Optimal. Leaf size=45 \[ \frac{b (c d-b e)}{c^3 (b+c x)}+\frac{(c d-2 b e) \log (b+c x)}{c^3}+\frac{e x}{c^2} \]
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Rubi [A] time = 0.0430754, antiderivative size = 45, 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} \[ \frac{b (c d-b e)}{c^3 (b+c x)}+\frac{(c d-2 b e) \log (b+c x)}{c^3}+\frac{e x}{c^2} \]
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 )^2} \, dx &=\int \left (\frac{e}{c^2}+\frac{b (-c d+b e)}{c^2 (b+c x)^2}+\frac{c d-2 b e}{c^2 (b+c x)}\right ) \, dx\\ &=\frac{e x}{c^2}+\frac{b (c d-b e)}{c^3 (b+c x)}+\frac{(c d-2 b e) \log (b+c x)}{c^3}\\ \end{align*}
Mathematica [A] time = 0.0259708, size = 41, normalized size = 0.91 \[ \frac{\frac{b (c d-b e)}{b+c x}+(c d-2 b e) \log (b+c x)+c e x}{c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 61, normalized size = 1.4 \begin{align*}{\frac{ex}{{c}^{2}}}-{\frac{{b}^{2}e}{{c}^{3} \left ( cx+b \right ) }}+{\frac{bd}{{c}^{2} \left ( cx+b \right ) }}-2\,{\frac{\ln \left ( cx+b \right ) be}{{c}^{3}}}+{\frac{\ln \left ( cx+b \right ) d}{{c}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11642, size = 68, normalized size = 1.51 \begin{align*} \frac{b c d - b^{2} e}{c^{4} x + b c^{3}} + \frac{e x}{c^{2}} + \frac{{\left (c d - 2 \, b e\right )} \log \left (c x + b\right )}{c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64778, size = 149, normalized size = 3.31 \begin{align*} \frac{c^{2} e x^{2} + b c e x + b c d - b^{2} e +{\left (b c d - 2 \, b^{2} e +{\left (c^{2} d - 2 \, b c e\right )} x\right )} \log \left (c x + b\right )}{c^{4} x + b c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.611978, size = 44, normalized size = 0.98 \begin{align*} - \frac{b^{2} e - b c d}{b c^{3} + c^{4} x} + \frac{e x}{c^{2}} - \frac{\left (2 b e - c d\right ) \log{\left (b + c x \right )}}{c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13689, size = 69, normalized size = 1.53 \begin{align*} \frac{x e}{c^{2}} + \frac{{\left (c d - 2 \, b e\right )} \log \left ({\left | c x + b \right |}\right )}{c^{3}} + \frac{b c d - b^{2} e}{{\left (c x + b\right )} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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