Optimal. Leaf size=555 \[ -\frac {45 b^3 f m n^3}{8 e x}-\frac {3 b^3 f^2 m n^3 \log (x)}{8 e^2}-\frac {21 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right )}{4 e x}+\frac {3 b^2 f^2 m n^2 \log \left (1+\frac {e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e^2}-\frac {9 b f m n \left (a+b \log \left (c x^n\right )\right )^2}{4 e x}+\frac {3 b f^2 m n \log \left (1+\frac {e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^3}{2 e x}+\frac {f^2 m \log \left (1+\frac {e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2}+\frac {3 b^3 f^2 m n^3 \log (e+f x)}{8 e^2}-\frac {3 b^3 n^3 \log \left (d (e+f x)^m\right )}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{2 x^2}-\frac {3 b^3 f^2 m n^3 \text {Li}_2\left (-\frac {e}{f x}\right )}{4 e^2}-\frac {3 b^2 f^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e}{f x}\right )}{2 e^2}-\frac {3 b f^2 m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {e}{f x}\right )}{2 e^2}-\frac {3 b^3 f^2 m n^3 \text {Li}_3\left (-\frac {e}{f x}\right )}{2 e^2}-\frac {3 b^2 f^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {e}{f x}\right )}{e^2}-\frac {3 b^3 f^2 m n^3 \text {Li}_4\left (-\frac {e}{f x}\right )}{e^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.59, antiderivative size = 555, normalized size of antiderivative = 1.00, number of steps
used = 22, number of rules used = 10, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2342, 2341,
2425, 46, 2380, 2379, 2438, 2421, 6724, 2430} \begin {gather*} -\frac {3 b^2 f^2 m n^2 \text {PolyLog}\left (2,-\frac {e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac {3 b^2 f^2 m n^2 \text {PolyLog}\left (3,-\frac {e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {3 b f^2 m n \text {PolyLog}\left (2,-\frac {e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}-\frac {3 b^3 f^2 m n^3 \text {PolyLog}\left (2,-\frac {e}{f x}\right )}{4 e^2}-\frac {3 b^3 f^2 m n^3 \text {PolyLog}\left (3,-\frac {e}{f x}\right )}{2 e^2}-\frac {3 b^3 f^2 m n^3 \text {PolyLog}\left (4,-\frac {e}{f x}\right )}{e^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{4 x^2}+\frac {3 b^2 f^2 m n^2 \log \left (\frac {e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e^2}-\frac {21 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right )}{4 e x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{2 x^2}+\frac {3 b f^2 m n \log \left (\frac {e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2}+\frac {f^2 m \log \left (\frac {e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2}-\frac {9 b f m n \left (a+b \log \left (c x^n\right )\right )^2}{4 e x}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^3}{2 e x}-\frac {3 b^3 n^3 \log \left (d (e+f x)^m\right )}{8 x^2}-\frac {3 b^3 f^2 m n^3 \log (x)}{8 e^2}+\frac {3 b^3 f^2 m n^3 \log (e+f x)}{8 e^2}-\frac {45 b^3 f m n^3}{8 e x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 46
Rule 2341
Rule 2342
Rule 2379
Rule 2380
Rule 2421
Rule 2425
Rule 2430
Rule 2438
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x^3} \, dx &=-\frac {3 b^3 n^3 \log \left (d (e+f x)^m\right )}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{2 x^2}-(f m) \int \left (-\frac {3 b^3 n^3}{8 x^2 (e+f x)}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^2 (e+f x)}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2 (e+f x)}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 x^2 (e+f x)}\right ) \, dx\\ &=-\frac {3 b^3 n^3 \log \left (d (e+f x)^m\right )}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{2 x^2}+\frac {1}{2} (f m) \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x^2 (e+f x)} \, dx+\frac {1}{4} (3 b f m n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (e+f x)} \, dx+\frac {1}{4} \left (3 b^2 f m n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x^2 (e+f x)} \, dx+\frac {1}{8} \left (3 b^3 f m n^3\right ) \int \frac {1}{x^2 (e+f x)} \, dx\\ &=-\frac {3 b^3 n^3 \log \left (d (e+f x)^m\right )}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{2 