Optimal. Leaf size=101 \[ -\frac{a B e^2-2 A c d e+3 B c d^2}{2 e^4 (d+e x)^2}+\frac{\left (a e^2+c d^2\right ) (B d-A e)}{3 e^4 (d+e x)^3}+\frac{c (3 B d-A e)}{e^4 (d+e x)}+\frac{B c \log (d+e x)}{e^4} \]
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Rubi [A] time = 0.072604, antiderivative size = 101, 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} \[ -\frac{a B e^2-2 A c d e+3 B c d^2}{2 e^4 (d+e x)^2}+\frac{\left (a e^2+c d^2\right ) (B d-A e)}{3 e^4 (d+e x)^3}+\frac{c (3 B d-A e)}{e^4 (d+e x)}+\frac{B c \log (d+e x)}{e^4} \]
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)^4} \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right )}{e^3 (d+e x)^4}+\frac{3 B c d^2-2 A c d e+a B e^2}{e^3 (d+e x)^3}+\frac{c (-3 B d+A e)}{e^3 (d+e x)^2}+\frac{B c}{e^3 (d+e x)}\right ) \, dx\\ &=\frac{(B d-A e) \left (c d^2+a e^2\right )}{3 e^4 (d+e x)^3}-\frac{3 B c d^2-2 A c d e+a B e^2}{2 e^4 (d+e x)^2}+\frac{c (3 B d-A e)}{e^4 (d+e x)}+\frac{B c \log (d+e x)}{e^4}\\ \end{align*}
Mathematica [A] time = 0.0508488, size = 98, normalized size = 0.97 \[ \frac{-2 A e \left (a e^2+c \left (d^2+3 d e x+3 e^2 x^2\right )\right )+B \left (c d \left (11 d^2+27 d e x+18 e^2 x^2\right )-a e^2 (d+3 e x)\right )+6 B c (d+e x)^3 \log (d+e x)}{6 e^4 (d+e x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 151, normalized size = 1.5 \begin{align*} -{\frac{Ac}{{e}^{3} \left ( ex+d \right ) }}+3\,{\frac{Bcd}{{e}^{4} \left ( ex+d \right ) }}+{\frac{Acd}{{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{aB}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{3\,Bc{d}^{2}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{aA}{3\,e \left ( ex+d \right ) ^{3}}}-{\frac{Ac{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{aBd}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}+{\frac{Bc{d}^{3}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}+{\frac{Bc\ln \left ( ex+d \right ) }{{e}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08839, size = 174, normalized size = 1.72 \begin{align*} \frac{11 \, B c d^{3} - 2 \, A c d^{2} e - B a d e^{2} - 2 \, A a e^{3} + 6 \,{\left (3 \, B c d e^{2} - A c e^{3}\right )} x^{2} + 3 \,{\left (9 \, B c d^{2} e - 2 \, A c d e^{2} - B a e^{3}\right )} x}{6 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} + \frac{B c \log \left (e x + d\right )}{e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73411, size = 344, normalized size = 3.41 \begin{align*} \frac{11 \, B c d^{3} - 2 \, A c d^{2} e - B a d e^{2} - 2 \, A a e^{3} + 6 \,{\left (3 \, B c d e^{2} - A c e^{3}\right )} x^{2} + 3 \,{\left (9 \, B c d^{2} e - 2 \, A c d e^{2} - B a e^{3}\right )} x + 6 \,{\left (B c e^{3} x^{3} + 3 \, B c d e^{2} x^{2} + 3 \, B c d^{2} e x + B c d^{3}\right )} \log \left (e x + d\right )}{6 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.6239, size = 138, normalized size = 1.37 \begin{align*} \frac{B c \log{\left (d + e x \right )}}{e^{4}} + \frac{- 2 A a e^{3} - 2 A c d^{2} e - B a d e^{2} + 11 B c d^{3} + x^{2} \left (- 6 A c e^{3} + 18 B c d e^{2}\right ) + x \left (- 6 A c d e^{2} - 3 B a e^{3} + 27 B c d^{2} e\right )}{6 d^{3} e^{4} + 18 d^{2} e^{5} x + 18 d e^{6} x^{2} + 6 e^{7} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19532, size = 139, normalized size = 1.38 \begin{align*} B c e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{{\left (6 \,{\left (3 \, B c d e - A c e^{2}\right )} x^{2} + 3 \,{\left (9 \, B c d^{2} - 2 \, A c d e - B a e^{2}\right )} x +{\left (11 \, B c d^{3} - 2 \, A c d^{2} e - B a d e^{2} - 2 \, A a e^{3}\right )} e^{\left (-1\right )}\right )} e^{\left (-3\right )}}{6 \,{\left (x e + d\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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