3.142 \(\int \frac{c+d x^2+e x^4+f x^6}{x^{10} (a+b x^2)^3} \, dx\)

Optimal. Leaf size=277 \[ -\frac{b^2 x \left (15 a^2 b e-11 a^3 f-19 a b^2 d+23 b^3 c\right )}{8 a^7 \left (a+b x^2\right )}-\frac{b^2 x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{4 a^6 \left (a+b x^2\right )^2}+\frac{3 a^2 b e+a^3 (-f)-6 a b^2 d+10 b^3 c}{3 a^6 x^3}-\frac{b \left (6 a^2 b e-3 a^3 f-10 a b^2 d+15 b^3 c\right )}{a^7 x}-\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (63 a^2 b e-35 a^3 f-99 a b^2 d+143 b^3 c\right )}{8 a^{15/2}}-\frac{a^2 e-3 a b d+6 b^2 c}{5 a^5 x^5}+\frac{3 b c-a d}{7 a^4 x^7}-\frac{c}{9 a^3 x^9} \]

[Out]

-c/(9*a^3*x^9) + (3*b*c - a*d)/(7*a^4*x^7) - (6*b^2*c - 3*a*b*d + a^2*e)/(5*a^5*x^5) + (10*b^3*c - 6*a*b^2*d +
3*a^2*b*e - a^3*f)/(3*a^6*x^3) - (b*(15*b^3*c - 10*a*b^2*d + 6*a^2*b*e - 3*a^3*f))/(a^7*x) - (b^2*(b^3*c - a*
b^2*d + a^2*b*e - a^3*f)*x)/(4*a^6*(a + b*x^2)^2) - (b^2*(23*b^3*c - 19*a*b^2*d + 15*a^2*b*e - 11*a^3*f)*x)/(8
*a^7*(a + b*x^2)) - (b^(3/2)*(143*b^3*c - 99*a*b^2*d + 63*a^2*b*e - 35*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*
a^(15/2))

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Rubi [A]  time = 0.603401, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {1805, 1802, 205} \[ -\frac{b^2 x \left (15 a^2 b e-11 a^3 f-19 a b^2 d+23 b^3 c\right )}{8 a^7 \left (a+b x^2\right )}-\frac{b^2 x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{4 a^6 \left (a+b x^2\right )^2}+\frac{3 a^2 b e+a^3 (-f)-6 a b^2 d+10 b^3 c}{3 a^6 x^3}-\frac{b \left (6 a^2 b e-3 a^3 f-10 a b^2 d+15 b^3 c\right )}{a^7 x}-\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (63 a^2 b e-35 a^3 f-99 a b^2 d+143 b^3 c\right )}{8 a^{15/2}}-\frac{a^2 e-3 a b d+6 b^2 c}{5 a^5 x^5}+\frac{3 b c-a d}{7 a^4 x^7}-\frac{c}{9 a^3 x^9} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2 + e*x^4 + f*x^6)/(x^10*(a + b*x^2)^3),x]

[Out]

-c/(9*a^3*x^9) + (3*b*c - a*d)/(7*a^4*x^7) - (6*b^2*c - 3*a*b*d + a^2*e)/(5*a^5*x^5) + (10*b^3*c - 6*a*b^2*d +
3*a^2*b*e - a^3*f)/(3*a^6*x^3) - (b*(15*b^3*c - 10*a*b^2*d + 6*a^2*b*e - 3*a^3*f))/(a^7*x) - (b^2*(b^3*c - a*
b^2*d + a^2*b*e - a^3*f)*x)/(4*a^6*(a + b*x^2)^2) - (b^2*(23*b^3*c - 19*a*b^2*d + 15*a^2*b*e - 11*a^3*f)*x)/(8
*a^7*(a + b*x^2)) - (b^(3/2)*(143*b^3*c - 99*a*b^2*d + 63*a^2*b*e - 35*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*
a^(15/2))

