Optimal. Leaf size=344 \[ -\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{2 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}-\frac {b f^2 m n \text {Li}_2\left (-\frac {e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {f^2 m \log \left (\frac {e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}+\frac {b f^2 m n \log \left (\frac {e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^2}{2 e x}-\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )}{2 e x}-\frac {b^2 n^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {b^2 f^2 m n^2 \text {Li}_2\left (-\frac {e}{f x}\right )}{2 e^2}-\frac {b^2 f^2 m n^2 \text {Li}_3\left (-\frac {e}{f x}\right )}{e^2}-\frac {b^2 f^2 m n^2 \log (x)}{4 e^2}+\frac {b^2 f^2 m n^2 \log (e+f x)}{4 e^2}-\frac {7 b^2 f m n^2}{4 e x} \]
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Rubi [A] time = 0.59, antiderivative size = 385, normalized size of antiderivative = 1.12, number of steps used = 19, number of rules used = 13, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2305, 2304, 2378, 44, 2351, 2301, 2317, 2391, 2353, 2302, 30, 2374, 6589} \[ \frac {b f^2 m n \text {PolyLog}\left (2,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {b^2 f^2 m n^2 \text {PolyLog}\left (2,-\frac {f x}{e}\right )}{2 e^2}-\frac {b^2 f^2 m n^2 \text {PolyLog}\left (3,-\frac {f x}{e}\right )}{e^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{2 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}-\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^3}{6 b e^2 n}-\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2}+\frac {f^2 m \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}+\frac {b f^2 m n \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^2}{2 e x}-\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )}{2 e x}-\frac {b^2 n^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {b^2 f^2 m n^2 \log (x)}{4 e^2}+\frac {b^2 f^2 m n^2 \log (e+f x)}{4 e^2}-\frac {7 b^2 f m n^2}{4 e x} \]
Antiderivative was successfully verified.
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Rule 30
Rule 44
Rule 2301
Rule 2302
Rule 2304
Rule 2305
Rule 2317
Rule 2351
Rule 2353
Rule 2374
Rule 2378
Rule 2391
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x^3} \, dx &=-\frac {b^2 n^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{2 x^2}-(f m) \int \left (-\frac {b^2 n^2}{4 x^2 (e+f x)}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{2 x^2 (e+f x)}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2 (e+f x)}\right ) \, dx\\ &=-\frac {b^2 n^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \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 )^2}{x^2 (e+f x)} \, dx+\frac {1}{2} (b f m n) \int \frac {a+b \log \left (c x^n\right )}{x^2 (e+f x)} \, dx+\frac {1}{4} \left (b^2 f m n^2\right ) \int \frac {1}{x^2 (e+f x)} \, dx\\ &=-\frac {b^2 n^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \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 )^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}{2} (b f m n) \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}{4} \left (b^2 f m n^2\right ) \int \left (\frac {1}{e x^2}-\frac {f}{e^2 x}+\frac {f^2}{e^2 (e+f x)}\right ) \, dx\\ &=-\frac {b^2 f m n^2}{4 e x}-\frac {b^2 f^2 m n^2 \log (x)}{4 e^2}+\frac {b^2 f^2 m n^2 \log (e+f x)}{4 e^2}-\frac {b^2 n^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \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 )^2}{x^2} \, dx}{2 e}-\frac {\left (f^2 m\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{2 e^2}+\frac {\left (f^3 