Optimal. Leaf size=110 \[ \frac{3 c d-b e}{b^4 x}+\frac{c (3 c d-2 b e)}{b^4 (b+c x)}+\frac{c (c d-b e)}{2 b^3 (b+c x)^2}+\frac{3 c \log (x) (2 c d-b e)}{b^5}-\frac{3 c (2 c d-b e) \log (b+c x)}{b^5}-\frac{d}{2 b^3 x^2} \]
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Rubi [A] time = 0.103272, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {631} \[ \frac{3 c d-b e}{b^4 x}+\frac{c (3 c d-2 b e)}{b^4 (b+c x)}+\frac{c (c d-b e)}{2 b^3 (b+c x)^2}+\frac{3 c \log (x) (2 c d-b e)}{b^5}-\frac{3 c (2 c d-b e) \log (b+c x)}{b^5}-\frac{d}{2 b^3 x^2} \]
Antiderivative was successfully verified.
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Rule 631
Rubi steps
\begin{align*} \int \frac{d+e x}{\left (b x+c x^2\right )^3} \, dx &=\int \left (\frac{d}{b^3 x^3}+\frac{-3 c d+b e}{b^4 x^2}-\frac{3 c (-2 c d+b e)}{b^5 x}+\frac{c^2 (-c d+b e)}{b^3 (b+c x)^3}+\frac{c^2 (-3 c d+2 b e)}{b^4 (b+c x)^2}+\frac{3 c^2 (-2 c d+b e)}{b^5 (b+c x)}\right ) \, dx\\ &=-\frac{d}{2 b^3 x^2}+\frac{3 c d-b e}{b^4 x}+\frac{c (c d-b e)}{2 b^3 (b+c x)^2}+\frac{c (3 c d-2 b e)}{b^4 (b+c x)}+\frac{3 c (2 c d-b e) \log (x)}{b^5}-\frac{3 c (2 c d-b e) \log (b+c x)}{b^5}\\ \end{align*}
Mathematica [A] time = 0.080662, size = 102, normalized size = 0.93 \[ \frac{-\frac{b \left (b^2 c x (9 e x-4 d)+b^3 (d+2 e x)+6 b c^2 x^2 (e x-3 d)-12 c^3 d x^3\right )}{x^2 (b+c x)^2}+6 c \log (x) (2 c d-b e)+6 c (b e-2 c d) \log (b+c x)}{2 b^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 138, normalized size = 1.3 \begin{align*} -{\frac{d}{2\,{b}^{3}{x}^{2}}}-{\frac{e}{{b}^{3}x}}+3\,{\frac{cd}{{b}^{4}x}}-3\,{\frac{c\ln \left ( x \right ) e}{{b}^{4}}}+6\,{\frac{{c}^{2}\ln \left ( x \right ) d}{{b}^{5}}}-2\,{\frac{ce}{{b}^{3} \left ( cx+b \right ) }}+3\,{\frac{d{c}^{2}}{{b}^{4} \left ( cx+b \right ) }}-{\frac{ce}{2\,{b}^{2} \left ( cx+b \right ) ^{2}}}+{\frac{d{c}^{2}}{2\,{b}^{3} \left ( cx+b \right ) ^{2}}}+3\,{\frac{c\ln \left ( cx+b \right ) e}{{b}^{4}}}-6\,{\frac{{c}^{2}\ln \left ( cx+b \right ) d}{{b}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12258, size = 184, normalized size = 1.67 \begin{align*} -\frac{b^{3} d - 6 \,{\left (2 \, c^{3} d - b c^{2} e\right )} x^{3} - 9 \,{\left (2 \, b c^{2} d - b^{2} c e\right )} x^{2} - 2 \,{\left (2 \, b^{2} c d - b^{3} e\right )} x}{2 \,{\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}} - \frac{3 \,{\left (2 \, c^{2} d - b c e\right )} \log \left (c x + b\right )}{b^{5}} + \frac{3 \,{\left (2 \, c^{2} d - b c e\right )} \log \left (x\right )}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.87802, size = 467, normalized size = 4.25 \begin{align*} -\frac{b^{4} d - 6 \,{\left (2 \, b c^{3} d - b^{2} c^{2} e\right )} x^{3} - 9 \,{\left (2 \, b^{2} c^{2} d - b^{3} c e\right )} x^{2} - 2 \,{\left (2 \, b^{3} c d - b^{4} e\right )} x + 6 \,{\left ({\left (2 \, c^{4} d - b c^{3} e\right )} x^{4} + 2 \,{\left (2 \, b c^{3} d - b^{2} c^{2} e\right )} x^{3} +{\left (2 \, b^{2} c^{2} d - b^{3} c e\right )} x^{2}\right )} \log \left (c x + b\right ) - 6 \,{\left ({\left (2 \, c^{4} d - b c^{3} e\right )} x^{4} + 2 \,{\left (2 \, b c^{3} d - b^{2} c^{2} e\right )} x^{3} +{\left (2 \, b^{2} c^{2} d - b^{3} c e\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (b^{5} c^{2} x^{4} + 2 \, b^{6} c x^{3} + b^{7} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.74491, size = 219, normalized size = 1.99 \begin{align*} - \frac{b^{3} d + x^{3} \left (6 b c^{2} e - 12 c^{3} d\right ) + x^{2} \left (9 b^{2} c e - 18 b c^{2} d\right ) + x \left (2 b^{3} e - 4 b^{2} c d\right )}{2 b^{6} x^{2} + 4 b^{5} c x^{3} + 2 b^{4} c^{2} x^{4}} - \frac{3 c \left (b e - 2 c d\right ) \log{\left (x + \frac{3 b^{2} c e - 6 b c^{2} d - 3 b c \left (b e - 2 c d\right )}{6 b c^{2} e - 12 c^{3} d} \right )}}{b^{5}} + \frac{3 c \left (b e - 2 c d\right ) \log{\left (x + \frac{3 b^{2} c e - 6 b c^{2} d + 3 b c \left (b e - 2 c d\right )}{6 b c^{2} e - 12 c^{3} d} \right )}}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25839, size = 178, normalized size = 1.62 \begin{align*} \frac{3 \,{\left (2 \, c^{2} d - b c e\right )} \log \left ({\left | x \right |}\right )}{b^{5}} - \frac{3 \,{\left (2 \, c^{3} d - b c^{2} e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} + \frac{12 \, c^{3} d x^{3} - 6 \, b c^{2} x^{3} e + 18 \, b c^{2} d x^{2} - 9 \, b^{2} c x^{2} e + 4 \, b^{2} c d x - 2 \, b^{3} x e - b^{3} d}{2 \,{\left (c x^{2} + b x\right )}^{2} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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