Rule 1805

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[((a*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )^3} \, dx &=-\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^6 \left (a+b x^2\right )^2}-\frac{\int \frac{-4 c+4 \left (\frac{b c}{a}-d\right ) x^2-\frac{4 \left (b^2 c-a b d+a^2 e\right ) x^4}{a^2}+\frac{4 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^6}{a^3}-\frac{4 b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^8}{a^4}+\frac{3 b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^{10}}{a^5}}{x^{10} \left (a+b x^2\right )^2} \, dx}{4 a}\\ &=-\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^6 \left (a+b x^2\right )^2}-\frac{b^2 \left (23 b^3 c-19 a b^2 d+15 a^2 b e-11 a^3 f\right ) x}{8 a^7 \left (a+b x^2\right )}+\frac{\int \frac{8 c-8 \left (\frac{2 b c}{a}-d\right ) x^2+8 \left (\frac{3 b^2 c}{a^2}-\frac{2 b d}{a}+e\right ) x^4-\frac{8 \left (4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f\right ) x^6}{a^3}+\frac{8 b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right ) x^8}{a^4}-\frac{b^2 \left (23 b^3 c-19 a b^2 d+15 a^2 b e-11 a^3 f\right ) x^{10}}{a^5}}{x^{10} \left (a+b x^2\right )} \, dx}{8 a^2}\\ &=-\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^6 \left (a+b x^2\right )^2}-\frac{b^2 \left (23 b^3 c-19 a b^2 d+15 a^2 b e-11 a^3 f\right ) x}{8 a^7 \left (a+b x^2\right )}+\frac{\int \left (\frac{8 c}{a x^{10}}+\frac{8 (-3 b c+a d)}{a^2 x^8}+\frac{8 \left (6 b^2 c-3 a b d+a^2 e\right )}{a^3 x^6}+\frac{8 \left (-10 b^3 c+6 a b^2 d-3 a^2 b e+a^3 f\right )}{a^4 x^4}-\frac{8 b \left (-15 b^3 c+10 a b^2 d-6 a^2 b e+3 a^3 f\right )}{a^5 x^2}+\frac{b^2 \left (-143 b^3 c+99 a b^2 d-63 a^2 b e+35 a^3 f\right )}{a^5 \left (a+b x^2\right )}\right ) \, dx}{8 a^2}\\ &=-\frac{c}{9 a^3 x^9}+\frac{3 b c-a d}{7 a^4 x^7}-\frac{6 b^2 c-3 a b d+a^2 e}{5 a^5 x^5}+\frac{10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{3 a^6 x^3}-\frac{b \left (15 b^3 c-10 a b^2 d+6 a^2 b e-3 a^3 f\right )}{a^7 x}-\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^6 \left (a+b x^2\right )^2}-\frac{b^2 \left (23 b^3 c-19 a b^2 d+15 a^2 b e-11 a^3 f\right ) x}{8 a^7 \left (a+b x^2\right )}-\frac{\left (b^2 \left (143 b^3 c-99 a b^2 d+63 a^2 b e-35 a^3 f\right )\right ) \int \frac{1}{a+b x^2} \, dx}{8 a^7}\\ &=-\frac{c}{9 a^3 x^9}+\frac{3 b c-a d}{7 a^4 x^7}-\frac{6 b^2 c-3 a b d+a^2 e}{5 a^5 x^5}+\frac{10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{3 a^6 x^3}-\frac{b \left (15 b^3 c-10 a b^2 d+6 a^2 b e-3 a^3 f\right )}{a^7 x}-\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^6 \left (a+b x^2\right )^2}-\frac{b^2 \left (23 b^3 c-19 a b^2 d+15 a^2 b e-11 a^3 f\right ) x}{8 a^7 \left (a+b x^2\right )}-\frac{b^{3/2} \left (143 b^3 c-99 a b^2 d+63 a^2 b e-35 a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{15/2}}\\ \end{align*}

Mathematica [A]  time = 0.148101, size = 276, normalized size = 1. \[ \frac{b^2 x \left (-15 a^2 b e+11 a^3 f+19 a b^2 d-23 b^3 c\right )}{8 a^7 \left (a+b x^2\right )}+\frac{b^2 x \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{4 a^6 \left (a+b x^2\right )^2}+\frac{3 a^2 b e+a^3 (-f)-6 a b^2 d+10 b^3 c}{3 a^6 x^3}+\frac{b \left (-6 a^2 b e+3 a^3 f+10 a b^2 d-15 b^3 c\right )}{a^7 x}+\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-63 a^2 b e+35 a^3 f+99 a b^2 d-143 b^3 c\right )}{8 a^{15/2}}-\frac{a^2 e-3 a b d+6 b^2 c}{5 a^5 x^5}+\frac{3 b c-a d}{7 a^4 x^7}-\frac{c}{9 a^3 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^10*(a + b*x^2)^3),x]

[Out]

-c/(9*a^3*x^9) + (3*b*c - a*d)/(7*a^4*x^7) - (6*b^2*c - 3*a*b*d + a^2*e)/(5*a^5*x^5) + (10*b^3*c - 6*a*b^2*d +
3*a^2*b*e - a^3*f)/(3*a^6*x^3) + (b*(-15*b^3*c + 10*a*b^2*d - 6*a^2*b*e + 3*a^3*f))/(a^7*x) + (b^2*(-(b^3*c)
+ a*b^2*d - a^2*b*e + a^3*f)*x)/(4*a^6*(a + b*x^2)^2) + (b^2*(-23*b^3*c + 19*a*b^2*d - 15*a^2*b*e + 11*a^3*f)*
x)/(8*a^7*(a + b*x^2)) + (b^(3/2)*(-143*b^3*c + 99*a*b^2*d - 63*a^2*b*e + 35*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]
])/(8*a^(15/2))