x^2}+\frac {1}{2} (f m) \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^3}{e x^2}-\frac {f \left (a+b \log \left (c x^n\right )\right )^3}{e^2 x}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^3}{e^2 (e+f x)}\right ) \, dx+\frac {1}{4} (3 b f m n) \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^2}{e x^2}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{e^2 x}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (e+f x)}\right ) \, dx+\frac {1}{4} \left (3 b^2 f m n^2\right ) \int \left (\frac {a+b \log \left (c x^n\right )}{e x^2}-\frac {f \left (a+b \log \left (c x^n\right )\right )}{e^2 x}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )}{e^2 (e+f x)}\right ) \, dx+\frac {1}{8} \left (3 b^3 f m n^3\right ) \int \left (\frac {1}{e x^2}-\frac {f}{e^2 x}+\frac {f^2}{e^2 (e+f x)}\right ) \, dx\\ &=-\frac {3 b^3 f m n^3}{8 e x}-\frac {3 b^3 f^2 m n^3 \log (x)}{8 e^2}+\frac {3 b^3 f^2 m n^3 \log (e+f x)}{8 e^2}-\frac {3 b^3 n^3 \log \left (d (e+f x)^m\right )}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{2 x^2}+\frac {(f m) \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx}{2 e}-\frac {\left (f^2 m\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx}{2 e^2}+\frac {\left (f^3 m\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{e+f x} \, dx}{2 e^2}+\frac {(3 b f m n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx}{4 e}-\frac {\left (3 b f^2 m n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{4 e^2}+\frac {\left (3 b f^3 m n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{e+f x} \, dx}{4 e^2}+\frac {\left (3 b^2 f m n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{4 e}-\frac {\left (3 b^2 f^2 m n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{4 e^2}+\frac {\left (3 b^2 f^3 m n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{e+f x} \, dx}{4 e^2}\\ &=-\frac {9 b^3 f m n^3}{8 e x}-\frac {3 b^3 f^2 m n^3 \log (x)}{8 e^2}-\frac {3 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right )}{4 e x}-\frac {3 b f^2 m n \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}-\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )^2}{4 e x}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^3}{2 e x}+\frac {3 b^3 f^2 m n^3 \log (e+f x)}{8 e^2}-\frac {3 b^3 n^3 \log \left (d (e+f x)^m\right )}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{2 x^2}+\frac {3 b^2 f^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{4 e^2}+\frac {3 b f^2 m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{4 e^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x}{e}\right )}{2 e^2}-\frac {\left (3 f^2 m\right ) \text {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{4 e^2}-\frac {\left (f^2 m\right ) \text {Subst}\left (\int x^3 \, dx,x,a+b \log \left (c x^n\right )\right )}{2 b e^2 n}+\frac {(3 b f m n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx}{2 e}-\frac {\left (3 b f^2 m n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{x} \, dx}{2 e^2}+\frac {\left (3 b^2 f m n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{2 e}-\frac {\left (3 b^2 f^2 m n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{x} \, dx}{2 e^2}-\frac {\left (3 b^3 f^2 m n^3\right ) \int \frac {\log \left (1+\frac {f x}{e}\right )}{x} \, dx}{4 e^2}\\ &=-\frac {21 b^3 f m n^3}{8 e x}-\frac {3 b^3 f^2 m n^3 \log (x)}{8 e^2}-\frac {9 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right )}{4 e x}-\frac {3 b f^2 m n \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}-\frac {9 b f m n \left (a+b \log \left (c x^n\right )\right )^2}{4 e x}-\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^3}{4 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^3}{2 e x}-\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^4}{8 b e^2 n}+\frac {3 b^3 f^2 m n^3 \log (e+f x)}{8 e^2}-\frac {3 b^3 n^3 \log \left (d (e+f x)^m\right )}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{2 x^2}+\frac {3 b^2 f^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{4 