Optimal. Leaf size=99 \[ \frac{e^2 x \left (b^2 e^2-4 b c d e+6 c^2 d^2\right )}{c^3}+\frac{e^3 x^2 (4 c d-b e)}{2 c^2}-\frac{(c d-b e)^4 \log (b+c x)}{b c^4}+\frac{d^4 \log (x)}{b}+\frac{e^4 x^3}{3 c} \]
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Rubi [A] time = 0.0908252, antiderivative size = 99, 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} \[ \frac{e^2 x \left (b^2 e^2-4 b c d e+6 c^2 d^2\right )}{c^3}+\frac{e^3 x^2 (4 c d-b e)}{2 c^2}-\frac{(c d-b e)^4 \log (b+c x)}{b c^4}+\frac{d^4 \log (x)}{b}+\frac{e^4 x^3}{3 c} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{(d+e x)^4}{b x+c x^2} \, dx &=\int \left (\frac{e^2 \left (6 c^2 d^2-4 b c d e+b^2 e^2\right )}{c^3}+\frac{d^4}{b x}+\frac{e^3 (4 c d-b e) x}{c^2}+\frac{e^4 x^2}{c}-\frac{(-c d+b e)^4}{b c^3 (b+c x)}\right ) \, dx\\ &=\frac{e^2 \left (6 c^2 d^2-4 b c d e+b^2 e^2\right ) x}{c^3}+\frac{e^3 (4 c d-b e) x^2}{2 c^2}+\frac{e^4 x^3}{3 c}+\frac{d^4 \log (x)}{b}-\frac{(c d-b e)^4 \log (b+c x)}{b c^4}\\ \end{align*}
Mathematica [A] time = 0.0454209, size = 90, normalized size = 0.91 \[ \frac{b c e^2 x \left (6 b^2 e^2-3 b c e (8 d+e x)+2 c^2 \left (18 d^2+6 d e x+e^2 x^2\right )\right )-6 (c d-b e)^4 \log (b+c x)+6 c^4 d^4 \log (x)}{6 b c^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 162, normalized size = 1.6 \begin{align*}{\frac{{e}^{4}{x}^{3}}{3\,c}}-{\frac{{e}^{4}{x}^{2}b}{2\,{c}^{2}}}+2\,{\frac{d{e}^{3}{x}^{2}}{c}}+{\frac{{e}^{4}{b}^{2}x}{{c}^{3}}}-4\,{\frac{{e}^{3}bdx}{{c}^{2}}}+6\,{\frac{{d}^{2}{e}^{2}x}{c}}+{\frac{{d}^{4}\ln \left ( x \right ) }{b}}-{\frac{{b}^{3}\ln \left ( cx+b \right ){e}^{4}}{{c}^{4}}}+4\,{\frac{{b}^{2}\ln \left ( cx+b \right ) d{e}^{3}}{{c}^{3}}}-6\,{\frac{b\ln \left ( cx+b \right ){d}^{2}{e}^{2}}{{c}^{2}}}+4\,{\frac{\ln \left ( cx+b \right ){d}^{3}e}{c}}-{\frac{\ln \left ( cx+b \right ){d}^{4}}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14139, size = 192, normalized size = 1.94 \begin{align*} \frac{d^{4} \log \left (x\right )}{b} + \frac{2 \, c^{2} e^{4} x^{3} + 3 \,{\left (4 \, c^{2} d e^{3} - b c e^{4}\right )} x^{2} + 6 \,{\left (6 \, c^{2} d^{2} e^{2} - 4 \, b c d e^{3} + b^{2} e^{4}\right )} x}{6 \, c^{3}} - \frac{{\left (c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, b^{2} c^{2} d^{2} e^{2} - 4 \, b^{3} c d e^{3} + b^{4} e^{4}\right )} \log \left (c x + b\right )}{b c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70411, size = 312, normalized size = 3.15 \begin{align*} \frac{2 \, b c^{3} e^{4} x^{3} + 6 \, c^{4} d^{4} \log \left (x\right ) + 3 \,{\left (4 \, b c^{3} d e^{3} - b^{2} c^{2} e^{4}\right )} x^{2} + 6 \,{\left (6 \, b c^{3} d^{2} e^{2} - 4 \, b^{2} c^{2} d e^{3} + b^{3} c e^{4}\right )} x - 6 \,{\left (c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, b^{2} c^{2} d^{2} e^{2} - 4 \, b^{3} c d e^{3} + b^{4} e^{4}\right )} \log \left (c x + b\right )}{6 \, b c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.8469, size = 165, normalized size = 1.67 \begin{align*} \frac{e^{4} x^{3}}{3 c} - \frac{x^{2} \left (b e^{4} - 4 c d e^{3}\right )}{2 c^{2}} + \frac{x \left (b^{2} e^{4} - 4 b c d e^{3} + 6 c^{2} d^{2} e^{2}\right )}{c^{3}} + \frac{d^{4} \log{\left (x \right )}}{b} - \frac{\left (b e - c d\right )^{4} \log{\left (x + \frac{b c^{3} d^{4} + \frac{b \left (b e - c d\right )^{4}}{c}}{b^{4} e^{4} - 4 b^{3} c d e^{3} + 6 b^{2} c^{2} d^{2} e^{2} - 4 b c^{3} d^{3} e + 2 c^{4} d^{4}} \right )}}{b c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24519, size = 184, normalized size = 1.86 \begin{align*} \frac{d^{4} \log \left ({\left | x \right |}\right )}{b} + \frac{2 \, c^{2} x^{3} e^{4} + 12 \, c^{2} d x^{2} e^{3} + 36 \, c^{2} d^{2} x e^{2} - 3 \, b c x^{2} e^{4} - 24 \, b c d x e^{3} + 6 \, b^{2} x e^{4}}{6 \, c^{3}} - \frac{{\left (c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, b^{2} c^{2} d^{2} e^{2} - 4 \, b^{3} c d e^{3} + b^{4} e^{4}\right )} \log \left ({\left | c x + b \right |}\right )}{b c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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