m\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{e+f x} \, dx}{2 e^2}+\frac {(b f m n) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{2 e}-\frac {\left (b f^2 m n\right ) \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{2 e^2}+\frac {\left (b f^3 m n\right ) \int \frac {a+b \log \left (c x^n\right )}{e+f x} \, dx}{2 e^2}\\ &=-\frac {3 b^2 f m n^2}{4 e x}-\frac {b^2 f^2 m n^2 \log (x)}{4 e^2}-\frac {b f m n \left (a+b \log \left (c x^n\right )\right )}{2 e x}-\frac {f^2 m \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 )^2}{2 e x}+\frac {b^2 f^2 m n^2 \log (e+f x)}{4 e^2}-\frac {b^2 n^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{2 x^2}+\frac {b f^2 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{2 e^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{2 e^2}-\frac {\left (f^2 m\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{2 b e^2 n}+\frac {(b f m n) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{e}-\frac {\left (b f^2 m n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{x} \, dx}{e^2}-\frac {\left (b^2 f^2 m n^2\right ) \int \frac {\log \left (1+\frac {f x}{e}\right )}{x} \, dx}{2 e^2}\\ &=-\frac {7 b^2 f m n^2}{4 e x}-\frac {b^2 f^2 m n^2 \log (x)}{4 e^2}-\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )}{2 e x}-\frac {f^2 m \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 )^2}{2 e x}-\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^3}{6 b e^2 n}+\frac {b^2 f^2 m n^2 \log (e+f x)}{4 e^2}-\frac {b^2 n^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{2 x^2}+\frac {b f^2 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{2 e^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{2 e^2}+\frac {b^2 f^2 m n^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{2 e^2}+\frac {b f^2 m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )}{e^2}-\frac {\left (b^2 f^2 m n^2\right ) \int \frac {\text {Li}_2\left (-\frac {f x}{e}\right )}{x} \, dx}{e^2}\\ &=-\frac {7 b^2 f m n^2}{4 e x}-\frac {b^2 f^2 m n^2 \log (x)}{4 e^2}-\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )}{2 e x}-\frac {f^2 m \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 )^2}{2 e x}-\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^3}{6 b e^2 n}+\frac {b^2 f^2 m n^2 \log (e+f x)}{4 e^2}-\frac {b^2 n^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{2 x^2}+\frac {b f^2 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{2 e^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{2 e^2}+\frac {b^2 f^2 m n^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{2 e^2}+\frac {b f^2 m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )}{e^2}-\frac {b^2 f^2 m n^2 \text {Li}_3\left (-\frac {f x}{e}\right )}{e^2}\\ \end {align*}
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Mathematica [B] time = 0.39, size = 796, normalized size = 2.31 \[ -\frac {2 b^2 f^2 m n^2 x^2 \log ^3(x)-3 b^2 f^2 m n^2 x^2 \log ^2(x)-6 a b f^2 m n x^2 \log ^2(x)-6 b^2 f^2 m n x^2 \log \left (c x^n\right ) \log ^2(x)-6 b^2 f^2 m n^2 x^2 \log (e+f x) \log ^2(x)+6 b^2 f^2 m n^2 x^2 \log \left (\frac {f x}{e}+1\right ) \log ^2(x)+3 b^2 f^2 m n^2 x^2 \log (x)+6 a^2 f^2 m x^2 \log (x)+6 a b f^2 m n x^2 \log (x)+6 b^2 f^2 m x^2 \log ^2\left (c x^n\right ) \log (x)+12 a b f^2 m x^2 \log \left (c x^n\right ) \log (x)+6 b^2 f^2 m n x^2 \log \left (c x^n\right ) \log (x)+6 b^2 f^2 m n^2 x^2 \log (e+f x) \log (x)+12 a b f^2 m n x^2 \log (e+f x) \log (x)+12 b^2 f^2 m n x^2 \log \left (c x^n\right ) \log (e+f x) \log (x)-6 b^2 f^2 m n^2 x^2 \log \left (\frac {f x}{e}+1\right ) \log (x)-12 a b f^2 m