Optimal. Leaf size=206 \[ -\frac{c \left (a B e^2-2 A c d e+5 B c d^2\right )}{2 e^6 (d+e x)^4}+\frac{2 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{5 e^6 (d+e x)^5}-\frac{\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{6 e^6 (d+e x)^6}+\frac{\left (a e^2+c d^2\right )^2 (B d-A e)}{7 e^6 (d+e x)^7}+\frac{c^2 (5 B d-A e)}{3 e^6 (d+e x)^3}-\frac{B c^2}{2 e^6 (d+e x)^2} \]
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Rubi [A] time = 0.139045, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {772} \[ -\frac{c \left (a B e^2-2 A c d e+5 B c d^2\right )}{2 e^6 (d+e x)^4}+\frac{2 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{5 e^6 (d+e x)^5}-\frac{\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{6 e^6 (d+e x)^6}+\frac{\left (a e^2+c d^2\right )^2 (B d-A e)}{7 e^6 (d+e x)^7}+\frac{c^2 (5 B d-A e)}{3 e^6 (d+e x)^3}-\frac{B c^2}{2 e^6 (d+e x)^2} \]
Antiderivative was successfully verified.
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Rule 772
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )^2}{(d+e x)^8} \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right )^2}{e^5 (d+e x)^8}+\frac{\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{e^5 (d+e x)^7}+\frac{2 c \left (-5 B c d^3+3 A c d^2 e-3 a B d e^2+a A e^3\right )}{e^5 (d+e x)^6}-\frac{2 c \left (-5 B c d^2+2 A c d e-a B e^2\right )}{e^5 (d+e x)^5}+\frac{c^2 (-5 B d+A e)}{e^5 (d+e x)^4}+\frac{B c^2}{e^5 (d+e x)^3}\right ) \, dx\\ &=\frac{(B d-A e) \left (c d^2+a e^2\right )^2}{7 e^6 (d+e x)^7}-\frac{\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{6 e^6 (d+e x)^6}+\frac{2 c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right )}{5 e^6 (d+e x)^5}-\frac{c \left (5 B c d^2-2 A c d e+a B e^2\right )}{2 e^6 (d+e x)^4}+\frac{c^2 (5 B d-A e)}{3 e^6 (d+e x)^3}-\frac{B c^2}{2 e^6 (d+e x)^2}\\ \end{align*}
Mathematica [A] time = 0.101338, size = 202, normalized size = 0.98 \[ -\frac{2 A e \left (15 a^2 e^4+2 a c e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+c^2 \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )\right )+B \left (5 a^2 e^4 (d+7 e x)+3 a c e^2 \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )+5 c^2 \left (21 d^3 e^2 x^2+35 d^2 e^3 x^3+7 d^4 e x+d^5+35 d e^4 x^4+21 e^5 x^5\right )\right )}{210 e^6 (d+e x)^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 249, normalized size = 1.2 \begin{align*} -{\frac{A{a}^{2}{e}^{5}+2\,A{d}^{2}ac{e}^{3}+A{d}^{4}{c}^{2}e-B{a}^{2}d{e}^{4}-2\,aBc{d}^{3}{e}^{2}-B{c}^{2}{d}^{5}}{7\,{e}^{6} \left ( ex+d \right ) ^{7}}}-{\frac{B{c}^{2}}{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}-{\frac{{c}^{2} \left ( Ae-5\,Bd \right ) }{3\,{e}^{6} \left ( ex+d \right ) ^{3}}}+{\frac{c \left ( 2\,Acde-aB{e}^{2}-5\,Bc{d}^{2} \right ) }{2\,{e}^{6} \left ( ex+d \right ) ^{4}}}-{\frac{2\,c \left ( aA{e}^{3}+3\,Ac{d}^{2}e-3\,aBd{e}^{2}-5\,Bc{d}^{3} \right ) }{5\,{e}^{6} \left ( ex+d \right ) ^{5}}}-{\frac{-4\,Adac{e}^{3}-4\,A{c}^{2}{d}^{3}e+B{e}^{4}{a}^{2}+6\,aBc{d}^{2}{e}^{2}+5\,B{c}^{2}{d}^{4}}{6\,{e}^{6} \left ( ex+d \right ) ^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04722, size = 428, normalized size = 2.08 \begin{align*} -\frac{105 \, B c^{2} e^{5} x^{5} + 5 \, B c^{2} d^{5} + 2 \, A c^{2} d^{4} e + 3 \, B a c d^{3} e^{2} + 4 \, A a c d^{2} e^{3} + 5 \, B a^{2} d e^{4} + 30 \, A a^{2} e^{5} + 35 \,{\left (5 \, B c^{2} d e^{4} + 2 \, A c^{2} e^{5}\right )} x^{4} + 35 \,{\left (5 \, B c^{2} d^{2} e^{3} + 2 \, A c^{2} d e^{4} + 3 \, B a c e^{5}\right )} x^{3} + 21 \,{\left (5 \, B c^{2} d^{3} e^{2} + 2 \, A c^{2} d^{2} e^{3} + 3 \, B a c d e^{4} + 4 \, A a c e^{5}\right )} x^{2} + 7 \,{\left (5 \, B c^{2} d^{4} e + 2 \, A c^{2} d^{3} e^{2} + 3 \, B a c d^{2} e^{3} + 4 \, A a c d e^{4} + 5 \, B a^{2} e^{5}\right )} x}{210 \,{\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71325, size = 680, normalized size = 3.3 \begin{align*} -\frac{105 \, B c^{2} e^{5} x^{5} + 5 \, B c^{2} d^{5} + 2 \, A c^{2} d^{4} e + 3 \, B a c d^{3} e^{2} + 4 \, A a c d^{2} e^{3} + 5 \, B a^{2} d e^{4} + 30 \, A a^{2} e^{5} + 35 \,{\left (5 \, B c^{2} d e^{4} + 2 \, A c^{2} e^{5}\right )} x^{4} + 35 \,{\left (5 \, B c^{2} d^{2} e^{3} + 2 \, A c^{2} d e^{4} + 3 \, B a c e^{5}\right )} x^{3} + 21 \,{\left (5 \, B c^{2} d^{3} e^{2} + 2 \, A c^{2} d^{2} e^{3} + 3 \, B a c d e^{4} + 4 \, A a c e^{5}\right )} x^{2} + 7 \,{\left (5 \, B c^{2} d^{4} e + 2 \, A c^{2} d^{3} e^{2} + 3 \, B a c d^{2} e^{3} + 4 \, A a c d e^{4} + 5 \, B a^{2} e^{5}\right )} x}{210 \,{\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [A] time = 1.33057, size = 327, normalized size = 1.59 \begin{align*} -\frac{{\left (105 \, B c^{2} x^{5} e^{5} + 175 \, B c^{2} d x^{4} e^{4} + 175 \, B c^{2} d^{2} x^{3} e^{3} + 105 \, B c^{2} d^{3} x^{2} e^{2} + 35 \, B c^{2} d^{4} x e + 5 \, B c^{2} d^{5} + 70 \, A c^{2} x^{4} e^{5} + 70 \, A c^{2} d x^{3} e^{4} + 42 \, A c^{2} d^{2} x^{2} e^{3} + 14 \, A c^{2} d^{3} x e^{2} + 2 \, A c^{2} d^{4} e + 105 \, B a c x^{3} e^{5} + 63 \, B a c d x^{2} e^{4} + 21 \, B a c d^{2} x e^{3} + 3 \, B a c d^{3} e^{2} + 84 \, A a c x^{2} e^{5} + 28 \, A a c d x e^{4} + 4 \, A a c d^{2} e^{3} + 35 \, B a^{2} x e^{5} + 5 \, B a^{2} d e^{4} + 30 \, A a^{2} e^{5}\right )} e^{\left (-6\right )}}{210 \,{\left (x e + d\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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