Optimal. Leaf size=137 \[ \frac{3 d^2 (c d-b e)}{b^4 x}+\frac{3 d (c d-b e)^2}{b^4 (b+c x)}+\frac{(c d-b e)^3}{2 b^3 c (b+c x)^2}+\frac{3 d \log (x) (c d-b e) (2 c d-b e)}{b^5}-\frac{3 d (c d-b e) (2 c d-b e) \log (b+c x)}{b^5}-\frac{d^3}{2 b^3 x^2} \]
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Rubi [A] time = 0.147119, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {698} \[ \frac{3 d^2 (c d-b e)}{b^4 x}+\frac{3 d (c d-b e)^2}{b^4 (b+c x)}+\frac{(c d-b e)^3}{2 b^3 c (b+c x)^2}+\frac{3 d \log (x) (c d-b e) (2 c d-b e)}{b^5}-\frac{3 d (c d-b e) (2 c d-b e) \log (b+c x)}{b^5}-\frac{d^3}{2 b^3 x^2} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{\left (b x+c x^2\right )^3} \, dx &=\int \left (\frac{d^3}{b^3 x^3}+\frac{3 d^2 (-c d+b e)}{b^4 x^2}+\frac{3 d (c d-b e) (2 c d-b e)}{b^5 x}+\frac{(-c d+b e)^3}{b^3 (b+c x)^3}-\frac{3 c d (-c d+b e)^2}{b^4 (b+c x)^2}+\frac{3 c d (c d-b e) (-2 c d+b e)}{b^5 (b+c x)}\right ) \, dx\\ &=-\frac{d^3}{2 b^3 x^2}+\frac{3 d^2 (c d-b e)}{b^4 x}+\frac{(c d-b e)^3}{2 b^3 c (b+c x)^2}+\frac{3 d (c d-b e)^2}{b^4 (b+c x)}+\frac{3 d (c d-b e) (2 c d-b e) \log (x)}{b^5}-\frac{3 d (c d-b e) (2 c d-b e) \log (b+c x)}{b^5}\\ \end{align*}
Mathematica [A] time = 0.15537, size = 138, normalized size = 1.01 \[ -\frac{-6 d \log (x) \left (b^2 e^2-3 b c d e+2 c^2 d^2\right )+6 d \left (b^2 e^2-3 b c d e+2 c^2 d^2\right ) \log (b+c x)+\frac{b^2 (b e-c d)^3}{c (b+c x)^2}+\frac{b^2 d^3}{x^2}+\frac{6 b d^2 (b e-c d)}{x}-\frac{6 b d (c d-b e)^2}{b+c x}}{2 b^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.058, size = 238, normalized size = 1.7 \begin{align*} -{\frac{{d}^{3}}{2\,{b}^{3}{x}^{2}}}+3\,{\frac{d\ln \left ( x \right ){e}^{2}}{{b}^{3}}}-9\,{\frac{{d}^{2}\ln \left ( x \right ) ce}{{b}^{4}}}+6\,{\frac{{d}^{3}\ln \left ( x \right ){c}^{2}}{{b}^{5}}}-3\,{\frac{{d}^{2}e}{{b}^{3}x}}+3\,{\frac{{d}^{3}c}{{b}^{4}x}}-{\frac{{e}^{3}}{2\,c \left ( cx+b \right ) ^{2}}}+{\frac{3\,d{e}^{2}}{2\,b \left ( cx+b \right ) ^{2}}}-{\frac{3\,{d}^{2}ec}{2\,{b}^{2} \left ( cx+b \right ) ^{2}}}+{\frac{{c}^{2}{d}^{3}}{2\,{b}^{3} \left ( cx+b \right ) ^{2}}}-3\,{\frac{d\ln \left ( cx+b \right ){e}^{2}}{{b}^{3}}}+9\,{\frac{{d}^{2}\ln \left ( cx+b \right ) ce}{{b}^{4}}}-6\,{\frac{{d}^{3}\ln \left ( cx+b \right ){c}^{2}}{{b}^{5}}}+3\,{\frac{d{e}^{2}}{{b}^{2} \left ( cx+b \right ) }}-6\,{\frac{{d}^{2}ec}{{b}^{3} \left ( cx+b \right ) }}+3\,{\frac{{c}^{2}{d}^{3}}{{b}^{4} \left ( cx+b \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08297, size = 293, normalized size = 2.14 \begin{align*} -\frac{b^{3} c d^{3} - 6 \,{\left (2 \, c^{4} d^{3} - 3 \, b c^{3} d^{2} e + b^{2} c^{2} d e^{2}\right )} x^{3} -{\left (18 \, b c^{3} d^{3} - 27 \, b^{2} c^{2} d^{2} e + 9 \, b^{3} c d e^{2} - b^{4} e^{3}\right )} x^{2} - 2 \,{\left (2 \, b^{2} c^{2} d^{3} - 3 \, b^{3} c d^{2} e\right )} x}{2 \,{\left (b^{4} c^{3} x^{4} + 2 \, b^{5} c^{2} x^{3} + b^{6} c x^{2}\right )}} - \frac{3 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} \log \left (c x + b\right )}{b^{5}} + \frac{3 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\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 = 1.93519, size = 770, normalized size = 5.62 \begin{align*} -\frac{b^{4} c d^{3} - 6 \,{\left (2 \, b c^{4} d^{3} - 3 \, b^{2} c^{3} d^{2} e + b^{3} c^{2} d e^{2}\right )} x^{3} -{\left (18 \, b^{2} c^{3} d^{3} - 27 \, b^{3} c^{2} d^{2} e + 9 \, b^{4} c d e^{2} - b^{5} e^{3}\right )} x^{2} - 2 \,{\left (2 \, b^{3} c^{2} d^{3} - 3 \, b^{4} c d^{2} e\right )} x + 6 \,{\left ({\left (2 \, c^{5} d^{3} - 3 \, b c^{4} d^{2} e + b^{2} c^{3} d e^{2}\right )} x^{4} + 2 \,{\left (2 \, b c^{4} d^{3} - 3 \, b^{2} c^{3} d^{2} e + b^{3} c^{2} d e^{2}\right )} x^{3} +{\left (2 \, b^{2} c^{3} d^{3} - 3 \, b^{3} c^{2} d^{2} e + b^{4} c d e^{2}\right )} x^{2}\right )} \log \left (c x + b\right ) - 6 \,{\left ({\left (2 \, c^{5} d^{3} - 3 \, b c^{4} d^{2} e + b^{2} c^{3} d e^{2}\right )} x^{4} + 2 \,{\left (2 \, b c^{4} d^{3} - 3 \, b^{2} c^{3} d^{2} e + b^{3} c^{2} d e^{2}\right )} x^{3} +{\left (2 \, b^{2} c^{3} d^{3} - 3 \, b^{3} c^{2} d^{2} e + b^{4} c d e^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (b^{5} c^{3} x^{4} + 2 \, b^{6} c^{2} x^{3} + b^{7} c x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 4.0019, size = 371, normalized size = 2.71 \begin{align*} \frac{- b^{3} c d^{3} + x^{3} \left (6 b^{2} c^{2} d e^{2} - 18 b c^{3} d^{2} e + 12 c^{4} d^{3}\right ) + x^{2} \left (- b^{4} e^{3} + 9 b^{3} c d e^{2} - 27 b^{2} c^{2} d^{2} e + 18 b c^{3} d^{3}\right ) + x \left (- 6 b^{3} c d^{2} e + 4 b^{2} c^{2} d^{3}\right )}{2 b^{6} c x^{2} + 4 b^{5} c^{2} x^{3} + 2 b^{4} c^{3} x^{4}} + \frac{3 d \left (b e - 2 c d\right ) \left (b e - c d\right ) \log{\left (x + \frac{3 b^{3} d e^{2} - 9 b^{2} c d^{2} e + 6 b c^{2} d^{3} - 3 b d \left (b e - 2 c d\right ) \left (b e - c d\right )}{6 b^{2} c d e^{2} - 18 b c^{2} d^{2} e + 12 c^{3} d^{3}} \right )}}{b^{5}} - \frac{3 d \left (b e - 2 c d\right ) \left (b e - c d\right ) \log{\left (x + \frac{3 b^{3} d e^{2} - 9 b^{2} c d^{2} e + 6 b c^{2} d^{3} + 3 b d \left (b e - 2 c d\right ) \left (b e - c d\right )}{6 b^{2} c d e^{2} - 18 b c^{2} d^{2} e + 12 c^{3} d^{3}} \right )}}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26788, size = 296, normalized size = 2.16 \begin{align*} \frac{3 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} - \frac{3 \,{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + b^{2} c d e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} + \frac{12 \, c^{4} d^{3} x^{3} - 18 \, b c^{3} d^{2} x^{3} e + 18 \, b c^{3} d^{3} x^{2} + 6 \, b^{2} c^{2} d x^{3} e^{2} - 27 \, b^{2} c^{2} d^{2} x^{2} e + 4 \, b^{2} c^{2} d^{3} x + 9 \, b^{3} c d x^{2} e^{2} - 6 \, b^{3} c d^{2} x e - b^{3} c d^{3} - b^{4} x^{2} e^{3}}{2 \,{\left (c x^{2} + b x\right )}^{2} b^{4} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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