b}^3+225a^2{x}^6{b}^4+90a^2{x}^7{b}^5+15a^2{x}^8{b}^6+56a^2{b}^2{x}^2+112a^2{b}^3{x}^3+140a^2{b}^4{x}^4+112a^2{b}^5{x}^5+56a^2{b}^6{x}^6+16a^2{b}^7{x}^7+2a^2{b}^8{x}^8-216a{x}^2+13608b^5{x}^5+6804b^6{x}^6+1944b^7{x}^7+243b^8{x}^8-360abx+15a^2{x}^2+16a^{2}bx+2a^2\right)}^2}\&comma;-\frac{\frac{243\left(-2835+243{F\left(x\right)}^2-2349{F\left(x\right)}^4-927{F\left(x\right)}^6+14175C{F\left(x\right)}^4-495{F\left(x\right)}^8-105{F\left(x\right)}^{10}-10{F\left(x\right)}^{12}+4806C{F\left(x\right)}^6+4131C{F\left(x\right)}^8+900C{F\left(x\right)}^{10}+105C{F\left(x\right)}^{12}-8505C^2{F\left(x\right)}^8-1917C^2{F\left(x\right)}^{10}-360C^2{F\left(x\right)}^{12}+405C^3{F\left(x\right)}^{12}\right)}{{F\left(x\right)}^4}\left(-72C{F\left(x\right)}^6-270C{F\left(x\right)}^4+54{F\left(x\right)}^4+36{F\left(x\right)}^2+135-15C{F\left(x\right)}^8+15{F\left(x\right)}^6+2{F\left(x\right)}^8+27C^2{F\left(x\right)}^8\right)}{{\left(-2{F\left(x\right)}^4+9C{F\left(x\right)}^4-9{F\left(x\right)}^2-27\right)}^4}=\frac{243\left(-10935+3240a-2405700b^3{x}^3-721710b^2{x}^2-131220bx+76545a{x}^4-42525a^2{x}^8-5412825b^4{x}^4+172530a{x}^{3}b+776385a{x}^4{b}^2+2070360a{x}^5{b}^3+3623130a{x}^6{b}^4+4347756a{x}^7{b}^5+3623130a{x}^8{b}^6+612360a{x}^{5}b+2143260a{x}^6{b}^2+4286520a{x}^7{b}^3+5358150a{x}^8{b}^4-12393a^2{x}^4-99144a^2{x}^{5}b-347004a^2{x}^6{b}^2-694008a^2{x}^7{b}^3-867510a^2{x}^8{b}^4-14418a^2{x}^6-86508a^2{x}^{7}b-216270a^2{x}^8{b}^2+213840a{b}^2{x}^2+712800a{b}^3{x}^3+1603800a{b}^4{x}^4+2566080a{b}^5{x}^5+2993760a{b}^6{x}^6+2566080a{b}^7{x}^7+1603800a{b}^8{x}^8-27000a^2{x}^{3}b-121500a^2{x}^4{b}^2-324000a^2{x}^5{b}^3-567000a^2{x}^6{b}^4-680400a^2{x}^7{b}^5-567000a^2{x}^8{b}^6-20790a^2{b}^2{x}^2-69300a^2{b}^3{x}^3-155925a^2{b}^4{x}^4-249480a^2{b}^5{x}^5-291060a^2{b}^6{x}^6-249480a^2{b}^7{x}^7-155925a^2{b}^8{x}^8+17253a{x}^2-8660520b^5{x}^5-10103940b^6{x}^6-8660520b^7{x}^7-5412825b^8{x}^8+38880abx-2700a^2{x}^2-3780a^{2}bx-315a^2+2835a^3{x}^{12}-243a^3{x}^{10}-486a^3{x}^{11}b-243a^3{x}^{12}{b}^2+2349a^3{x}^8+9396a^3{x}^{9}b+14094a^3{x}^{10}{b}^2+9396a^3{x}^{11}{b}^3+2349a^3{x}^{12}{b}^4+5562a^3{x}^{7}b+13905a^3{x}^8{b}^2+18540a^3{x}^9{b}^3+13905a^3{x}^{10}{b}^4+5562a^3{x}^{11}{b}^5+927a^3{x}^{12}{b}^6-170100a^2{x}^{9}b-255150a^2{x}^{10}{b}^2-170100a^2{x}^{11}{b}^3-