n x^2 \log \left (\frac {f x}{e}+1\right ) \log (x)-12 b^2 f^2 m n x^2 \log \left (c x^n\right ) \log \left (\frac {f x}{e}+1\right ) \log (x)+6 b^2 e f m x \log ^2\left (c x^n\right )+21 b^2 e f m n^2 x+6 a^2 e f m x+18 a b e f m n x+12 a b e f m x \log \left (c x^n\right )+18 b^2 e f m n x \log \left (c x^n\right )-3 b^2 f^2 m n^2 x^2 \log (e+f x)-6 a^2 f^2 m x^2 \log (e+f x)-6 a b f^2 m n x^2 \log (e+f x)-6 b^2 f^2 m x^2 \log ^2\left (c x^n\right ) \log (e+f x)-12 a b f^2 m x^2 \log \left (c x^n\right ) \log (e+f x)-6 b^2 f^2 m n x^2 \log \left (c x^n\right ) \log (e+f x)+6 a^2 e^2 \log \left (d (e+f x)^m\right )+3 b^2 e^2 n^2 \log \left (d (e+f x)^m\right )+6 b^2 e^2 \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )+6 a b e^2 n \log \left (d (e+f x)^m\right )+12 a b e^2 \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+6 b^2 e^2 n \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-6 b f^2 m n x^2 \left (2 a+b n+2 b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )+12 b^2 f^2 m n^2 x^2 \text {Li}_3\left (-\frac {f x}{e}\right )}{12 e^2 x^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}\right )} \log \left ({\left (f x + e\right )}^{m} d\right )}{x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x + e\right )}^{m} d\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 5.04, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{2} \ln \left (d \left (f x +e \right )^{m}\right )}{x^{3}}\, dx \]
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 \[ \frac {2 \, {\left (b^{2} f^{2} m x^{2} \log \left (f x + e\right ) - b^{2} f^{2} m x^{2} \log \relax (x) - b^{2} e f m x - b^{2} e^{2} \log \relax (d)\right )} \log \left (x^{n}\right )^{2} - {\left (2 \, b^{2} e^{2} \log \left (x^{n}\right )^{2} + 2 \, a^{2} e^{2} + 2 \, {\left (e^{2} n + 2 \, e^{2} \log \relax (c)\right )} a b + {\left (e^{2} n^{2} + 2 \, e^{2} n \log \relax (c) + 2 \, e^{2} \log \relax (c)^{2}\right )} b^{2} + 2 \, {\left (2 \, a b e^{2} + {\left (e^{2} n + 2 \, e^{2} \log \relax (c)\right )} b^{2}\right )} \log \left (x^{n}\right )\right )} \log \left ({\left (f x + e\right )}^{m}\right )}{4 \, e^{2} x^{2}} - \int -\frac {4 \, b^{2} e^{3} \log \relax (c)^{2} \log \relax (d) + 8 \, a b e^{3} \log \relax (c) \log \relax (d) + 4 \, a^{2} e^{3} \log \relax (d) + {\left (2 \, {\left (e^{2} f m + 2 \, e^{2} f \log \relax (d)\right )} a^{2} + 2 \, {\left (e^{2} f m n + 2 \, {\left (e^{2} f m + 2 \, e^{2} f \log \relax (d)\right )} \log \relax (c)\right )} a b + {\left (e^{2} f m n^{2} + 2 \, e^{2} f m n \log \relax (c) + 2 \, {\left (e^{2} f m + 2 \, e^{2} f \log \relax (d)\right )} \log \relax (c)^{2}\right )} b^{2}\right )} x + 2 \, {\left (2 \, b^{2} e f^{2} m n x^{2} + 4 \, a b e^{3} \log \relax (d) + 2 \, {\left (e^{3} n \log \relax (d) + 2 \, e^{3} \log \relax (c) \log \relax (d)\right )} b^{2} + {\left (2 \, {\left (e^{2} f m + 2 \, e^{2} f \log \relax (d)\right )} a b + {\left (3 \, e^{2} f m n + 2 \, e^{2} f n \log \relax (d) + 2 \, {\left (e^{2} f m + 2 \, e^{2} f \log \relax (d)\right )} \log \relax (c)\right )} b^{2}\right )} x - 2 \, {\left (b^{2} f^{3} m n x^{3} + b^{2} e f^{2} m n x^{2}\right )} \log \left (f x + e\right ) + 2 \, {\left (b^{2} f^{3} m n x^{3} + b^{2} e f^{2} m n x^{2}\right )} \log \relax (x)\right )} \log \left (x^{n}\right )}{4 \, {\left (e^{2} f x^{4} + e^{3} x^{3}\right )}}\,{d x} \]
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 \[ \int \frac {\ln \left (d\,{\left (e+f\,x\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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