Optimal. Leaf size=66 \[ \frac{b^2 (c d-b e) \log (b+c x)}{c^4}+\frac{x^2 (c d-b e)}{2 c^2}-\frac{b x (c d-b e)}{c^3}+\frac{e x^3}{3 c} \]
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Rubi [A] time = 0.0574302, antiderivative size = 66, 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} \[ \frac{b^2 (c d-b e) \log (b+c x)}{c^4}+\frac{x^2 (c d-b e)}{2 c^2}-\frac{b x (c d-b e)}{c^3}+\frac{e x^3}{3 c} \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin{align*} \int \frac{x^3 (d+e x)}{b x+c x^2} \, dx &=\int \left (\frac{b (-c d+b e)}{c^3}+\frac{(c d-b e) x}{c^2}+\frac{e x^2}{c}-\frac{b^2 (-c d+b e)}{c^3 (b+c x)}\right ) \, dx\\ &=-\frac{b (c d-b e) x}{c^3}+\frac{(c d-b e) x^2}{2 c^2}+\frac{e x^3}{3 c}+\frac{b^2 (c d-b e) \log (b+c x)}{c^4}\\ \end{align*}
Mathematica [A] time = 0.0212502, size = 61, normalized size = 0.92 \[ \frac{c x \left (6 b^2 e-3 b c (2 d+e x)+c^2 x (3 d+2 e x)\right )+6 b^2 (c d-b e) \log (b+c x)}{6 c^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 76, normalized size = 1.2 \begin{align*}{\frac{e{x}^{3}}{3\,c}}-{\frac{b{x}^{2}e}{2\,{c}^{2}}}+{\frac{d{x}^{2}}{2\,c}}+{\frac{{b}^{2}ex}{{c}^{3}}}-{\frac{bdx}{{c}^{2}}}-{\frac{{b}^{3}\ln \left ( cx+b \right ) e}{{c}^{4}}}+{\frac{{b}^{2}\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 = 1.09317, size = 93, normalized size = 1.41 \begin{align*} \frac{2 \, c^{2} e x^{3} + 3 \,{\left (c^{2} d - b c e\right )} x^{2} - 6 \,{\left (b c d - b^{2} e\right )} x}{6 \, c^{3}} + \frac{{\left (b^{2} c d - b^{3} 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 = 1.75917, size = 149, normalized size = 2.26 \begin{align*} \frac{2 \, c^{3} e x^{3} + 3 \,{\left (c^{3} d - b c^{2} e\right )} x^{2} - 6 \,{\left (b c^{2} d - b^{2} c e\right )} x + 6 \,{\left (b^{2} c d - b^{3} e\right )} \log \left (c x + b\right )}{6 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.620153, size = 58, normalized size = 0.88 \begin{align*} - \frac{b^{2} \left (b e - c d\right ) \log{\left (b + c x \right )}}{c^{4}} + \frac{e x^{3}}{3 c} - \frac{x^{2} \left (b e - c d\right )}{2 c^{2}} + \frac{x \left (b^{2} e - b c d\right )}{c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13357, size = 100, normalized size = 1.52 \begin{align*} \frac{2 \, c^{2} x^{3} e + 3 \, c^{2} d x^{2} - 3 \, b c x^{2} e - 6 \, b c d x + 6 \, b^{2} x e}{6 \, c^{3}} + \frac{{\left (b^{2} c d - b^{3} e\right )} \log \left ({\left | c x + b \right |}\right )}{c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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