Optimal. Leaf size=411 \[ -\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}+\frac {6 b^2 f m n^2 \text {Li}_2\left (-\frac {e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac {6 b^2 f m n^2 \text {Li}_3\left (-\frac {e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {6 b^2 f m n^2 \log \left (\frac {e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x}+\frac {3 b f m n \text {Li}_2\left (-\frac {e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {3 b f m n \log \left (\frac {e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {f m \log \left (\frac {e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{e}-\frac {6 b^3 n^3 \log \left (d (e+f x)^m\right )}{x}+\frac {6 b^3 f m n^3 \text {Li}_2\left (-\frac {e}{f x}\right )}{e}+\frac {6 b^3 f m n^3 \text {Li}_3\left (-\frac {e}{f x}\right )}{e}+\frac {6 b^3 f m n^3 \text {Li}_4\left (-\frac {e}{f x}\right )}{e}+\frac {6 b^3 f m n^3 \log (x)}{e}-\frac {6 b^3 f m n^3 \log (e+f x)}{e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.70, antiderivative size = 459, normalized size of antiderivative = 1.12, number of steps used = 22, number of rules used = 15, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used = {2305, 2304, 2378, 36, 29, 31, 2344, 2301, 2317, 2391, 2302, 30, 2374, 6589, 2383} \[ -\frac {6 b^2 f m n^2 \text {PolyLog}\left (2,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac {6 b^2 f m n^2 \text {PolyLog}\left (3,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {3 b f m n \text {PolyLog}\left (2,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {6 b^3 f m n^3 \text {PolyLog}\left (2,-\frac {f x}{e}\right )}{e}+\frac {6 b^3 f m n^3 \text {PolyLog}\left (3,-\frac {f x}{e}\right )}{e}-\frac {6 b^3 f m n^3 \text {PolyLog}\left (4,-\frac {f x}{e}\right )}{e}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}-\frac {6 b^2 f m n^2 \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}+\frac {f m \left (a+b \log \left (c x^n\right )\right )^4}{4 b e n}+\frac {f m \left (a+b \log \left (c x^n\right )\right )^3}{e}-\frac {f m \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{e}+\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {3 b f m n \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {6 b^3 n^3 \log \left (d (e+f x)^m\right )}{x}+\frac {6 b^3 f m n^3 \log (x)}{e}-\frac {6 b^3 f m n^3 \log (e+f x)}{e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 29
Rule 30
Rule 31
Rule 36
Rule 2301
Rule 2302
Rule 2304
Rule 2305
Rule 2317
Rule 2344
Rule 2374
Rule 2378
Rule 2383
Rule 2391
Rule 6589
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^2} \, dx &=-\frac {6 b^3 n^3 \log \left (d (e+f x)^m\right )}{x}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x}-(f m) \int \left (-\frac {6 b^3 n^3}{x (e+f x)}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{x (e+f x)}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2}{x (e+f x)}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (e+f x)}\right ) \, dx\\ &=-\frac {6 b^3 n^3 \log \left (d (e+f x)^m\right )}{x}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x}+(f m) \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (e+f x)} \, dx+(3 b f m n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (e+f x)} \, dx+\left (6 b^2 f m n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x (e+f x)} \, dx+\left (6 b^3 f m n^3\right ) \int \frac {1}{x (e+f x)} \, dx\\ &=-\frac {6 b^3 n^3 \log \left (d (e+f x)^m\right )}{x}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x}+\frac {(f m) \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx}{e}-\frac {\left (f^2 m\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{e+f x} \, dx}{e}+\frac {(3 b f m n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{e}-\frac {\left (3 b f^2 m n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{e+f