Optimal. Leaf size=102 \[ \frac {b e m n \log (x)}{d}-\frac {b e n \log \left (1+\frac {d}{e x}\right ) \log \left (f x^m\right )}{d}-\frac {b e m n \log (d+e x)}{d}-\left (\frac {m}{x}+\frac {\log \left (f x^m\right )}{x}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b e m n \text {Li}_2\left (-\frac {d}{e x}\right )}{d} \]
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Rubi [A]
time = 0.05, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2473, 2379,
2438, 36, 29, 31} \begin {gather*} \frac {b e m n \text {PolyLog}\left (2,-\frac {d}{e x}\right )}{d}-\left (\frac {\log \left (f x^m\right )}{x}+\frac {m}{x}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {b e n \log \left (\frac {d}{e x}+1\right ) \log \left (f x^m\right )}{d}+\frac {b e m n \log (x)}{d}-\frac {b e m n \log (d+e x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2379
Rule 2438
Rule 2473
Rubi steps
\begin {align*} \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x^2} \, dx &=-\left (\frac {m}{x}+\frac {\log \left (f x^m\right )}{x}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+(b e n) \int \frac {\log \left (f x^m\right )}{x (d+e x)} \, dx+(b e m n) \int \frac {1}{x (d+e x)} \, dx\\ &=-\left (\frac {m}{x}+\frac {\log \left (f x^m\right )}{x}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {(b e n) \int \frac {\log \left (f x^m\right )}{x} \, dx}{d}-\frac {\left (b e^2 n\right ) \int \frac {\log \left (f x^m\right )}{d+e x} \, dx}{d}+\frac {(b e m n) \int \frac {1}{x} \, dx}{d}-\frac {\left (b e^2 m n\right ) \int \frac {1}{d+e x} \, dx}{d}\\ &=\frac {b e m n \log (x)}{d}+\frac {b e n \log ^2\left (f x^m\right )}{2 d m}-\frac {b e m n \log (d+e x)}{d}-\left (\frac {m}{x}+\frac {\log \left (f x^m\right )}{x}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {b e n \log \left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )}{d}+\frac {(b e m n) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d}\\ &=\frac {b e m n \log (x)}{d}+\frac {b e n \log ^2\left (f x^m\right )}{2 d m}-\frac {b e m n \log (d+e x)}{d}-\left (\frac {m}{x}+\frac {\log \left (f x^m\right )}{x}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {b e n \log \left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )}{d}-\frac {b e m n \text {Li}_2\left (-\frac {e x}{d}\right )}{d}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 111, normalized size = 1.09 \begin {gather*} -\frac {b e m n x \log ^2(x)+2 \left (m+\log \left (f x^m\right )\right ) \left (a d+b e n x \log (d+e x)+b d \log \left (c (d+e x)^n\right )\right )-2 b e n x \log (x) \left (m+\log \left (f x^m\right )+m \log (d+e x)-m \log \left (1+\frac {e x}{d}\right )\right )+2 b e m n x \text {Li}_2\left (-\frac {e x}{d}\right )}{2 d x} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.60, size = 1859, normalized size = 18.23
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1859\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 175, normalized size = 1.72 \begin {gather*} -\frac {1}{2} \, {\left (\frac {2 \, {\left (\log \left (x\right ) \log \left (\frac {x e}{d} + 1\right ) + {\rm Li}_2\left (-\frac {x e}{d}\right )\right )} b n e}{d} + \frac {2 \, b n e \log \left (x e + d\right )}{d} - \frac {2 \, b n x e \log \left (x e + d\right ) \log \left (x\right ) - b n x e \log \left (x\right )^{2} + 2 \, b n x e \log \left (x\right ) - 2 \, b d \log \left ({\left (x e + d\right )}^{n}\right ) - 2 \, b d \log \left (c\right ) - 2 \, a d}{d x}\right )} m - {\left (b n {\left (\frac {\log \left (x e + d\right )}{d} - \frac {\log \left (x\right )}{d}\right )} e + \frac {b \log \left ({\left (x e + d\right )}^{n} c\right )}{x} + \frac {a}{x}\right )} \log \left (f x^{m}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\ln \left (f\,x^m\right )\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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