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 = 0.03, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, 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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.91, size = 38, normalized size = 1.06 \[ -\frac {2 \, c e x + c d + b e}{2 \, {\left (c^{4} x^{2} + 2 \, b c^{3} x + b^{2} c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 26, normalized size = 0.72 \[ -\frac {2 \, c x e + c d + b e}{2 \, {\left (c x + b\right )}^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 35, normalized size = 0.97 \[ -\frac {e}{\left (c x +b \right ) c^{2}}-\frac {-b e +c d}{2 \left (c x +b \right )^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.92, size = 38, normalized size = 1.06 \[ -\frac {2 \, c e x + c d + b e}{2 \, {\left (c^{4} x^{2} + 2 \, b c^{3} x + b^{2} c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.03, size = 39, normalized size = 1.08 \[ -\frac {\frac {b\,e+c\,d}{2\,c^2}+\frac {e\,x}{c}}{b^2+2\,b\,c\,x+c^2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.28, size = 39, normalized size = 1.08 \[ \frac {- b e - c d - 2 c e x}{2 b^{2} c^{2} + 4 b c^{3} x + 2 c^{4} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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