Optimal. Leaf size=55 \[ -\frac{c d-2 b e}{c^3 (b+c x)}+\frac{b (c d-b e)}{2 c^3 (b+c x)^2}+\frac{e \log (b+c x)}{c^3} \]
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Rubi [A] time = 0.0466188, antiderivative size = 55, 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} \[ -\frac{c d-2 b e}{c^3 (b+c x)}+\frac{b (c d-b e)}{2 c^3 (b+c x)^2}+\frac{e \log (b+c x)}{c^3} \]
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 )^3} \, dx &=\int \left (\frac{b (-c d+b e)}{c^2 (b+c x)^3}+\frac{c d-2 b e}{c^2 (b+c x)^2}+\frac{e}{c^2 (b+c x)}\right ) \, dx\\ &=\frac{b (c d-b e)}{2 c^3 (b+c x)^2}-\frac{c d-2 b e}{c^3 (b+c x)}+\frac{e \log (b+c x)}{c^3}\\ \end{align*}
Mathematica [A] time = 0.0163263, size = 54, normalized size = 0.98 \[ \frac{3 b^2 e-b c (d-4 e x)+2 e (b+c x)^2 \log (b+c x)-2 c^2 d x}{2 c^3 (b+c x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 70, normalized size = 1.3 \begin{align*} 2\,{\frac{be}{{c}^{3} \left ( cx+b \right ) }}-{\frac{d}{{c}^{2} \left ( cx+b \right ) }}+{\frac{e\ln \left ( cx+b \right ) }{{c}^{3}}}-{\frac{{b}^{2}e}{2\,{c}^{3} \left ( cx+b \right ) ^{2}}}+{\frac{bd}{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 = 1.01698, size = 85, normalized size = 1.55 \begin{align*} -\frac{b c d - 3 \, b^{2} e + 2 \,{\left (c^{2} d - 2 \, b c e\right )} x}{2 \,{\left (c^{5} x^{2} + 2 \, b c^{4} x + b^{2} c^{3}\right )}} + \frac{e \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 = 1.73018, size = 174, normalized size = 3.16 \begin{align*} -\frac{b c d - 3 \, b^{2} e + 2 \,{\left (c^{2} d - 2 \, b c e\right )} x - 2 \,{\left (c^{2} e x^{2} + 2 \, b c e x + b^{2} e\right )} \log \left (c x + b\right )}{2 \,{\left (c^{5} x^{2} + 2 \, b c^{4} x + b^{2} c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.89074, size = 63, normalized size = 1.15 \begin{align*} \frac{3 b^{2} e - b c d + x \left (4 b c e - 2 c^{2} d\right )}{2 b^{2} c^{3} + 4 b c^{4} x + 2 c^{5} x^{2}} + \frac{e \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 = 1.13753, size = 74, normalized size = 1.35 \begin{align*} \frac{e \log \left ({\left | c x + b \right |}\right )}{c^{3}} - \frac{2 \,{\left (c d - 2 \, b e\right )} x + \frac{b c d - 3 \, b^{2} e}{c}}{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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