Optimal. Leaf size=150 \[ \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{3 b n}+\frac {2 b n r \text {Li}_3\left (-\frac {f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{m^2}-\frac {r \text {Li}_2\left (-\frac {f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{m}-\frac {r \log \left (\frac {f x^m}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b n}-\frac {2 b^2 n^2 r \text {Li}_4\left (-\frac {f x^m}{e}\right )}{m^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.25, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2375, 2337, 2374, 2383, 6589} \[ \frac {2 b n r \text {PolyLog}\left (3,-\frac {f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{m^2}-\frac {r \text {PolyLog}\left (2,-\frac {f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{m}-\frac {2 b^2 n^2 r \text {PolyLog}\left (4,-\frac {f x^m}{e}\right )}{m^3}+\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{3 b n}-\frac {r \log \left (\frac {f x^m}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2337
Rule 2374
Rule 2375
Rule 2383
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^m\right )^r\right )}{x} \, dx &=\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{3 b n}-\frac {(f m r) \int \frac {x^{-1+m} \left (a+b \log \left (c x^n\right )\right )^3}{e+f x^m} \, dx}{3 b n}\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{3 b n}-\frac {r \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x^m}{e}\right )}{3 b n}+r \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^m}{e}\right )}{x} \, dx\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{3 b n}-\frac {r \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x^m}{e}\right )}{3 b n}-\frac {r \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f x^m}{e}\right )}{m}+\frac {(2 b n r) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x^m}{e}\right )}{x} \, dx}{m}\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{3 b n}-\frac {r \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x^m}{e}\right )}{3 b n}-\frac {r \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f x^m}{e}\right )}{m}+\frac {2 b n r \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f x^m}{e}\right )}{m^2}-\frac {\left (2 b^2 n^2 r\right ) \int \frac {\text {Li}_3\left (-\frac {f x^m}{e}\right )}{x} \, dx}{m^2}\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{3 b n}-\frac {r \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x^m}{e}\right )}{3 b n}-\frac {r \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f x^m}{e}\right )}{m}+\frac {2 b n r \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f x^m}{e}\right )}{m^2}-\frac {2 b^2 n^2 r \text {Li}_4\left (-\frac {f x^m}{e}\right )}{m^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.38, size = 741, normalized size = 4.94 \[ a^2 \log (x) \log \left (d \left (e+f x^m\right )^r\right )-a^2 r \log (x) \log \left (e+f x^m\right )+\frac {a^2 r \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{m}+2 a b \log (x) \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^r\right )+\frac {b n r \log (x) \text {Li}_2\left (-\frac {e x^{-m}}{f}\right ) \left (2 \left (a+b \log \left (c x^n\right )\right )-b n \log (x)\right )}{m}+\frac {r \text {Li}_2\left (\frac {f x^m}{e}+1\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^2}{m}-2 a b r \log (x) \log \left (c x^n\right ) \log \left (e+f x^m\right )+\frac {2 a b r \log \left (c x^n\right ) \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{m}-a b n \log ^2(x) \log \left (d \left (e+f x^m\right )^r\right )+\frac {2 a b n r \text {Li}_3\left (-\frac {e x^{-m}}{f}\right )}{m^2}-a b n r \log ^2(x) \log \left (\frac {e x^{-m}}{f}+1\right )+2 a b n r \log ^2(x) \log \left (e+f x^m\right )-\frac {2 a b n r \log (x) \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{m}-\frac {1}{3} a b m n r \log ^3(x)-b^2 n \log ^2(x) \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^r\right )+b^2 \log (x) \log ^2\left (c x^n\right ) \log \left (d \left (e+f x^m\right )^r\right )+\frac {2 b^2 n r \log \left (c x^n\right ) \text {Li}_3\left (-\frac {e x^{-m}}{f}\right )}{m^2}-b^2 n r \log ^2(x) \log \left (c x^n\right ) \log \left (\frac {e x^{-m}}{f}+1\right )+2 b^2 n r \log ^2(x) \log \left (c x^n\right ) \log \left (e+f x^m\right )-b^2 r \log (x) \log ^2\left (c x^n\right ) \log \left (e+f