e^2}+\frac {3 b f^2 m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{4 e^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x}{e}\right )}{2 e^2}+\frac {3 b^3 f^2 m n^3 \text {Li}_2\left (-\frac {f x}{e}\right )}{4 e^2}+\frac {3 b^2 f^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )}{2 e^2}+\frac {3 b f^2 m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{2 e^2}+\frac {\left (3 b^2 f m n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{e}-\frac {\left (3 b^2 f^2 m n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )}{x} \, dx}{e^2}-\frac {\left (3 b^3 f^2 m n^3\right ) \int \frac {\text {Li}_2\left (-\frac {f x}{e}\right )}{x} \, dx}{2 e^2}\\ &=-\frac {45 b^3 f m n^3}{8 e x}-\frac {3 b^3 f^2 m n^3 \log (x)}{8 e^2}-\frac {21 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right )}{4 e x}-\frac {3 b f^2 m n \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}-\frac {9 b f m n \left (a+b \log \left (c x^n\right )\right )^2}{4 e x}-\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^3}{4 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^3}{2 e x}-\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^4}{8 b e^2 n}+\frac {3 b^3 f^2 m n^3 \log (e+f x)}{8 e^2}-\frac {3 b^3 n^3 \log \left (d (e+f x)^m\right )}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{2 x^2}+\frac {3 b^2 f^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{4 e^2}+\frac {3 b f^2 m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{4 e^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x}{e}\right )}{2 e^2}+\frac {3 b^3 f^2 m n^3 \text {Li}_2\left (-\frac {f x}{e}\right )}{4 e^2}+\frac {3 b^2 f^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )}{2 e^2}+\frac {3 b f^2 m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{2 e^2}-\frac {3 b^3 f^2 m n^3 \text {Li}_3\left (-\frac {f x}{e}\right )}{2 e^2}-\frac {3 b^2 f^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f x}{e}\right )}{e^2}+\frac {\left (3 b^3 f^2 m n^3\right ) \int \frac {\text {Li}_3\left (-\frac {f x}{e}\right )}{x} \, dx}{e^2}\\ &=-\frac {45 b^3 f m n^3}{8 e x}-\frac {3 b^3 f^2 m n^3 \log (x)}{8 e^2}-\frac {21 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right )}{4 e x}-\frac {3 b f^2 m n \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}-\frac {9 b f m n \left (a+b \log \left (c x^n\right )\right )^2}{4 e x}-\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^3}{4 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^3}{2 e x}-\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^4}{8 b e^2 n}+\frac {3 b^3 f^2 m n^3 \log (e+f x)}{8 e^2}-\frac {3 b^3 n^3 \log \left (d (e+f x)^m\right )}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{2 x^2}+\frac {3 b^2 f^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{4 e^2}+\frac {3 b f^2 m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{4 e^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x}{e}\right )}{2 e^2}+\frac {3 b^3 f^2 m n^3 \text {Li}_2\left (-\frac {f x}{e}\right )}{4 e^2}+\frac {3 b^2 f^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )}{2 e^2}+\frac {3 b f^2 m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{2 e^2}-\frac {3 b^3 f^2 m n^3 \text {Li}_3\left (-\frac {f x}{e}\right )}{2 e^2}-\frac {3 b^2 f^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f x}{e}\right )}{e^2}+\frac {3 b^3 f^2 m n^3 \text {Li}_4\left (-\frac {f x}{e}\right )}{e^2}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1736\) vs. \(2(555)=1110\).