x} \, dx}{e}+\frac {\left (6 b^2 f m n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{e}-\frac {\left (6 b^2 f^2 m n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{e+f x} \, dx}{e}+\frac {\left (6 b^3 f m n^3\right ) \int \frac {1}{x} \, dx}{e}-\frac {\left (6 b^3 f^2 m n^3\right ) \int \frac {1}{e+f x} \, dx}{e}\\ &=\frac {6 b^3 f m n^3 \log (x)}{e}+\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {6 b^3 f m n^3 \log (e+f x)}{e}-\frac {6 b^3 n^3 \log \left (d (e+f x)^m\right )}{x}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x}-\frac {6 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{e}-\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{e}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x}{e}\right )}{e}+\frac {(3 f m) \operatorname {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{e}+\frac {(f m) \operatorname {Subst}\left (\int x^3 \, dx,x,a+b \log \left (c x^n\right )\right )}{b e n}+\frac {(3 b f m n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{x} \, dx}{e}+\frac {\left (6 b^2 f 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}{e}+\frac {\left (6 b^3 f m n^3\right ) \int \frac {\log \left (1+\frac {f x}{e}\right )}{x} \, dx}{e}\\ &=\frac {6 b^3 f m n^3 \log (x)}{e}+\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )^2}{e}+\frac {f m \left (a+b \log \left (c x^n\right )\right )^3}{e}+\frac {f m \left (a+b \log \left (c x^n\right )\right )^4}{4 b e n}-\frac {6 b^3 f m n^3 \log (e+f x)}{e}-\frac {6 b^3 n^3 \log \left (d (e+f x)^m\right )}{x}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x}-\frac {6 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{e}-\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{e}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x}{e}\right )}{e}-\frac {6 b^3 f m n^3 \text {Li}_2\left (-\frac {f x}{e}\right )}{e}-\frac {6 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )}{e}-\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{e}+\frac {\left (6 b^2 f 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}+\frac {\left (6 b^3 f m n^3\right ) \int \frac {\text {Li}_2\left (-\frac {f x}{e}\right )}{x} \, dx}{e}\\ &=\frac {6 b^3 f m n^3 \log (x)}{e}+\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )^2}{e}+\frac {f m \left (a+b \log \left (c x^n\right )\right )^3}{e}+\frac {f m \left (a+b \log \left (c x^n\right )\right )^4}{4 b e n}-\frac {6 b^3 f m n^3 \log (e+f x)}{e}-\frac {6 b^3 n^3 \log \left (d (e+f x)^m\right )}{x}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x}-\frac {6 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{e}-\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{e}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x}{e}\right )}{e}-\frac {6 b^3 f m n^3 \text {Li}_2\left (-\frac {f x}{e}\right )}{e}-\frac {6 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )}{e}-\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{e}+\frac {6 b^3 f m n^3 \text {Li}_3\left (-\frac {f x}{e}\right )}{e}+\frac {6 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f x}{e}\right )}{e}-\frac {\left (6 b^3 f m n^3\right ) \int \frac {\text {Li}_3\left (-\frac {f x}{e}\right )}{x} \, dx}{e}\\ &=\frac {6 b^3 f m n^3 \log (x)}{e}+\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )^2}{e}+\frac {f m \left (a+b \log \left (c x^n\right )\right )^3}{e}+\frac {f m \left (a+b \log \left (c x^n\right )\right )^4}{4 b e n}-\frac {6 b^3 f m n^3 \log (e+f x)}{e}-\frac {6 b^3 n^3 \log \left (d (e+f x)^m\right )}{x}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x}-\frac {6 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{e}-\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{e}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x}{e}\right )}{e}-\frac {6 b^3 f m n^3 \text {Li}_2\left (-\frac {f x}{e}\right )}{e}-\frac {6 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )}{e}-\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{e}+\frac {6 b^3 f m n^3 \text {Li}_3\left (-\frac {f x}{e}\right )}{e}+\frac {6 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f x}{e}\right )}{e}-\frac {6 b^3 f m n^3 \text {Li}_4\left (-\frac {f x}{e}\right )}{e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.69, size = 1347, normalized size = 3.28 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{3} \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (c x^{n}\right ) + a^{3}\right )} \log \left ({\left (f x + e\right )}^{m} d\right )}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x + e\right )}^{m} d\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 2.66, size = 42181, normalized size = 102.63 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (b^{3} f m x \log \left (f x + e\right ) - b^{3} f m x \log \relax (x) + b^{3} e \log \relax (d)\right )} \log \left (x^{n}\right )^{3} + {\left (b^{3} e \log \left (x^{n}\right )^{3} + 3 \, {\left (e n + e \log \relax (c)\right )} a^{2} b + 3 \, {\left (2 \, e n^{2} + 2 \, e n \log \relax (c) + e \log \relax (c)^{2}\right )} a b^{2} + {\left (6 \, e n^{3} + 6 \, e n^{2} \log \relax (c) + 3 \, e n \log \relax (c)^{2} + e \log \relax (c)^{3}\right )} b^{3} + a^{3} e + 3 \, {\left ({\left (e n + e \log \relax (c)\right )} b^{3} + a b^{2} e\right )} \log \left (x^{n}\right )^{2} + 3 \, {\left (2 \, {\left (e n + e \log \relax (c)\right )} a b^{2} + {\left (2 \, e n^{2} + 2 \, e n \log \relax (c) + e \log \relax (c)^{2}\right )} b^{3} + a^{2} b e\right )} \log \left (x^{n}\right )\right )} \log \left ({\left (f x + e\right )}^{m}\right )}{e x} + \int \frac {b^{3} e^{2} \log \relax (c)^{3} \log \relax (d) + 3 \, a b^{2} e^{2} \log \relax (c)^{2} \log \relax (d) + 3 \, a^{2} b e^{2} \log \relax (c) \log \relax (d) + a^{3} e^{2} \log \relax (d) + 3 \, {\left (a b^{2} e^{2} \log \relax (d) + {\left (e^{2} n \log \relax (d) + e^{2} \log \relax (c) \log \relax (d)\right )} b^{3} + {\left ({\left (e f m + e f \log \relax (d)\right )} a b^{2} + {\left (e f m n + e f n \log \relax (d) + {\left (e f m + e f \log \relax (d)\right )} \log \relax (c)\right )} b^{3}\right )} x + {\left (b^{3} f^{2} m n x^{2} + b^{3} e f m n x\right )} \log \left (f x + e\right ) - {\left (b^{3} f^{2} m n x^{2} + b^{3} e f m n x\right )} \log \relax (x)\right )} \log \left (x^{n}\right )^{2} + {\left ({\left (e f m + e f \log \relax (d)\right )} a^{3} + 3 \, {\left (e f m n + {\left (e f m + e f \log \relax (d)\right )} \log \relax (c)\right )} a^{2} b + 3 \, {\left (2 \, e f m n^{2} + 2 \, e f m n \log \relax (c) + {\left (e f m + e f \log \relax (d)\right )} \log \relax (c)^{2}\right )} a b^{2} + {\left (6 \, e f m n^{3} + 6 \, e f m n^{2} \log \relax (c) + 3 \, e f m n \log \relax (c)^{2} + {\left (e f m + e f \log \relax (d)\right )} \log \relax (c)^{3}\right )} b^{3}\right )} x + 3 \, {\left (b^{3} e^{2} \log \relax (c)^{2} \log \relax (d) + 2 \, a b^{2} e^{2} \log \relax (c) \log \relax (d) + a^{2} b e^{2} \log \relax (d) + {\left ({\left (e f m + e f \log \relax (d)\right )} a^{2} b + 2 \, {\left (e f m n + {\left (e f m + e f \log \relax (d)\right )} \log \relax (c)\right )} a b^{2} + {\left (2 \, e f m n^{2} + 2 \, e f m n \log \relax (c) + {\left (e f m + e f \log \relax (d)\right )} \log \relax (c)^{2}\right )} b^{3}\right )} x\right )} \log \left (x^{n}\right )}{e f x^{3} + e^{2} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (d\,{\left (e+f\,x\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x^2} \,d x \]
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]
________________________________________________________________________________________