Optimal. Leaf size=69 \[ -\frac{b^2 (c d-b e)}{c^4 (b+c x)}+\frac{x (c d-2 b e)}{c^3}-\frac{b (2 c d-3 b e) \log (b+c x)}{c^4}+\frac{e x^2}{2 c^2} \]
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Rubi [A] time = 0.0686247, antiderivative size = 69, 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} \[ -\frac{b^2 (c d-b e)}{c^4 (b+c x)}+\frac{x (c d-2 b e)}{c^3}-\frac{b (2 c d-3 b e) \log (b+c x)}{c^4}+\frac{e x^2}{2 c^2} \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin{align*} \int \frac{x^4 (d+e x)}{\left (b x+c x^2\right )^2} \, dx &=\int \left (\frac{c d-2 b e}{c^3}+\frac{e x}{c^2}-\frac{b^2 (-c d+b e)}{c^3 (b+c x)^2}+\frac{b (-2 c d+3 b e)}{c^3 (b+c x)}\right ) \, dx\\ &=\frac{(c d-2 b e) x}{c^3}+\frac{e x^2}{2 c^2}-\frac{b^2 (c d-b e)}{c^4 (b+c x)}-\frac{b (2 c d-3 b e) \log (b+c x)}{c^4}\\ \end{align*}
Mathematica [A] time = 0.0500609, size = 66, normalized size = 0.96 \[ \frac{\frac{2 b^2 (b e-c d)}{b+c x}+2 c x (c d-2 b e)+2 b (3 b e-2 c d) \log (b+c x)+c^2 e x^2}{2 c^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 84, normalized size = 1.2 \begin{align*}{\frac{e{x}^{2}}{2\,{c}^{2}}}-2\,{\frac{bex}{{c}^{3}}}+{\frac{dx}{{c}^{2}}}+{\frac{{b}^{3}e}{{c}^{4} \left ( cx+b \right ) }}-{\frac{{b}^{2}d}{{c}^{3} \left ( cx+b \right ) }}+3\,{\frac{{b}^{2}\ln \left ( cx+b \right ) e}{{c}^{4}}}-2\,{\frac{b\ln \left ( cx+b \right ) d}{{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0835, size = 101, normalized size = 1.46 \begin{align*} -\frac{b^{2} c d - b^{3} e}{c^{5} x + b c^{4}} + \frac{c e x^{2} + 2 \,{\left (c d - 2 \, b e\right )} x}{2 \, c^{3}} - \frac{{\left (2 \, b c d - 3 \, b^{2} e\right )} \log \left (c x + b\right )}{c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85371, size = 240, normalized size = 3.48 \begin{align*} \frac{c^{3} e x^{3} - 2 \, b^{2} c d + 2 \, b^{3} e +{\left (2 \, c^{3} d - 3 \, b c^{2} e\right )} x^{2} + 2 \,{\left (b c^{2} d - 2 \, b^{2} c e\right )} x - 2 \,{\left (2 \, b^{2} c d - 3 \, b^{3} e +{\left (2 \, b c^{2} d - 3 \, b^{2} c e\right )} x\right )} \log \left (c x + b\right )}{2 \,{\left (c^{5} x + b c^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.612942, size = 66, normalized size = 0.96 \begin{align*} \frac{b \left (3 b e - 2 c d\right ) \log{\left (b + c x \right )}}{c^{4}} + \frac{b^{3} e - b^{2} c d}{b c^{4} + c^{5} x} + \frac{e x^{2}}{2 c^{2}} - \frac{x \left (2 b e - c d\right )}{c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20667, size = 109, normalized size = 1.58 \begin{align*} -\frac{{\left (2 \, b c d - 3 \, b^{2} e\right )} \log \left ({\left | c x + b \right |}\right )}{c^{4}} + \frac{c^{2} x^{2} e + 2 \, c^{2} d x - 4 \, b c x e}{2 \, c^{4}} - \frac{b^{2} c d - b^{3} e}{{\left (c x + b\right )} c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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