Optimal. Leaf size=373 \[ -\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac {b e^2 m n \text {Li}_2\left (-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+\frac {b e^2 m n \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^2}-\frac {e^2 m \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 f^2}+\frac {e m x \left (a+b \log \left (c x^n\right )\right )^2}{2 f}+\frac {1}{2} b m n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {3 a b e m n x}{2 f}-\frac {3 b^2 e m n x \log \left (c x^n\right )}{2 f}+\frac {1}{4} b^2 n^2 x^2 \log \left (d (e+f x)^m\right )+\frac {b^2 e^2 m n^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{2 f^2}+\frac {b^2 e^2 m n^2 \text {Li}_3\left (-\frac {f x}{e}\right )}{f^2}-\frac {b^2 e^2 m n^2 \log (e+f x)}{4 f^2}+\frac {7 b^2 e m n^2 x}{4 f}-\frac {3}{8} b^2 m n^2 x^2 \]
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Rubi [A] time = 0.53, antiderivative size = 373, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 12, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2305, 2304, 2378, 43, 2351, 2295, 2317, 2391, 2353, 2296, 2374, 6589} \[ -\frac {b e^2 m n \text {PolyLog}\left (2,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+\frac {b^2 e^2 m n^2 \text {PolyLog}\left (2,-\frac {f x}{e}\right )}{2 f^2}+\frac {b^2 e^2 m n^2 \text {PolyLog}\left (3,-\frac {f x}{e}\right )}{f^2}-\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+\frac {b e^2 m n \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^2}-\frac {e^2 m \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 f^2}+\frac {e m x \left (a+b \log \left (c x^n\right )\right )^2}{2 f}+\frac {1}{2} b m n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {3 a b e m n x}{2 f}-\frac {3 b^2 e m n x \log \left (c x^n\right )}{2 f}+\frac {1}{4} b^2 n^2 x^2 \log \left (d (e+f x)^m\right )-\frac {b^2 e^2 m n^2 \log (e+f x)}{4 f^2}+\frac {7 b^2 e m n^2 x}{4 f}-\frac {3}{8} b^2 m n^2 x^2 \]
Antiderivative was successfully verified.
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Rule 43
Rule 2295
Rule 2296
Rule 2304
Rule 2305
Rule 2317
Rule 2351
Rule 2353
Rule 2374
Rule 2378
Rule 2391
Rule 6589
Rubi steps
\begin {align*} \int x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right ) \, dx &=\frac {1}{4} b^2 n^2 x^2 \log \left (d (e+f x)^m\right )-\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-(f m) \int \left (\frac {b^2 n^2 x^2}{4 (e+f x)}-\frac {b n x^2 \left (a+b \log \left (c x^n\right )\right )}{2 (e+f x)}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 (e+f x)}\right ) \, dx\\ &=\frac {1}{4} b^2 n^2 x^2 \log \left (d (e+f x)^m\right )-\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac {1}{2} (f m) \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{e+f x} \, dx+\frac {1}{2} (b f m n) \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{e+f x} \, dx-\frac {1}{4} \left (b^2 f m n^2\right ) \int \frac {x^2}{e+f x} \, dx\\ &=\frac {1}{4} b^2 n^2 x^2 \log \left (d (e+f x)^m\right )-\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac {1}{2} (f m) \int \left (-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{f^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{f}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{f^2 (e+f x)}\right ) \, dx+\frac {1}{2} (b f m n) \int \left (-\frac {e \left (a+b \log \left (c x^n\right )\right )}{f^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{f}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{f^2 (e+f x)}\right ) \, dx-\frac {1}{4} \left (b^2 f m n^2\right ) \int \left (-\frac {e}{f^2}+\frac {x}{f}+\frac {e^2}{f^2 (e+f x)}\right ) \, dx\\ &=\frac {b^2 e m n^2 x}{4 f}-\frac {1}{8} b^2 m n^2 x^2-\frac {b^2 e^2 m n^2 \log (e+f x)}{4 f^2}+\frac {1}{4} b^2 n^2 x^2 \log \left (d (e+f x)^m\right )-\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac {1}{2} m \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx+\frac {(e m) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{2 f}-\frac {\left (e^2 m\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{e+f x} \, dx}{2 f}+\frac {1}{2} (b m n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac {(b e m n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{2 f}+\frac {\left (b e^2 m n\right ) \int \frac {a+b \log \left (c x^n\right )}{e+f x} \, dx}{2 f}\\ &=-\frac {a b e m n x}{2 f}+\frac {b^2 e m n^2 x}{4 f}-\frac {1}{4} b^2 m n^2 x^2+\frac {1}{4} b m n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {e m x \left (a+b \log \left (c x^n\right )\right )^2}{2 f}-\frac {1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {b^2 e^2 m n^2 \log (e+f x)}{4 f^2}+\frac {1}{4} b^2 n^2 x^2 \log \left (d (e+f x)^m\right )-\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+\frac {b e^2 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{2 f^2}-\frac {e^2 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{2 f^2}+\frac {1}{2} (b m n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac {\left (b e^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}{f^2}-\frac {(b e m n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{f}-\frac {\left (b^2 e m n\right ) \int \log \left (c x^n\right ) \, dx}{2 f}-\frac {\left (b^2 e^2 m n^2\right ) \int \frac {\log \left (1+\frac {f x}{e}\right )}{x} \, dx}{2 f^2}\\ &=-\frac {3 a b e m n x}{2 f}+\frac {3 b^2 e m n^2 x}{4 f}-\frac {3}{8} b^2 m n^2 x^2-\frac {b^2 e m n x \log \left (c x^n\right )}{2 f}+\frac {1}{2} b m n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {e m x \left (a+b \log \left (c x^n\right )\right )^2}{2 f}-\frac {1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {b^2 e^2 m n^2 \log (e+f x)}{4 f^2}+\frac {1}{4} b^2 n^2 x^2 \log \left (d (e+f x)^m\right )-\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+\frac {b e^2 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{2 f^2}-\frac {e^2 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{2 f^2}+\frac {b^2 e^2 m n^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{2 f^2}-\frac {b e^2 m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )}{f^2}-\frac {\left (b^2 e m n\right ) \int \log \left (c x^n\right ) \, dx}{f}+\frac {\left (b^2 e^2 m n^2\right ) \int \frac {\text {Li}_2\left (-\frac {f x}{e}\right )}{x} \, dx}{f^2}\\ &=-\frac {3 a b e m n x}{2 f}+\frac {7 b^2 e m n^2 x}{4 f}-\frac {3}{8} b^2 m n^2 x^2-\frac {3 b^2 e m n x \log \left (c x^n\right )}{2 f}+\frac {1}{2} b m n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {e m x \left (a+b \log \left (c x^n\right )\right )^2}{2 f}-\frac {1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {b^2 e^2 m n^2 \log (e+f x)}{4 f^2}+\frac {1}{4} b^2 n^2 x^2 \log \left (d (e+f x)^m\right )-\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+\frac {b e^2 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{2 f^2}-\frac {e^2 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{2 f^2}+\frac {b^2 e^2 m n^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{2 f^2}-\frac {b e^2 m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )}{f^2}+\frac {b^2 e^2 m n^2 \text {Li}_3\left (-\frac {f x}{e}\right )}{f^2}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 674, normalized size = 1.81 \[ \frac {4 a^2 f^2 x^2 \log \left (d (e+f x)^m\right )-4 a^2 e^2 m \log (e+f x)+4 a^2 e f m x-2 a^2 f^2 m x^2+8 a b f^2 x^2 \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+4 b e^2 m n \text {Li}_2\left (-\frac {f x}{e}\right ) \left (-2 a-2 b \log \left (c x^n\right )+b n\right )-8 a b e^2 m \log \left (c x^n\right ) \log (e+f x)+8 a b e f m x \log \left (c x^n\right )-4 a b f^2 m x^2 \log \left (c x^n\right )-4 a b f^2 n x^2 \log \left (d (e+f x)^m\right )+4 a b e^2 m n \log (e+f x)+8 a b e^2 m n \log (x) \log (e+f x)-8 a b e^2 m n \log (x) \log \left (\frac {f x}{e}+1\right )-12 a b e f m n x+4 a b f^2 m n x^2+4 b^2 f^2 x^2 \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )-4 b^2 f^2 n x^2 \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-4 b^2 e^2 m \log ^2\left (c x^n\right ) \log (e+f x)+4 b^2 e^2 m n \log \left (c x^n\right ) \log (e+f x)+8 b^2 e^2 m n \log (x) \log \left (c x^n\right ) \log (e+f x)-8 b^2 e^2 m n \log (x) \log \left (c x^n\right ) \log \left (\frac {f x}{e}+1\right )+4 b^2 e f m x \log ^2\left (c x^n\right )-12 b^2 e f m n x \log \left (c x^n\right )-2 b^2 f^2 m x^2 \log ^2\left (c x^n\right )+4 b^2 f^2 m n x^2 \log \left (c x^n\right )+2 b^2 f^2 n^2 x^2 \log \left (d (e+f x)^m\right )+8 b^2 e^2 m n^2 \text {Li}_3\left (-\frac {f x}{e}\right )-4 b^2 e^2 m n^2 \log ^2(x) \log (e+f x)+4 b^2 e^2 m n^2 \log ^2(x) \log \left (\frac {f x}{e}+1\right )-2 b^2 e^2 m n^2 \log (e+f x)-4 b^2 e^2 m n^2 \log (x) \log (e+f x)+4 b^2 e^2 m n^2 \log (x) \log \left (\frac {f x}{e}+1\right )+14 b^2 e f m n^2 x-3 b^2 f^2 m n^2 x^2}{8 f^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} x \log \left (c x^{n}\right )^{2} + 2 \, a b x \log \left (c x^{n}\right ) + a^{2} x\right )} \log \left ({\left (f x + e\right )}^{m} d\right ), x\right ) \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \log \left ({\left (f x + e\right )}^{m} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 6.90, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right )^{2} x \ln \left (d \left (f x +e \right )^{m}\right )\, dx \]
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 \[ \frac {{\left (2 \, b^{2} e f m x - 2 \, b^{2} e^{2} m \log \left (f x + e\right ) - {\left (f^{2} m - 2 \, f^{2} \log \relax (d)\right )} b^{2} x^{2}\right )} \log \left (x^{n}\right )^{2} + {\left (2 \, b^{2} f^{2} x^{2} \log \left (x^{n}\right )^{2} + 2 \, {\left (2 \, a b f^{2} - {\left (f^{2} n - 2 \, f^{2} \log \relax (c)\right )} b^{2}\right )} x^{2} \log \left (x^{n}\right ) + {\left (2 \, a^{2} f^{2} - 2 \, {\left (f^{2} n - 2 \, f^{2} \log \relax (c)\right )} a b + {\left (f^{2} n^{2} - 2 \, f^{2} n \log \relax (c) + 2 \, f^{2} \log \relax (c)^{2}\right )} b^{2}\right )} x^{2}\right )} \log \left ({\left (f x + e\right )}^{m}\right )}{4 \, f^{2}} + \int -\frac {{\left (2 \, {\left (f^{3} m - 2 \, f^{3} \log \relax (d)\right )} a^{2} - 2 \, {\left (f^{3} m n - 2 \, {\left (f^{3} m - 2 \, f^{3} \log \relax (d)\right )} \log \relax (c)\right )} a b + {\left (f^{3} m n^{2} - 2 \, f^{3} m n \log \relax (c) + 2 \, {\left (f^{3} m - 2 \, f^{3} \log \relax (d)\right )} \log \relax (c)^{2}\right )} b^{2}\right )} x^{3} - 4 \, {\left (b^{2} e f^{2} \log \relax (c)^{2} \log \relax (d) + 2 \, a b e f^{2} \log \relax (c) \log \relax (d) + a^{2} e f^{2} \log \relax (d)\right )} x^{2} + 2 \, {\left (2 \, b^{2} e^{2} f m n x + 2 \, {\left ({\left (f^{3} m - 2 \, f^{3} \log \relax (d)\right )} a b - {\left (f^{3} m n - f^{3} n \log \relax (d) - {\left (f^{3} m - 2 \, f^{3} \log \relax (d)\right )} \log \relax (c)\right )} b^{2}\right )} x^{3} - {\left (4 \, a b e f^{2} \log \relax (d) - {\left (e f^{2} m n + 2 \, e f^{2} n \log \relax (d) - 4 \, e f^{2} \log \relax (c) \log \relax (d)\right )} b^{2}\right )} x^{2} - 2 \, {\left (b^{2} e^{2} f m n x + b^{2} e^{3} m n\right )} \log \left (f x + e\right )\right )} \log \left (x^{n}\right )}{4 \, {\left (f^{3} x^{2} + e f^{2} x\right )}}\,{d x} \]
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 \[ \int x\,\ln \left (d\,{\left (e+f\,x\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]
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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