Optimal. Leaf size=118 \[ -\frac{(c d-b e)^5}{b^2 c^4 (b+c x)}+\frac{(c d-b e)^4 (3 b e+2 c d) \log (b+c x)}{b^3 c^4}-\frac{d^4 \log (x) (2 c d-5 b e)}{b^3}-\frac{d^5}{b^2 x}+\frac{e^4 x (5 c d-2 b e)}{c^3}+\frac{e^5 x^2}{2 c^2} \]
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Rubi [A] time = 0.13772, antiderivative size = 118, 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} \[ -\frac{(c d-b e)^5}{b^2 c^4 (b+c x)}+\frac{(c d-b e)^4 (3 b e+2 c d) \log (b+c x)}{b^3 c^4}-\frac{d^4 \log (x) (2 c d-5 b e)}{b^3}-\frac{d^5}{b^2 x}+\frac{e^4 x (5 c d-2 b e)}{c^3}+\frac{e^5 x^2}{2 c^2} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{(d+e x)^5}{\left (b x+c x^2\right )^2} \, dx &=\int \left (\frac{e^4 (5 c d-2 b e)}{c^3}+\frac{d^5}{b^2 x^2}+\frac{d^4 (-2 c d+5 b e)}{b^3 x}+\frac{e^5 x}{c^2}-\frac{(-c d+b e)^5}{b^2 c^3 (b+c x)^2}+\frac{(-c d+b e)^4 (2 c d+3 b e)}{b^3 c^3 (b+c x)}\right ) \, dx\\ &=-\frac{d^5}{b^2 x}+\frac{e^4 (5 c d-2 b e) x}{c^3}+\frac{e^5 x^2}{2 c^2}-\frac{(c d-b e)^5}{b^2 c^4 (b+c x)}-\frac{d^4 (2 c d-5 b e) \log (x)}{b^3}+\frac{(c d-b e)^4 (2 c d+3 b e) \log (b+c x)}{b^3 c^4}\\ \end{align*}
Mathematica [A] time = 0.0568775, size = 116, normalized size = 0.98 \[ \frac{(b e-c d)^5}{b^2 c^4 (b+c x)}+\frac{(c d-b e)^4 (3 b e+2 c d) \log (b+c x)}{b^3 c^4}+\frac{d^4 \log (x) (5 b e-2 c d)}{b^3}-\frac{d^5}{b^2 x}+\frac{e^4 x (5 c d-2 b e)}{c^3}+\frac{e^5 x^2}{2 c^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.061, size = 251, normalized size = 2.1 \begin{align*}{\frac{{e}^{5}{x}^{2}}{2\,{c}^{2}}}-2\,{\frac{{e}^{5}xb}{{c}^{3}}}+5\,{\frac{d{e}^{4}x}{{c}^{2}}}-{\frac{{d}^{5}}{{b}^{2}x}}+5\,{\frac{{d}^{4}\ln \left ( x \right ) e}{{b}^{2}}}-2\,{\frac{{d}^{5}\ln \left ( x \right ) c}{{b}^{3}}}+3\,{\frac{{b}^{2}\ln \left ( cx+b \right ){e}^{5}}{{c}^{4}}}-10\,{\frac{b\ln \left ( cx+b \right ) d{e}^{4}}{{c}^{3}}}+10\,{\frac{\ln \left ( cx+b \right ){d}^{2}{e}^{3}}{{c}^{2}}}-5\,{\frac{\ln \left ( cx+b \right ){d}^{4}e}{{b}^{2}}}+2\,{\frac{c\ln \left ( cx+b \right ){d}^{5}}{{b}^{3}}}+{\frac{{b}^{3}{e}^{5}}{{c}^{4} \left ( cx+b \right ) }}-5\,{\frac{{b}^{2}d{e}^{4}}{{c}^{3} \left ( cx+b \right ) }}+10\,{\frac{b{d}^{2}{e}^{3}}{{c}^{2} \left ( cx+b \right ) }}-10\,{\frac{{d}^{3}{e}^{2}}{c \left ( cx+b \right ) }}+5\,{\frac{{d}^{4}e}{b \left ( cx+b \right ) }}-{\frac{c{d}^{5}}{{b}^{2} \left ( cx+b \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.997527, size = 292, normalized size = 2.47 \begin{align*} -\frac{b c^{4} d^{5} +{\left (2 \, c^{5} d^{5} - 5 \, b c^{4} d^{4} e + 10 \, b^{2} c^{3} d^{3} e^{2} - 10 \, b^{3} c^{2} d^{2} e^{3} + 5 \, b^{4} c d e^{4} - b^{5} e^{5}\right )} x}{b^{2} c^{5} x^{2} + b^{3} c^{4} x} - \frac{{\left (2 \, c d^{5} - 5 \, b d^{4} e\right )} \log \left (x\right )}{b^{3}} + \frac{c e^{5} x^{2} + 2 \,{\left (5 \, c d e^{4} - 2 \, b e^{5}\right )} x}{2 \, c^{3}} + \frac{{\left (2 \, c^{5} d^{5} - 5 \, b c^{4} d^{4} e + 10 \, b^{3} c^{2} d^{2} e^{3} - 