Optimal. Leaf size=99 \[ \frac {e^2 x \left (b^2 e^2-4 b c d e+6 c^2 d^2\right )}{c^3}-\frac {(c d-b e)^4 \log (b+c x)}{b c^4}+\frac {e^3 x^2 (4 c d-b e)}{2 c^2}+\frac {d^4 \log (x)}{b}+\frac {e^4 x^3}{3 c} \]
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Rubi [A] time = 0.09, antiderivative size = 99, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.04, 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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fricas [A] time = 1.02, size = 151, normalized size = 1.53 \[ \frac {2 \, b c^{3} e^{4} x^{3} + 6 \, c^{4} d^{4} \log \relax (x) + 3 \, {\left (4 \, b c^{3} d e^{3} - b^{2} c^{2} e^{4}\right )} x^{2} + 6 \, {\left (6 \, b c^{3} d^{2} e^{2} - 4 \, b^{2} c^{2} d e^{3} + b^{3} c e^{4}\right )} x - 6 \, {\left (c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, b^{2} c^{2} d^{2} e^{2} - 4 \, b^{3} c d e^{3} + b^{4} e^{4}\right )} \log \left (c x + b\right )}{6 \, b c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 136, normalized size = 1.37 \[ \frac {d^{4} \log \left ({\left | x \right |}\right )}{b} + \frac {2 \, c^{2} x^{3} e^{4} + 12 \, c^{2} d x^{2} e^{3} + 36 \, c^{2} d^{2} x e^{2} - 3 \, b c x^{2} e^{4} - 24 \, b c d x e^{3} + 6 \, b^{2} x e^{4}}{6 \, c^{3}} - \frac {{\left (c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, b^{2} c^{2} d^{2} e^{2} - 4 \, b^{3} c d e^{3} + b^{4} e^{4}\right )} \log \left ({\left | c x + b \right |}\right )}{b c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 162, normalized size = 1.64 \[ \frac {e^{4} x^{3}}{3 c}-\frac {b \,e^{4} x^{2}}{2 c^{2}}+\frac {2 d \,e^{3} x^{2}}{c}-\frac {b^{3} e^{4} \ln \left (c x +b \right )}{c^{4}}+\frac {4 b^{2} d \,e^{3} \ln \left (c x +b \right )}{c^{3}}+\frac {b^{2} e^{4} x}{c^{3}}-\frac {6 b \,d^{2} e^{2} \ln \left (c x +b \right )}{c^{2}}-\frac {4 b d \,e^{3} x}{c^{2}}+\frac {d^{4} \ln \relax (x )}{b}-\frac {d^{4} \ln \left (c x +b \right )}{b}+\frac {4 d^{3} e \ln \left (c x +b \right )}{c}+\frac {6 d^{2} e^{2} x}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.41, size = 142, normalized size = 1.43 \[ \frac {d^{4} \log \relax (x)}{b} + \frac {2 \, c^{2} e^{4} x^{3} + 3 \, {\left (4 \, c^{2} d e^{3} - b c e^{4}\right )} x^{2} + 6 \, {\left (6 \, c^{2} d^{2} e^{2} - 4 \, b c d e^{3} + b^{2} e^{4}\right )} x}{6 \, c^{3}} - \frac {{\left (c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, b^{2} c^{2} d^{2} e^{2} - 4 \, b^{3} c d e^{3} + b^{4} e^{4}\right )} \log \left (c x + b\right )}{b c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 106, normalized size = 1.07 \[ x\,\left (\frac {b\,\left (\frac {b\,e^4}{c^2}-\frac {4\,d\,e^3}{c}\right )}{c}+\frac {6\,d^2\,e^2}{c}\right )-x^2\,\left (\frac {b\,e^4}{2\,c^2}-\frac {2\,d\,e^3}{c}\right )+\frac {e^4\,x^3}{3\,c}+\frac {d^4\,\ln \relax (x)}{b}-\frac {\ln \left (b+c\,x\right )\,{\left (b\,e-c\,d\right )}^4}{b\,c^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.80, size = 165, normalized size = 1.67 \[ x^{2} \left (- \frac {b e^{4}}{2 c^{2}} + \frac {2 d e^{3}}{c}\right ) + x \left (\frac {b^{2} e^{4}}{c^{3}} - \frac {4 b d e^{3}}{c^{2}} + \frac {6 d^{2} e^{2}}{c}\right ) + \frac {e^{4} x^{3}}{3 c} + \frac {d^{4} \log {\relax (x )}}{b} - \frac {\left (b e - c d\right )^{4} \log {\left (x + \frac {b c^{3} d^{4} + \frac {b \left (b e - c d\right )^{4}}{c}}{b^{4} e^{4} - 4 b^{3} c d e^{3} + 6 b^{2} c^{2} d^{2} e^{2} - 4 b c^{3} d^{3} e + 2 c^{4} d^{4}} \right )}}{b c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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