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Maple [A]  time = 0.02, size = 401, normalized size = 1.5 \begin{align*} -{\frac{143\,{b}^{5}c}{8\,{a}^{7}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{c}{9\,{a}^{3}{x}^{9}}}-{\frac{25\,{b}^{5}cx}{8\,{a}^{6} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{99\,{b}^{4}d}{8\,{a}^{6}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{17\,{b}^{3}ex}{8\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{21\,{b}^{4}dx}{8\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{35\,{b}^{2}f}{8\,{a}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{63\,{b}^{3}e}{8\,{a}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,bc}{7\,{a}^{4}{x}^{7}}}+{\frac{3\,bd}{5\,{a}^{4}{x}^{5}}}-{\frac{6\,{b}^{2}c}{5\,{a}^{5}{x}^{5}}}+{\frac{be}{{a}^{4}{x}^{3}}}-2\,{\frac{{b}^{2}d}{{a}^{5}{x}^{3}}}+{\frac{10\,{b}^{3}c}{3\,{a}^{6}{x}^{3}}}+3\,{\frac{fb}{{a}^{4}x}}-6\,{\frac{e{b}^{2}}{{a}^{5}x}}+10\,{\frac{d{b}^{3}}{{a}^{6}x}}-15\,{\frac{c{b}^{4}}{{a}^{7}x}}-{\frac{e}{5\,{a}^{3}{x}^{5}}}-{\frac{f}{3\,{a}^{3}{x}^{3}}}-{\frac{d}{7\,{a}^{3}{x}^{7}}}+{\frac{11\,{b}^{3}{x}^{3}f}{8\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{23\,{b}^{6}{x}^{3}c}{8\,{a}^{7} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{13\,{b}^{2}fx}{8\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{19\,{b}^{5}{x}^{3}d}{8\,{a}^{6} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{15\,{b}^{4}{x}^{3}e}{8\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^6+e*x^4+d*x^2+c)/x^10/(b*x^2+a)^3,x)

[Out]

-143/8/a^7*b^5/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*c-1/9*c/a^3/x^9-25/8/a^6*b^5/(b*x^2+a)^2*c*x+99/8/a^6*b^4/(
a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*d-17/8/a^4*b^3/(b*x^2+a)^2*e*x+21/8/a^5*b^4/(b*x^2+a)^2*d*x+35/8/a^4*b^2/(a
*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*f-63/8/a^5*b^3/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*e+3/7/a^4/x^7*b*c+3/5/a^4
/x^5*b*d-6/5/a^5/x^5*b^2*c+1/a^4/x^3*b*e-2/a^5/x^3*b^2*d+10/3/a^6/x^3*b^3*c+3*b/a^4/x*f-6*b^2/a^5/x*e+10*b^3/a
^6/x*d-15*b^4/a^7/x*c-1/5/a^3/x^5*e-1/3/a^3/x^3*f-1/7/a^3/x^7*d+11/8/a^4*b^3/(b*x^2+a)^2*x^3*f-23/8/a^7*b^6/(b
*x^2+a)^2*x^3*c+13/8/a^3*b^2/(b*x^2+a)^2*f*x+19/8/a^6*b^5/(b*x^2+a)^2*x^3*d-15/8/a^5*b^4/(b*x^2+a)^2*x^3*e

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^6+e*x^4+d*x^2+c)/x^10/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.3172, size = 1778, normalized size = 6.42 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^6+e*x^4+d*x^2+c)/x^10/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