10 \, b^{4} c d e^{4} + 3 \, b^{5} e^{5}\right )} \log \left (c x + b\right )}{b^{3} c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.75372, size = 707, normalized size = 5.99 \begin{align*} \frac{b^{3} c^{3} e^{5} x^{4} - 2 \, b^{2} c^{4} d^{5} +{\left (10 \, b^{3} c^{3} d e^{4} - 3 \, b^{4} c^{2} e^{5}\right )} x^{3} + 2 \,{\left (5 \, b^{4} c^{2} d e^{4} - 2 \, b^{5} c e^{5}\right )} x^{2} - 2 \,{\left (2 \, b c^{5} d^{5} - 5 \, b^{2} c^{4} d^{4} e + 10 \, b^{3} c^{3} d^{3} e^{2} - 10 \, b^{4} c^{2} d^{2} e^{3} + 5 \, b^{5} c d e^{4} - b^{6} e^{5}\right )} x + 2 \,{\left ({\left (2 \, c^{6} d^{5} - 5 \, b c^{5} d^{4} e + 10 \, b^{3} c^{3} d^{2} e^{3} - 10 \, b^{4} c^{2} d e^{4} + 3 \, b^{5} c e^{5}\right )} x^{2} +{\left (2 \, b c^{5} d^{5} - 5 \, b^{2} c^{4} d^{4} e + 10 \, b^{4} c^{2} d^{2} e^{3} - 10 \, b^{5} c d e^{4} + 3 \, b^{6} e^{5}\right )} x\right )} \log \left (c x + b\right ) - 2 \,{\left ({\left (2 \, c^{6} d^{5} - 5 \, b c^{5} d^{4} e\right )} x^{2} +{\left (2 \, b c^{5} d^{5} - 5 \, b^{2} c^{4} d^{4} e\right )} x\right )} \log \left (x\right )}{2 \,{\left (b^{3} c^{5} x^{2} + b^{4} c^{4} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 8.45732, size = 379, normalized size = 3.21 \begin{align*} \frac{- b c^{4} d^{5} + x \left (b^{5} e^{5} - 5 b^{4} c d e^{4} + 10 b^{3} c^{2} d^{2} e^{3} - 10 b^{2} c^{3} d^{3} e^{2} + 5 b c^{4} d^{4} e - 2 c^{5} d^{5}\right )}{b^{3} c^{4} x + b^{2} c^{5} x^{2}} + \frac{e^{5} x^{2}}{2 c^{2}} - \frac{x \left (2 b e^{5} - 5 c d e^{4}\right )}{c^{3}} + \frac{d^{4} \left (5 b e - 2 c d\right ) \log{\left (x + \frac{- 5 b^{2} c^{3} d^{4} e + 2 b c^{4} d^{5} + b c^{3} d^{4} \left (5 b e - 2 c d\right )}{3 b^{5} e^{5} - 10 b^{4} c d e^{4} + 10 b^{3} c^{2} d^{2} e^{3} - 10 b c^{4} d^{4} e + 4 c^{5} d^{5}} \right )}}{b^{3}} + \frac{\left (b e - c d\right )^{4} \left (3 b e + 2 c d\right ) \log{\left (x + \frac{- 5 b^{2} c^{3} d^{4} e + 2 b c^{4} d^{5} + \frac{b \left (b e - c d\right )^{4} \left (3 b e + 2 c d\right )}{c}}{3 b^{5} e^{5} - 10 b^{4} c d e^{4} + 10 b^{3} c^{2} d^{2} e^{3} - 10 b c^{4} d^{4} e + 4 c^{5} d^{5}} \right )}}{b^{3} c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30331, size = 282, normalized size = 2.39 \begin{align*} -\frac{{\left (2 \, c d^{5} - 5 \, b d^{4} e\right )} \log \left ({\left | x \right |}\right )}{b^{3}} + \frac{c^{2} x^{2} e^{5} + 10 \, c^{2} d x e^{4} - 4 \, b c x e^{5}}{2 \, c^{4}} + \frac{{\left (2 \, c^{5} d^{5} - 5 \, b c^{4} d^{4} e + 10 \, b^{3} c^{2} d^{2} e^{3} - 10 \, b^{4} c d e^{4} + 3 \, b^{5} e^{5}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c^{4}} - \frac{b c^{4} d^{5} +{\left (2 \, c^{5} d^{5} - 5 \, b c^{4} d^{4} e + 10 \, b^{2} c^{3} d^{3} e^{2} - 10 \, b^{3} c^{2} d^{2} e^{3} + 5 \, b^{4} c d e^{4} - b^{5} e^{5}\right )} x}{{\left (c x + b\right )} b^{2} c^{4} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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