42525a^2{x}^{12}{b}^4+3960a^3{x}^{5}b+13860a^3{x}^6{b}^2+27720a^3{x}^7{b}^3+34650a^3{x}^8{b}^4+27720a^3{x}^9{b}^5+13860a^3{x}^{10}{b}^6+3960a^3{x}^{11}{b}^7+495a^3{x}^{12}{b}^8+1050a^3{x}^{3}b+4725a^3{x}^4{b}^2+12600a^3{x}^5{b}^3+22050a^3{x}^6{b}^4+26460a^3{x}^7{b}^5+22050a^3{x}^8{b}^6+12600a^3{x}^9{b}^7+4725a^3{x}^{10}{b}^8+1050a^3{x}^{11}{b}^9+105a^3{x}^{12}{b}^{10}+927a^3{x}^6+495a^3{x}^4-2405700b^9{x}^9-721710b^{10}{x}^{10}+105a^3{x}^2-131220b^{11}{x}^{11}+10a^3+120a^{3}bx+660a^3{b}^2{x}^2+2200a^3{b}^3{x}^3+4950a^3{b}^4{x}^4+7920a^3{b}^5{x}^5+9240a^3{b}^6{x}^6+7920a^3{b}^7{x}^7+4950a^3{b}^8{x}^8+2200a^3{b}^9{x}^9+660a^3{b}^{10}{x}^{10}+120a^3{b}^{11}{x}^{11}+10a^3{b}^{12}{x}^{12}-288360a^2{x}^9{b}^3-216270a^2{x}^{10}{b}^4-86508a^2{x}^{11}{b}^5-14418a^2{x}^{12}{b}^6-694008a^2{x}^9{b}^5-347004a^2{x}^{10}{b}^6-99144a^2{x}^{11}{b}^7-12393a^2{x}^{12}{b}^8-324000a^2{x}^9{b}^7-121500a^2{x}^{10}{b}^8-27000a^2{x}^{11}{b}^9-2700a^2{x}^{12}{b}^{10}-69300a^2{b}^9{x}^9-20790a^2{b}^{10}{x}^{10}-3780a^2{b}^{11}{x}^{11}-315a^2{b}^{12}{x}^{12}+4286520a{x}^9{b}^5+2143260a{x}^{10}{b}^6+612360a{x}^{11}{b}^7+76545a{x}^{12}{b}^8+2070360a{x}^9{b}^7-10935b^{12}{x}^{12}+776385a{x}^{10}{b}^8+172530a{x}^{11}{b}^9+17253a{x}^{12}{b}^{10}+712800a{b}^9{x}^9+213840a{b}^{10}{x}^{10}+38880a{b}^{11}{x}^{11}+3240a{b}^{12}{x}^{12}\right)\left(243-45a+13608b^3{x}^3+6804b^2{x}^2+1944bx-810a{x}^4+135a^2{x}^8+17010b^4{x}^4-1296a{x}^{3}b-3240a{x}^4{b}^2-4320a{x}^5{b}^3-3240a{x}^6{b}^4-1296a{x}^7{b}^5-216a{x}^8{b}^6-3240a{x}^{5}b-4860a{x}^6{b}^2-3240a{x}^7{b}^3-810a{x}^8{b}^4+54a^2{x}^4+216a^2{x}^{5}b+324a^2{x}^6{b}^2+216a^2{x}^7{b}^3+54a^2{x}^8{b}^4+36a^2{x}^6+72a^2{x}^{7}b+36a^2{x}^8{b}^2-1260a{b}^2{x}^2-2520a{b}^3{x}^3-3150a{b}^4{x}^4-2520a{b}^5{x}^5-1260a{b}^6{x}^6-360a{b}^7{x}^7-45a{b}^8{x}^8+90a^2{x}^{3}b+225a^2{x}^4{b}^2+300a^2{x}^5{b}^3+225a^2{x}^6{b}^4+90a^2{x}^7{b}^5+15a^2{x}^8{b}^6+56a^2{b}^2{x}^2+112a^2{b}^3{x}^3+140a^2{b}^4{x}^4+112a^2{b}^5{x}^5+56a^2{b}^6{x}^6+16a^2{b}^7{x}^7+2a^2{b}^8{x}^8-216a{x}^2+13608b^5{x}^5+6804b^6{x}^6+1944b^7{x}^7+243b^8{x}^8-360abx+15a^2{x}^2+16a^{2}bx+2a^2\right)}{a{\left(1+bx\right)}^4{\left(2a+8abx+12a{b}^2{x}^2+8a{b}^3{x}^3+2a{b}^4{x}^4-27-108bx-162b^2{x}^2-108b^3{x}^3-27b^4{x}^4+9a{x}^2+18a{x}^{3}b+9a{x}^4{b}^2+27a{x}^4\right)}^4}\right]\&colon;$