Optimal. Leaf size=36 \[ -\frac{c d-b e}{2 c^2 (b+c x)^2}-\frac{e}{c^2 (b+c x)} \]
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Rubi [A] time = 0.0292562, antiderivative size = 36, 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 = {765} \[ -\frac{c d-b e}{2 c^2 (b+c x)^2}-\frac{e}{c^2 (b+c x)} \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin{align*} \int \frac{x^3 (d+e x)}{\left (b x+c x^2\right )^3} \, dx &=\int \left (\frac{c d-b e}{c (b+c x)^3}+\frac{e}{c (b+c x)^2}\right ) \, dx\\ &=-\frac{c d-b e}{2 c^2 (b+c x)^2}-\frac{e}{c^2 (b+c x)}\\ \end{align*}
Mathematica [A] time = 0.0087728, size = 26, normalized size = 0.72 \[ -\frac{b e+c (d+2 e x)}{2 c^2 (b+c x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 35, normalized size = 1. \begin{align*} -{\frac{e}{{c}^{2} \left ( cx+b \right ) }}-{\frac{-be+cd}{2\,{c}^{2} \left ( cx+b \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08501, size = 51, normalized size = 1.42 \begin{align*} -\frac{2 \, c e x + c d + b e}{2 \,{\left (c^{4} x^{2} + 2 \, b c^{3} x + b^{2} c^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67242, size = 81, normalized size = 2.25 \begin{align*} -\frac{2 \, c e x + c d + b e}{2 \,{\left (c^{4} x^{2} + 2 \, b c^{3} x + b^{2} c^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.678642, size = 39, normalized size = 1.08 \begin{align*} - \frac{b e + c d + 2 c e x}{2 b^{2} c^{2} + 4 b c^{3} x + 2 c^{4} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15295, size = 35, normalized size = 0.97 \begin{align*} -\frac{2 \, c x e + c d + b e}{2 \,{\left (c x + b\right )}^{2} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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