Optimal. Leaf size=108 \[ -\frac{a B e^2-2 A c d e+3 B c d^2}{4 e^4 (d+e x)^4}+\frac{\left (a e^2+c d^2\right ) (B d-A e)}{5 e^4 (d+e x)^5}+\frac{c (3 B d-A e)}{3 e^4 (d+e x)^3}-\frac{B c}{2 e^4 (d+e x)^2} \]
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Rubi [A] time = 0.0706327, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {772} \[ -\frac{a B e^2-2 A c d e+3 B c d^2}{4 e^4 (d+e x)^4}+\frac{\left (a e^2+c d^2\right ) (B d-A e)}{5 e^4 (d+e x)^5}+\frac{c (3 B d-A e)}{3 e^4 (d+e x)^3}-\frac{B c}{2 e^4 (d+e x)^2} \]
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 )}{(d+e x)^6} \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right )}{e^3 (d+e x)^6}+\frac{3 B c d^2-2 A c d e+a B e^2}{e^3 (d+e x)^5}+\frac{c (-3 B d+A e)}{e^3 (d+e x)^4}+\frac{B c}{e^3 (d+e x)^3}\right ) \, dx\\ &=\frac{(B d-A e) \left (c d^2+a e^2\right )}{5 e^4 (d+e x)^5}-\frac{3 B c d^2-2 A c d e+a B e^2}{4 e^4 (d+e x)^4}+\frac{c (3 B d-A e)}{3 e^4 (d+e x)^3}-\frac{B c}{2 e^4 (d+e x)^2}\\ \end{align*}
Mathematica [A] time = 0.0455319, size = 90, normalized size = 0.83 \[ -\frac{2 A e \left (6 a e^2+c \left (d^2+5 d e x+10 e^2 x^2\right )\right )+3 B \left (a e^2 (d+5 e x)+c \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )\right )}{60 e^4 (d+e x)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 110, normalized size = 1. \begin{align*} -{\frac{Bc}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{c \left ( Ae-3\,Bd \right ) }{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{-2\,Acde+aB{e}^{2}+3\,Bc{d}^{2}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{aA{e}^{3}+Ac{d}^{2}e-aBd{e}^{2}-Bc{d}^{3}}{5\,{e}^{4} \left ( ex+d \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.31273, size = 200, normalized size = 1.85 \begin{align*} -\frac{30 \, B c e^{3} x^{3} + 3 \, B c d^{3} + 2 \, A c d^{2} e + 3 \, B a d e^{2} + 12 \, A a e^{3} + 10 \,{\left (3 \, B c d e^{2} + 2 \, A c e^{3}\right )} x^{2} + 5 \,{\left (3 \, B c d^{2} e + 2 \, A c d e^{2} + 3 \, B a e^{3}\right )} x}{60 \,{\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63789, size = 324, normalized size = 3. \begin{align*} -\frac{30 \, B c e^{3} x^{3} + 3 \, B c d^{3} + 2 \, A c d^{2} e + 3 \, B a d e^{2} + 12 \, A a e^{3} + 10 \,{\left (3 \, B c d e^{2} + 2 \, A c e^{3}\right )} x^{2} + 5 \,{\left (3 \, B c d^{2} e + 2 \, A c d e^{2} + 3 \, B a e^{3}\right )} x}{60 \,{\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 22.5898, size = 163, normalized size = 1.51 \begin{align*} - \frac{12 A a e^{3} + 2 A c d^{2} e + 3 B a d e^{2} + 3 B c d^{3} + 30 B c e^{3} x^{3} + x^{2} \left (20 A c e^{3} + 30 B c d e^{2}\right ) + x \left (10 A c d e^{2} + 15 B a e^{3} + 15 B c d^{2} e\right )}{60 d^{5} e^{4} + 300 d^{4} e^{5} x + 600 d^{3} e^{6} x^{2} + 600 d^{2} e^{7} x^{3} + 300 d e^{8} x^{4} + 60 e^{9} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19429, size = 128, normalized size = 1.19 \begin{align*} -\frac{{\left (30 \, B c x^{3} e^{3} + 30 \, B c d x^{2} e^{2} + 15 \, B c d^{2} x e + 3 \, B c d^{3} + 20 \, A c x^{2} e^{3} + 10 \, A c d x e^{2} + 2 \, A c d^{2} e + 15 \, B a x e^{3} + 3 \, B a d e^{2} + 12 \, A a e^{3}\right )} e^{\left (-4\right )}}{60 \,{\left (x e + d\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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