[-1/5040*(630*(143*b^6*c - 99*a*b^5*d + 63*a^2*b^4*e - 35*a^3*b^3*f)*x^12 + 1050*(143*a*b^5*c - 99*a^2*b^4*d +
63*a^3*b^3*e - 35*a^4*b^2*f)*x^10 + 336*(143*a^2*b^4*c - 99*a^3*b^3*d + 63*a^4*b^2*e - 35*a^5*b*f)*x^8 + 560*
a^6*c - 48*(143*a^3*b^3*c - 99*a^4*b^2*d + 63*a^5*b*e - 35*a^6*f)*x^6 + 16*(143*a^4*b^2*c - 99*a^5*b*d + 63*a^
6*e)*x^4 - 80*(13*a^5*b*c - 9*a^6*d)*x^2 + 315*((143*b^6*c - 99*a*b^5*d + 63*a^2*b^4*e - 35*a^3*b^3*f)*x^13 +
2*(143*a*b^5*c - 99*a^2*b^4*d + 63*a^3*b^3*e - 35*a^4*b^2*f)*x^11 + (143*a^2*b^4*c - 99*a^3*b^3*d + 63*a^4*b^2
*e - 35*a^5*b*f)*x^9)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/(a^7*b^2*x^13 + 2*a^8*b*x^11
+ a^9*x^9), -1/2520*(315*(143*b^6*c - 99*a*b^5*d + 63*a^2*b^4*e - 35*a^3*b^3*f)*x^12 + 525*(143*a*b^5*c - 99*
a^2*b^4*d + 63*a^3*b^3*e - 35*a^4*b^2*f)*x^10 + 168*(143*a^2*b^4*c - 99*a^3*b^3*d + 63*a^4*b^2*e - 35*a^5*b*f)
*x^8 + 280*a^6*c - 24*(143*a^3*b^3*c - 99*a^4*b^2*d + 63*a^5*b*e - 35*a^6*f)*x^6 + 8*(143*a^4*b^2*c - 99*a^5*b
*d + 63*a^6*e)*x^4 - 40*(13*a^5*b*c - 9*a^6*d)*x^2 + 315*((143*b^6*c - 99*a*b^5*d + 63*a^2*b^4*e - 35*a^3*b^3*
f)*x^13 + 2*(143*a*b^5*c - 99*a^2*b^4*d + 63*a^3*b^3*e - 35*a^4*b^2*f)*x^11 + (143*a^2*b^4*c - 99*a^3*b^3*d +
63*a^4*b^2*e - 35*a^5*b*f)*x^9)*sqrt(b/a)*arctan(x*sqrt(b/a)))/(a^7*b^2*x^13 + 2*a^8*b*x^11 + a^9*x^9)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**6+e*x**4+d*x**2+c)/x**10/(b*x**2+a)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.21618, size = 406, normalized size = 1.47 \begin{align*} -\frac{{\left (143 \, b^{5} c - 99 \, a b^{4} d - 35 \, a^{3} b^{2} f + 63 \, a^{2} b^{3} e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{7}} - \frac{23 \, b^{6} c x^{3} - 19 \, a b^{5} d x^{3} - 11 \, a^{3} b^{3} f x^{3} + 15 \, a^{2} b^{4} x^{3} e + 25 \, a b^{5} c x - 21 \, a^{2} b^{4} d x - 13 \, a^{4} b^{2} f x + 17 \, a^{3} b^{3} x e}{8 \,{\left (b x^{2} + a\right )}^{2} a^{7}} - \frac{4725 \, b^{4} c x^{8} - 3150 \, a b^{3} d x^{8} - 945 \, a^{3} b f x^{8} + 1890 \, a^{2} b^{2} x^{8} e - 1050 \, a b^{3} c x^{6} + 630 \, a^{2} b^{2} d x^{6} + 105 \, a^{4} f x^{6} - 315 \, a^{3} b x^{6} e + 378 \, a^{2} b^{2} c x^{4} - 189 \, a^{3} b d x^{4} + 63 \, a^{4} x^{4} e - 135 \, a^{3} b c x^{2} + 45 \, a^{4} d x^{2} + 35 \, a^{4} c}{315 \, a^{7} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^6+e*x^4+d*x^2+c)/x^10/(b*x^2+a)^3,x, algorithm="giac")

[Out]

-1/8*(143*b^5*c - 99*a*b^4*d - 35*a^3*b^2*f + 63*a^2*b^3*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^7) - 1/8*(23*b^
6*c*x^3 - 19*a*b^5*d*x^3 - 11*a^3*b^3*f*x^3 + 15*a^2*b^4*x^3*e + 25*a*b^5*c*x - 21*a^2*b^4*d*x - 13*a^4*b^2*f*
x + 17*a^3*b^3*x*e)/((b*x^2 + a)^2*a^7) - 1/315*(4725*b^4*c*x^8 - 3150*a*b^3*d*x^8 - 945*a^3*b*f*x^8 + 1890*a^
2*b^2*x^8*e - 1050*a*b^3*c*x^6 + 630*a^2*b^2*d*x^6 + 105*a^4*f*x^6 - 315*a^3*b*x^6*e + 378*a^2*b^2*c*x^4 - 189
*a^3*b*d*x^4 + 63*a^4*x^4*e - 135*a^3*b*c*x^2 + 45*a^4*d*x^2 + 35*a^4*c)/(a^7*x^9)