Optimal. Leaf size=64 \[ \frac {e^2 (3 c d-b e) x}{c^2}+\frac {e^3 x^2}{2 c}+\frac {d^3 \log (x)}{b}-\frac {(c d-b e)^3 \log (b+c x)}{b c^3} \]
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Rubi [A]
time = 0.04, 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 = {712}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 712
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 = 0.02, size = 59, normalized size = 0.92 \begin {gather*} \frac {b c e^2 x (6 c d-2 b e+c e x)+2 c^3 d^3 \log (x)-2 (c d-b e)^3 \log (b+c x)}{2 b c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 85, normalized size = 1.33
method | result | size |
default | \(-\frac {e^{2} \left (-\frac {1}{2} c e \,x^{2}+b e x -3 c d x \right )}{c^{2}}+\frac {\left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right ) \ln \left (c x +b \right )}{c^{3} b}+\frac {d^{3} \ln \left (x \right )}{b}\) | \(85\) |
norman | \(\frac {e^{3} x^{2}}{2 c}-\frac {e^{2} \left (b e -3 c d \right ) x}{c^{2}}+\frac {d^{3} \ln \left (x \right )}{b}+\frac {\left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right ) \ln \left (c x +b \right )}{c^{3} b}\) | \(88\) |
risch | \(\frac {e^{3} x^{2}}{2 c}-\frac {e^{3} b x}{c^{2}}+\frac {3 d \,e^{2} x}{c}+\frac {b^{2} \ln \left (-c x -b \right ) e^{3}}{c^{3}}-\frac {3 b \ln \left (-c x -b \right ) d \,e^{2}}{c^{2}}+\frac {3 \ln \left (-c x -b \right ) d^{2} e}{c}-\frac {\ln \left (-c x -b \right ) d^{3}}{b}+\frac {d^{3} \ln \left (x \right )}{b}\) | \(115\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 87, normalized size = 1.36 \begin {gather*} \frac {d^{3} \log \left (x\right )}{b} + \frac {c x^{2} e^{3} + 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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.29, size = 91, normalized size = 1.42 \begin {gather*} \frac {2 \, c^{3} d^{3} \log \left (x\right ) + 6 \, b c^{2} d x e^{2} + {\left (b c^{2} x^{2} - 2 \, b^{2} c x\right )} e^{3} - 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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 112 vs.
\(2 (54) = 108\).
time = 0.78, size = 112, normalized size = 1.75 \begin {gather*} 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 {\left (x \right )}}{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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.54, size = 87, normalized size = 1.36 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \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 \left (x\right )}{b}+\frac {\ln \left (b+c\,x\right )\,{\left (b\,e-c\,d\right )}^3}{b\,c^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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