Optimal. Leaf size=144 \[ \frac{\log (x) \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{b^5}-\frac{\left (b^2 e^2-6 b c d e+6 c^2 d^2\right ) \log (b+c x)}{b^5}+\frac{d (3 c d-2 b e)}{b^4 x}+\frac{(c d-b e) (3 c d-b e)}{b^4 (b+c x)}+\frac{(c d-b e)^2}{2 b^3 (b+c x)^2}-\frac{d^2}{2 b^3 x^2} \]
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Rubi [A] time = 0.132759, antiderivative size = 144, 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} \[ \frac{\log (x) \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{b^5}-\frac{\left (b^2 e^2-6 b c d e+6 c^2 d^2\right ) \log (b+c x)}{b^5}+\frac{d (3 c d-2 b e)}{b^4 x}+\frac{(c d-b e) (3 c d-b e)}{b^4 (b+c x)}+\frac{(c d-b e)^2}{2 b^3 (b+c x)^2}-\frac{d^2}{2 b^3 x^2} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{\left (b x+c x^2\right )^3} \, dx &=\int \left (\frac{d^2}{b^3 x^3}+\frac{d (-3 c d+2 b e)}{b^4 x^2}+\frac{6 c^2 d^2-6 b c d e+b^2 e^2}{b^5 x}-\frac{c (-c d+b e)^2}{b^3 (b+c x)^3}+\frac{c (c d-b e) (-3 c d+b e)}{b^4 (b+c x)^2}-\frac{c \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )}{b^5 (b+c x)}\right ) \, dx\\ &=-\frac{d^2}{2 b^3 x^2}+\frac{d (3 c d-2 b e)}{b^4 x}+\frac{(c d-b e)^2}{2 b^3 (b+c x)^2}+\frac{(c d-b e) (3 c d-b e)}{b^4 (b+c x)}+\frac{\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) \log (x)}{b^5}-\frac{\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) \log (b+c x)}{b^5}\\ \end{align*}
Mathematica [A] time = 0.0939355, size = 144, normalized size = 1. \[ \frac{\frac{2 b \left (b^2 e^2-4 b c d e+3 c^2 d^2\right )}{b+c x}+2 \log (x) \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-2 \left (b^2 e^2-6 b c d e+6 c^2 d^2\right ) \log (b+c x)+\frac{b^2 (c d-b e)^2}{(b+c x)^2}-\frac{b^2 d^2}{x^2}-\frac{2 b d (2 b e-3 c d)}{x}}{2 b^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 207, normalized size = 1.4 \begin{align*} -{\frac{{d}^{2}}{2\,{b}^{3}{x}^{2}}}+{\frac{\ln \left ( x \right ){e}^{2}}{{b}^{3}}}-6\,{\frac{\ln \left ( x \right ) cde}{{b}^{4}}}+6\,{\frac{\ln \left ( x \right ){c}^{2}{d}^{2}}{{b}^{5}}}-2\,{\frac{de}{{b}^{3}x}}+3\,{\frac{c{d}^{2}}{{b}^{4}x}}-{\frac{\ln \left ( cx+b \right ){e}^{2}}{{b}^{3}}}+6\,{\frac{\ln \left ( cx+b \right ) cde}{{b}^{4}}}-6\,{\frac{\ln \left ( cx+b \right ){c}^{2}{d}^{2}}{{b}^{5}}}+{\frac{{e}^{2}}{{b}^{2} \left ( cx+b \right ) }}-4\,{\frac{dec}{{b}^{3} \left ( cx+b \right ) }}+3\,{\frac{{c}^{2}{d}^{2}}{{b}^{4} \left ( cx+b \right ) }}+{\frac{{e}^{2}}{2\,b \left ( cx+b \right ) ^{2}}}-{\frac{dec}{{b}^{2} \left ( cx+b \right ) ^{2}}}+{\frac{{c}^{2}{d}^{2}}{2\,{b}^{3} \left ( cx+b \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11236, size = 243, normalized size = 1.69 \begin{align*} -\frac{b^{3} d^{2} - 2 \,{\left (6 \, c^{3} d^{2} - 6 \, b c^{2} d e + b^{2} c e^{2}\right )} x^{3} - 3 \,{\left (6 \, b c^{2} d^{2} - 6 \, b^{2} c d e + b^{3} e^{2}\right )} x^{2} - 4 \,{\left (b^{2} c d^{2} - b^{3} d e\right )} x}{2 \,{\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}} - \frac{{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} \log \left (c x + b\right )}{b^{5}} + \frac{{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} 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 = 1.80002, size = 664, normalized size = 4.61 \begin{align*} -\frac{b^{4} d^{2} - 2 \,{\left (6 \, b c^{3} d^{2} - 6 \, b^{2} c^{2} d e + b^{3} c e^{2}\right )} x^{3} - 3 \,{\left (6 \, b^{2} c^{2} d^{2} - 6 \, b^{3} c d e + b^{4} e^{2}\right )} x^{2} - 4 \,{\left (b^{3} c d^{2} - b^{4} d e\right )} x + 2 \,{\left ({\left (6 \, c^{4} d^{2} - 6 \, b c^{3} d e + b^{2} c^{2} e^{2}\right )} x^{4} + 2 \,{\left (6 \, b c^{3} d^{2} - 6 \, b^{2} c^{2} d e + b^{3} c e^{2}\right )} x^{3} +{\left (6 \, b^{2} c^{2} d^{2} - 6 \, b^{3} c d e + b^{4} e^{2}\right )} x^{2}\right )} \log \left (c x + b\right ) - 2 \,{\left ({\left (6 \, c^{4} d^{2} - 6 \, b c^{3} d e + b^{2} c^{2} e^{2}\right )} x^{4} + 2 \,{\left (6 \, b c^{3} d^{2} - 6 \, b^{2} c^{2} d e + b^{3} c e^{2}\right )} x^{3} +{\left (6 \, b^{2} c^{2} d^{2} - 6 \, b^{3} c d e + b^{4} e^{2}\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 = 2.5886, size = 345, normalized size = 2.4 \begin{align*} \frac{- b^{3} d^{2} + x^{3} \left (2 b^{2} c e^{2} - 12 b c^{2} d e + 12 c^{3} d^{2}\right ) + x^{2} \left (3 b^{3} e^{2} - 18 b^{2} c d e + 18 b c^{2} d^{2}\right ) + x \left (- 4 b^{3} d e + 4 b^{2} c d^{2}\right )}{2 b^{6} x^{2} + 4 b^{5} c x^{3} + 2 b^{4} c^{2} x^{4}} + \frac{\left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right ) \log{\left (x + \frac{b^{3} e^{2} - 6 b^{2} c d e + 6 b c^{2} d^{2} - b \left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{2 b^{2} c e^{2} - 12 b c^{2} d e + 12 c^{3} d^{2}} \right )}}{b^{5}} - \frac{\left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right ) \log{\left (x + \frac{b^{3} e^{2} - 6 b^{2} c d e + 6 b c^{2} d^{2} + b \left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{2 b^{2} c e^{2} - 12 b c^{2} d e + 12 c^{3} d^{2}} \right )}}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26516, size = 246, normalized size = 1.71 \begin{align*} \frac{{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} - \frac{{\left (6 \, c^{3} d^{2} - 6 \, b c^{2} d e + b^{2} c e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} + \frac{12 \, c^{3} d^{2} x^{3} - 12 \, b c^{2} d x^{3} e + 18 \, b c^{2} d^{2} x^{2} + 2 \, b^{2} c x^{3} e^{2} - 18 \, b^{2} c d x^{2} e + 4 \, b^{2} c d^{2} x + 3 \, b^{3} x^{2} e^{2} - 4 \, b^{3} d x e - b^{3} d^{2}}{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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