Optimal. Leaf size=156 \[ -\frac {1}{4} \left (\frac {2 \log \left (f x^m\right )}{x^2}+\frac {m}{x^2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b e^2 n \log \left (\frac {d}{e x}+1\right ) \log \left (f x^m\right )}{2 d^2}-\frac {b e^2 m n \text {Li}_2\left (-\frac {d}{e x}\right )}{2 d^2}-\frac {b e^2 m n \log (x)}{4 d^2}+\frac {b e^2 m n \log (d+e x)}{4 d^2}-\frac {b e n \log \left (f x^m\right )}{2 d x}-\frac {3 b e m n}{4 d x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.15, antiderivative size = 175, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2426, 44, 2351, 2304, 2301, 2317, 2391} \[ \frac {b e^2 m n \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{2 d^2}-\frac {1}{4} \left (\frac {2 \log \left (f x^m\right )}{x^2}+\frac {m}{x^2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {b e^2 n \log ^2\left (f x^m\right )}{4 d^2 m}+\frac {b e^2 n \log \left (\frac {e x}{d}+1\right ) \log \left (f x^m\right )}{2 d^2}-\frac {b e^2 m n \log (x)}{4 d^2}+\frac {b e^2 m n \log (d+e x)}{4 d^2}-\frac {b e n \log \left (f x^m\right )}{2 d x}-\frac {3 b e m n}{4 d x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 2301
Rule 2304
Rule 2317
Rule 2351
Rule 2391
Rule 2426
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^3} \, dx &=-\frac {1}{4} \left (\frac {m}{x^2}+\frac {2 \log \left (f x^m\right )}{x^2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {1}{2} (b e n) \int \frac {\log \left (f x^m\right )}{x^2 (d+e x)} \, dx+\frac {1}{4} (b e m n) \int \frac {1}{x^2 (d+e x)} \, dx\\ &=-\frac {1}{4} \left (\frac {m}{x^2}+\frac {2 \log \left (f x^m\right )}{x^2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {1}{2} (b e n) \int \left (\frac {\log \left (f x^m\right )}{d x^2}-\frac {e \log \left (f x^m\right )}{d^2 x}+\frac {e^2 \log \left (f x^m\right )}{d^2 (d+e x)}\right ) \, dx+\frac {1}{4} (b e m n) \int \left (\frac {1}{d x^2}-\frac {e}{d^2 x}+\frac {e^2}{d^2 (d+e x)}\right ) \, dx\\ &=-\frac {b e m n}{4 d x}-\frac {b e^2 m n \log (x)}{4 d^2}+\frac {b e^2 m n \log (d+e x)}{4 d^2}-\frac {1}{4} \left (\frac {m}{x^2}+\frac {2 \log \left (f x^m\right )}{x^2}\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^2} \, dx}{2 d}-\frac {\left (b e^2 n\right ) \int \frac {\log \left (f x^m\right )}{x} \, dx}{2 d^2}+\frac {\left (b e^3 n\right ) \int \frac {\log \left (f x^m\right )}{d+e x} \, dx}{2 d^2}\\ &=-\frac {3 b e m n}{4 d x}-\frac {b e^2 m n \log (x)}{4 d^2}-\frac {b e n \log \left (f x^m\right )}{2 d x}-\frac {b e^2 n \log ^2\left (f x^m\right )}{4 d^2 m}+\frac {b e^2 m n \log (d+e x)}{4 d^2}-\frac {1}{4} \left (\frac {m}{x^2}+\frac {2 \log \left (f x^m\right )}{x^2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b e^2 n \log \left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )}{2 d^2}-\frac {\left (b e^2 m n\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{2 d^2}\\ &=-\frac {3 b e m n}{4 d x}-\frac {b e^2 m n \log (x)}{4 d^2}-\frac {b e n \log \left (f x^m\right )}{2 d x}-\frac {b e^2 n \log ^2\left (f x^m\right )}{4 d^2 m}+\frac {b e^2 m n \log (d+e x)}{4 d^2}-\frac {1}{4} \left (\frac {m}{x^2}+\frac {2 \log \left (f x^m\right )}{x^2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b e^2 n \log \left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )}{2 d^2}+\frac {b e^2 m n \text {Li}_2\left (-\frac {e x}{d}\right )}{2 d^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.13, size = 204, normalized size = 1.31 \[ -\frac {2 a d^2 \log \left (f x^m\right )+a d^2 m+2 b d^2 \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )+b d^2 m \log \left (c (d+e x)^n\right )-2 b e^2 n x^2 \log (d+e x) \log \left (f x^m\right )+b e^2 n x^2 \log (x) \left (2 m \log (d+e x)-2 m \log \left (\frac {e x}{d}+1\right )+2 \log \left (f x^m\right )+m\right )-2 b e^2 m n x^2 \text {Li}_2\left (-\frac {e x}{d}\right )-b e^2 m n x^2 \log (d+e x)+2 b d e n x \log \left (f x^m\right )+3 b d e m n x-b e^2 m n x^2 \log ^2(x)}{4 d^2 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) \log \left (f x^{m}\right ) + a \log \left (f x^{m}\right )}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} \log \left (f x^{m}\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.77, size = 2051, normalized size = 13.15 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.68, size = 198, normalized size = 1.27 \[ \frac {1}{4} \, {\left (\frac {2 \, {\left (\log \left (\frac {e x}{d} + 1\right ) \log \relax (x) + {\rm Li}_2\left (-\frac {e x}{d}\right )\right )} b e^{2} n}{d^{2}} + \frac {b e^{2} n \log \left (e x + d\right )}{d^{2}} - \frac {2 \, b e^{2} n x^{2} \log \left (e x + d\right ) \log \relax (x) - b e^{2} n x^{2} \log \relax (x)^{2} + b e^{2} n x^{2} \log \relax (x) + 3 \, b d e n x + b d^{2} \log \left ({\left (e x + d\right )}^{n}\right ) + b d^{2} \log \relax (c) + a d^{2}}{d^{2} x^{2}}\right )} m + \frac {1}{2} \, {\left (b e n {\left (\frac {e \log \left (e x + d\right )}{d^{2}} - \frac {e \log \relax (x)}{d^{2}} - \frac {1}{d x}\right )} - \frac {b \log \left ({\left (e x + d\right )}^{n} c\right )}{x^{2}} - \frac {a}{x^{2}}\right )} \log \left (f x^{m}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\ln \left (f\,x^m\right )\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________