time = 0.47, size = 1736, normalized size = 3.13 \begin {gather*} -\frac {4 a^3 e f m x+18 a^2 b e f m n x+42 a b^2 e f m n^2 x+45 b^3 e f m n^3 x+4 a^3 f^2 m x^2 \log (x)+6 a^2 b f^2 m n x^2 \log (x)+6 a b^2 f^2 m n^2 x^2 \log (x)+3 b^3 f^2 m n^3 x^2 \log (x)-6 a^2 b f^2 m n x^2 \log ^2(x)-6 a b^2 f^2 m n^2 x^2 \log ^2(x)-3 b^3 f^2 m n^3 x^2 \log ^2(x)+4 a b^2 f^2 m n^2 x^2 \log ^3(x)+2 b^3 f^2 m n^3 x^2 \log ^3(x)-b^3 f^2 m n^3 x^2 \log ^4(x)+12 a^2 b e f m x \log \left (c x^n\right )+36 a b^2 e f m n x \log \left (c x^n\right )+42 b^3 e f m n^2 x \log \left (c x^n\right )+12 a^2 b f^2 m x^2 \log (x) \log \left (c x^n\right )+12 a b^2 f^2 m n x^2 \log (x) \log \left (c x^n\right )+6 b^3 f^2 m n^2 x^2 \log (x) \log \left (c x^n\right )-12 a b^2 f^2 m n x^2 \log ^2(x) \log \left (c x^n\right )-6 b^3 f^2 m n^2 x^2 \log ^2(x) \log \left (c x^n\right )+4 b^3 f^2 m n^2 x^2 \log ^3(x) \log \left (c x^n\right )+12 a b^2 e f m x \log ^2\left (c x^n\right )+18 b^3 e f m n x \log ^2\left (c x^n\right )+12 a b^2 f^2 m x^2 \log (x) \log ^2\left (c x^n\right )+6 b^3 f^2 m n x^2 \log (x) \log ^2\left (c x^n\right )-6 b^3 f^2 m n x^2 \log ^2(x) \log ^2\left (c x^n\right )+4 b^3 e f m x \log ^3\left (c x^n\right )+4 b^3 f^2 m x^2 \log (x) \log ^3\left (c x^n\right )-4 a^3 f^2 m x^2 \log (e+f x)-6 a^2 b f^2 m n x^2 \log (e+f x)-6 a b^2 f^2 m n^2 x^2 \log (e+f x)-3 b^3 f^2 m n^3 x^2 \log (e+f x)+12 a^2 b f^2 m n x^2 \log (x) \log (e+f x)+12 a b^2 f^2 m n^2 x^2 \log (x) \log (e+f x)+6 b^3 f^2 m n^3 x^2 \log (x) \log (e+f x)-12 a b^2 f^2 m n^2 x^2 \log ^2(x) \log (e+f x)-6 b^3 f^2 m n^3 x^2 \log ^2(x) \log (e+f x)+4 b^3 f^2 m n^3 x^2 \log ^3(x) \log (e+f x)-12 a^2 b f^2 m x^2 \log \left (c x^n\right ) \log (e+f x)-12 a b^2 f^2 m n x^2 \log \left (c x^n\right ) \log (e+f x)-6 b^3 f^2 m n^2 x^2 \log \left (c x^n\right ) \log (e+f x)+24 a b^2 f^2 m n x^2 \log (x) \log \left (c x^n\right ) \log (e+f x)+12 b^3 f^2 m n^2 x^2 \log (x) \log \left (c x^n\right ) \log (e+f x)-12 b^3 f^2 m n^2 x^2 \log ^2(x) \log \left (c x^n\right ) \log (e+f x)-12 a b^2 f^2 m x^2 \log ^2\left (c x^n\right ) \log (e+f x)-6 b^3 f^2 m n x^2 \log ^2\left (c x^n\right ) \log (e+f x)+12 b^3 f^2 m n x^2 \log (x) \log ^2\left (c x^n\right ) \log (e+f x)-4 b^3 f^2 m x^2 \log ^3\left (c x^n\right ) \log (e+f x)+4 a^3 e^2 \log \left (d (e+f x)^m\right )+6 a^2 b e^2 n \log \left (d (e+f x)^m\right )+6 a b^2 e^2 n^2 \log \left (d (e+f x)^m\right )+3 b^3 e^2 n^3 \log \left (d (e+f x)^m\right )+12 a^2 b e^2 \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+12 a b^2 e^2 n \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+6 b^3 e^2 n^2 \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+12 a b^2 e^2 \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )+6 b^3 e^2 n \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )+4 b^3 e^2 \log ^3\left (c x^n\right ) \log \left (d (e+f x)^m\right )-12 a^2 b f^2 m n x^2 \log (x) \log \left (1+\frac {f x}{e}\right )-12 a b^2 f^2 m n^2 x^2 \log (x) \log \left (1+\frac {f x}{e}\right )-6 b^3 f^2 m n^3 x^2 \log (x) \log \left (1+\frac {f x}{e}\right )+12 a b^2 f^2 m n^2 x^2 \log ^2(x) \log \left (1+\frac {f x}{e}\right )+6 b^3 f^2 m n^3 x^2 \log ^2(x) \log \left (1+\frac {f x}{e}\right )-4 b^3 f^2 m n^3 x^2 \log ^3(x) \log \left (1+\frac {f x}{e}\right )-24 a b^2 f^2 m n x^2 \log (x) \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )-12 b^3 f^2 m n^2 x^2 \log (x) \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )+12 b^3 f^2 m n^2 x^2 \log ^2(x) \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )-12 b^3 f^2 m n x^2 \log (x) \log ^2\left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )-6 b f^2 m n x^2 \left (2 a^2+2 a b n+b^2 n^2+2 b (2 a+b n) \log \left (c x^n\right )+2 b^2 \log ^2\left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )+12 b^2 f^2 m n^2 x^2 \left (2 a+b n+2 b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f x}{e}\right )-24 b^3 f^2 m n^3 x^2 \text {Li}_4\left (-\frac {f x}{e}\right )}{8 e^2 x^2} \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 = 1.27, size = 46399, normalized size = 83.60
method | result | size |
risch | \(\text {Expression too large to display}\) | \(46399\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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 = -0.00 \begin {gather*} \int \frac {\ln \left (d\,{\left (e+f\,x\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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