x^m\right )+\frac {b^2 r \log ^2\left (c x^n\right ) \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{m}-\frac {2 b^2 n r \log (x) \log \left (c x^n\right ) \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{m}-\frac {1}{3} b^2 m n r \log ^3(x) \log \left (c x^n\right )+\frac {1}{3} b^2 n^2 \log ^3(x) \log \left (d \left (e+f x^m\right )^r\right )+\frac {2 b^2 n^2 r \text {Li}_4\left (-\frac {e x^{-m}}{f}\right )}{m^3}+\frac {2}{3} b^2 n^2 r \log ^3(x) \log \left (\frac {e x^{-m}}{f}+1\right )-b^2 n^2 r \log ^3(x) \log \left (e+f x^m\right )+\frac {b^2 n^2 r \log ^2(x) \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{m}+\frac {1}{4} b^2 m n^2 r \log ^4(x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [C] time = 0.77, size = 406, normalized size = 2.71 \[ \frac {b^{2} m^{3} n^{2} \log \relax (d) \log \relax (x)^{3} - 6 \, b^{2} n^{2} r {\rm polylog}\left (4, -\frac {f x^{m}}{e}\right ) + 3 \, {\left (b^{2} m^{3} n \log \relax (c) + a b m^{3} n\right )} \log \relax (d) \log \relax (x)^{2} + 3 \, {\left (b^{2} m^{3} \log \relax (c)^{2} + 2 \, a b m^{3} \log \relax (c) + a^{2} m^{3}\right )} \log \relax (d) \log \relax (x) - 3 \, {\left (b^{2} m^{2} n^{2} r \log \relax (x)^{2} + b^{2} m^{2} r \log \relax (c)^{2} + 2 \, a b m^{2} r \log \relax (c) + a^{2} m^{2} r + 2 \, {\left (b^{2} m^{2} n r \log \relax (c) + a b m^{2} n r\right )} \log \relax (x)\right )} {\rm Li}_2\left (-\frac {f x^{m} + e}{e} + 1\right ) + {\left (b^{2} m^{3} n^{2} r \log \relax (x)^{3} + 3 \, {\left (b^{2} m^{3} n r \log \relax (c) + a b m^{3} n r\right )} \log \relax (x)^{2} + 3 \, {\left (b^{2} m^{3} r \log \relax (c)^{2} + 2 \, a b m^{3} r \log \relax (c) + a^{2} m^{3} r\right )} \log \relax (x)\right )} \log \left (f x^{m} + e\right ) - {\left (b^{2} m^{3} n^{2} r \log \relax (x)^{3} + 3 \, {\left (b^{2} m^{3} n r \log \relax (c) + a b m^{3} n r\right )} \log \relax (x)^{2} + 3 \, {\left (b^{2} m^{3} r \log \relax (c)^{2} + 2 \, a b m^{3} r \log \relax (c) + a^{2} m^{3} r\right )} \log \relax (x)\right )} \log \left (\frac {f x^{m} + e}{e}\right ) + 6 \, {\left (b^{2} m n^{2} r \log \relax (x) + b^{2} m n r \log \relax (c) + a b m n r\right )} {\rm polylog}\left (3, -\frac {f x^{m}}{e}\right )}{3 \, m^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{m} + e\right )}^{r} d\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{2} \ln \left (d \left (f \,x^{m}+e \right )^{r}\right )}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, {\left (b^{2} n^{2} \log \relax (x)^{3} + 3 \, b^{2} \log \relax (x) \log \left (x^{n}\right )^{2} - 3 \, {\left (b^{2} n \log \relax (c) + a b n\right )} \log \relax (x)^{2} - 3 \, {\left (b^{2} n \log \relax (x)^{2} - 2 \, {\left (b^{2} \log \relax (c) + a b\right )} \log \relax (x)\right )} \log \left (x^{n}\right ) + 3 \, {\left (b^{2} \log \relax (c)^{2} + 2 \, a b \log \relax (c) + a^{2}\right )} \log \relax (x)\right )} \log \left ({\left (f x^{m} + e\right )}^{r}\right ) - \int -\frac {3 \, b^{2} e \log \relax (c)^{2} \log \relax (d) + 6 \, a b e \log \relax (c) \log \relax (d) + 3 \, a^{2} e \log \relax (d) + 3 \, {\left (b^{2} e \log \relax (d) - {\left (b^{2} f m r \log \relax (x) - b^{2} f \log \relax (d)\right )} x^{m}\right )} \log \left (x^{n}\right )^{2} - {\left (b^{2} f m n^{2} r \log \relax (x)^{3} - 3 \, b^{2} f \log \relax (c)^{2} \log \relax (d) - 6 \, a b f \log \relax (c) \log \relax (d) - 3 \, a^{2} f \log \relax (d) - 3 \, {\left (b^{2} f m n r \log \relax (c) + a b f m n r\right )} \log \relax (x)^{2} + 3 \, {\left (b^{2} f m r \log \relax (c)^{2} + 2 \, a b f m r \log \relax (c) + a^{2} f m r\right )} \log \relax (x)\right )} x^{m} + 3 \, {\left (2 \, b^{2} e \log \relax (c) \log \relax (d) + 2 \, a b e \log \relax (d) + {\left (b^{2} f m n r \log \relax (x)^{2} + 2 \, b^{2} f \log \relax (c) \log \relax (d) + 2 \, a b f \log \relax (d) - 2 \, {\left (b^{2} f m r \log \relax (c) + a b f m r\right )} \log \relax (x)\right )} x^{m}\right )} \log \left (x^{n}\right )}{3 \, {\left (f x x^{m} + e x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\ln \left (d\,{\left (e+f\,x^m\right )}^r\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x} \,d x \]
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]
________________________________________________________________________________________