Optimal. Leaf size=64 \[ -\frac {(c d-b e)^3 \log (b+c x)}{b c^3}+\frac {e^2 x (3 c d-b e)}{c^2}+\frac {d^3 \log (x)}{b}+\frac {e^3 x^2}{2 c} \]
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Rubi [A] time = 0.06, antiderivative size = 64, 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} \[ \frac {e^2 x (3 c d-b e)}{c^2}-\frac {(c d-b e)^3 \log (b+c x)}{b c^3}+\frac {d^3 \log (x)}{b}+\frac {e^3 x^2}{2 c} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{b x+c x^2} \, dx &=\int \left (\frac {e^2 (3 c d-b e)}{c^2}+\frac {d^3}{b x}+\frac {e^3 x}{c}+\frac {(-c d+b e)^3}{b c^2 (b+c x)}\right ) \, dx\\ &=\frac {e^2 (3 c d-b e) x}{c^2}+\frac {e^3 x^2}{2 c}+\frac {d^3 \log (x)}{b}-\frac {(c d-b e)^3 \log (b+c x)}{b c^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 59, normalized size = 0.92 \[ \frac {b c e^2 x (-2 b e+6 c d+c e x)-2 (c d-b e)^3 \log (b+c x)+2 c^3 d^3 \log (x)}{2 b c^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 97, normalized size = 1.52 \[ \frac {b c^{2} e^{3} x^{2} + 2 \, c^{3} d^{3} \log \relax (x) + 2 \, {\left (3 \, b c^{2} d e^{2} - b^{2} c e^{3}\right )} x - 2 \, {\left (c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \log \left (c x + b\right )}{2 \, b c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 87, normalized size = 1.36 \[ \frac {d^{3} \log \left ({\left | x \right |}\right )}{b} + \frac {c x^{2} e^{3} + 6 \, c d x e^{2} - 2 \, b x e^{3}}{2 \, c^{2}} - \frac {{\left (c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \log \left ({\left | c x + b \right |}\right )}{b c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 103, normalized size = 1.61 \[ \frac {e^{3} x^{2}}{2 c}+\frac {b^{2} e^{3} \ln \left (c x +b \right )}{c^{3}}-\frac {3 b d \,e^{2} \ln \left (c x +b \right )}{c^{2}}-\frac {b \,e^{3} x}{c^{2}}+\frac {d^{3} \ln \relax (x )}{b}-\frac {d^{3} \ln \left (c x +b \right )}{b}+\frac {3 d^{2} e \ln \left (c x +b \right )}{c}+\frac {3 d \,e^{2} x}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 91, normalized size = 1.42 \[ \frac {d^{3} \log \relax (x)}{b} + \frac {c e^{3} x^{2} + 2 \, {\left (3 \, c d e^{2} - b e^{3}\right )} x}{2 \, c^{2}} - \frac {{\left (c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \log \left (c x + b\right )}{b c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 65, normalized size = 1.02 \[ \frac {e^3\,x^2}{2\,c}-x\,\left (\frac {b\,e^3}{c^2}-\frac {3\,d\,e^2}{c}\right )+\frac {d^3\,\ln \relax (x)}{b}+\frac {\ln \left (b+c\,x\right )\,{\left (b\,e-c\,d\right )}^3}{b\,c^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.28, size = 112, normalized size = 1.75 \[ x \left (- \frac {b e^{3}}{c^{2}} + \frac {3 d e^{2}}{c}\right ) + \frac {e^{3} x^{2}}{2 c} + \frac {d^{3} \log {\relax (x )}}{b} + \frac {\left (b e - c d\right )^{3} \log {\left (x + \frac {- b c^{2} d^{3} + \frac {b \left (b e - c d\right )^{3}}{c}}{b^{3} e^{3} - 3 b^{2} c d e^{2} + 3 b c^{2} d^{2} e - 2 c^{3} d^{3}} \right )}}{b c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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