Optimal. Leaf size=452 \[ \frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {2 b e^3 m n \text {Li}_2\left (-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}+\frac {e^3 m \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 f^3}-\frac {2 b e^3 m n \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^3}-\frac {e^2 m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f^2}+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac {5 b e m n x^2 \left (a+b \log \left (c x^n\right )\right )}{18 f}-\frac {1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {4}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {8 a b e^2 m n x}{9 f^2}+\frac {8 b^2 e^2 m n x \log \left (c x^n\right )}{9 f^2}+\frac {2}{27} b^2 n^2 x^3 \log \left (d (e+f x)^m\right )-\frac {2 b^2 e^3 m n^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{9 f^3}-\frac {2 b^2 e^3 m n^2 \text {Li}_3\left (-\frac {f x}{e}\right )}{3 f^3}+\frac {2 b^2 e^3 m n^2 \log (e+f x)}{27 f^3}-\frac {26 b^2 e^2 m n^2 x}{27 f^2}+\frac {19 b^2 e m n^2 x^2}{108 f}-\frac {2}{27} b^2 m n^2 x^3 \]
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Rubi [A] time = 0.68, antiderivative size = 452, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 12, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2305, 2304, 2378, 43, 2351, 2295, 2317, 2391, 2353, 2296, 2374, 6589} \[ \frac {2 b e^3 m n \text {PolyLog}\left (2,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}-\frac {2 b^2 e^3 m n^2 \text {PolyLog}\left (2,-\frac {f x}{e}\right )}{9 f^3}-\frac {2 b^2 e^3 m n^2 \text {PolyLog}\left (3,-\frac {f x}{e}\right )}{3 f^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {e^3 m \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 f^3}-\frac {2 b e^3 m n \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^3}-\frac {e^2 m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f^2}+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac {5 b e m n x^2 \left (a+b \log \left (c x^n\right )\right )}{18 f}-\frac {1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {4}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {8 a b e^2 m n x}{9 f^2}+\frac {8 b^2 e^2 m n x \log \left (c x^n\right )}{9 f^2}+\frac {2}{27} b^2 n^2 x^3 \log \left (d (e+f x)^m\right )-\frac {26 b^2 e^2 m n^2 x}{27 f^2}+\frac {2 b^2 e^3 m n^2 \log (e+f x)}{27 f^3}+\frac {19 b^2 e m n^2 x^2}{108 f}-\frac {2}{27} b^2 m n^2 x^3 \]
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^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right ) \, dx &=\frac {2}{27} b^2 n^2 x^3 \log \left (d (e+f x)^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-(f m) \int \left (\frac {2 b^2 n^2 x^3}{27 (e+f x)}-\frac {2 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{9 (e+f x)}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 (e+f x)}\right ) \, dx\\ &=\frac {2}{27} b^2 n^2 x^3 \log \left (d (e+f x)^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac {1}{3} (f m) \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{e+f x} \, dx+\frac {1}{9} (2 b f m n) \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{e+f x} \, dx-\frac {1}{27} \left (2 b^2 f m n^2\right ) \int \frac {x^3}{e+f x} \, dx\\ &=\frac {2}{27} b^2 n^2 x^3 \log \left (d (e+f x)^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac {1}{3} (f m) \int \left (\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{f^3}-\frac {e x \left (a+b \log \left (c x^n\right )\right )^2}{f^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{f}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )^2}{f^3 (e+f x)}\right ) \, dx+\frac {1}{9} (2 b f m n) \int \left (\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{f^3}-\frac {e x \left (a+b \log \left (c x^n\right )\right )}{f^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{f}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{f^3 (e+f x)}\right ) \, dx-\frac {1}{27} \left (2 b^2 f m n^2\right ) \int \left (\frac {e^2}{f^3}-\frac {e x}{f^2}+\frac {x^2}{f}-\frac {e^3}{f^3 (e+f x)}\right ) \, dx\\ &=-\frac {2 b^2 e^2 m n^2 x}{27 f^2}+\frac {b^2 e m n^2 x^2}{27 f}-\frac {2}{81} b^2 m n^2 x^3+\frac {2 b^2 e^3 m n^2 \log (e+f x)}{27 f^3}+\frac {2}{27} b^2 n^2 x^3 \log \left (d (e+f x)^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac {1}{3} m \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx-\frac {\left (e^2 m\right ) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{3 f^2}+\frac {\left (e^3 m\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{e+f x} \, dx}{3 f^2}+\frac {(e m) \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{3 f}+\frac {1}{9} (2 b m n) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac {\left (2 b e^2 m n\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{9 f^2}-\frac {\left (2 b e^3 m n\right ) \int \frac {a+b \log \left (c x^n\right )}{e+f x} \, dx}{9 f^2}-\frac {(2 b e m n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{9 f}\\ &=\frac {2 a b e^2 m n x}{9 f^2}-\frac {2 b^2 e^2 m n^2 x}{27 f^2}+\frac {5 b^2 e m n^2 x^2}{54 f}-\frac {4}{81} b^2 m n^2 x^3-\frac {b e m n x^2 \left (a+b \log \left (c x^n\right )\right )}{9 f}+\frac {2}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {e^2 m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f^2}+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac {1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2 b^2 e^3 m n^2 \log (e+f x)}{27 f^3}+\frac {2}{27} b^2 n^2 x^3 \log \left (d (e+f x)^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac {2 b e^3 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{9 f^3}+\frac {e^3 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{3 f^3}+\frac {1}{9} (2 b m n) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac {\left (2 b e^3 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}{3 f^3}+\frac {\left (2 b e^2 m n\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 f^2}+\frac {\left (2 b^2 e^2 m n\right ) \int \log \left (c x^n\right ) \, dx}{9 f^2}-\frac {(b e m n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 f}+\frac {\left (2 b^2 e^3 m n^2\right ) \int \frac {\log \left (1+\frac {f x}{e}\right )}{x} \, dx}{9 f^3}\\ &=\frac {8 a b e^2 m n x}{9 f^2}-\frac {8 b^2 e^2 m n^2 x}{27 f^2}+\frac {19 b^2 e m n^2 x^2}{108 f}-\frac {2}{27} b^2 m n^2 x^3+\frac {2 b^2 e^2 m n x \log \left (c x^n\right )}{9 f^2}-\frac {5 b e m n x^2 \left (a+b \log \left (c x^n\right )\right )}{18 f}+\frac {4}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {e^2 m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f^2}+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac {1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2 b^2 e^3 m n^2 \log (e+f x)}{27 f^3}+\frac {2}{27} b^2 n^2 x^3 \log \left (d (e+f x)^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac {2 b e^3 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{9 f^3}+\frac {e^3 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{3 f^3}-\frac {2 b^2 e^3 m n^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{9 f^3}+\frac {2 b e^3 m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )}{3 f^3}+\frac {\left (2 b^2 e^2 m n\right ) \int \log \left (c x^n\right ) \, dx}{3 f^2}-\frac {\left (2 b^2 e^3 m n^2\right ) \int \frac {\text {Li}_2\left (-\frac {f x}{e}\right )}{x} \, dx}{3 f^3}\\ &=\frac {8 a b e^2 m n x}{9 f^2}-\frac {26 b^2 e^2 m n^2 x}{27 f^2}+\frac {19 b^2 e m n^2 x^2}{108 f}-\frac {2}{27} b^2 m n^2 x^3+\frac {8 b^2 e^2 m n x \log \left (c x^n\right )}{9 f^2}-\frac {5 b e m n x^2 \left (a+b \log \left (c x^n\right )\right )}{18 f}+\frac {4}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {e^2 m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f^2}+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac {1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2 b^2 e^3 m n^2 \log (e+f x)}{27 f^3}+\frac {2}{27} b^2 n^2 x^3 \log \left (d (e+f x)^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac {2 b e^3 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{9 f^3}+\frac {e^3 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{3 f^3}-\frac {2 b^2 e^3 m n^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{9 f^3}+\frac {2 b e^3 m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )}{3 f^3}-\frac {2 b^2 e^3 m n^2 \text {Li}_3\left (-\frac {f x}{e}\right )}{3 f^3}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 788, normalized size = 1.74 \[ \frac {36 a^2 f^3 x^3 \log \left (d (e+f x)^m\right )+36 a^2 e^3 m \log (e+f x)-36 a^2 e^2 f m x+18 a^2 e f^2 m x^2-12 a^2 f^3 m x^3+72 a b f^3 x^3 \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+24 b e^3 m n \text {Li}_2\left (-\frac {f x}{e}\right ) \left (3 a+3 b \log \left (c x^n\right )-b n\right )+72 a b e^3 m \log \left (c x^n\right ) \log (e+f x)-72 a b e^2 f m x \log \left (c x^n\right )+36 a b e f^2 m x^2 \log \left (c x^n\right )-24 a b f^3 m x^3 \log \left (c x^n\right )-24 a b f^3 n x^3 \log \left (d (e+f x)^m\right )-24 a b e^3 m n \log (e+f x)-72 a b e^3 m n \log (x) \log (e+f x)+72 a b e^3 m n \log (x) \log \left (\frac {f x}{e}+1\right )+96 a b e^2 f m n x-30 a b e f^2 m n x^2+16 a b f^3 m n x^3+36 b^2 f^3 x^3 \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )-24 b^2 f^3 n x^3 \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+36 b^2 e^3 m \log ^2\left (c x^n\right ) \log (e+f x)-24 b^2 e^3 m n \log \left (c x^n\right ) \log (e+f x)-72 b^2 e^3 m n \log (x) \log \left (c x^n\right ) \log (e+f x)+72 b^2 e^3 m n \log (x) \log \left (c x^n\right ) \log \left (\frac {f x}{e}+1\right )-36 b^2 e^2 f m x \log ^2\left (c x^n\right )+96 b^2 e^2 f m n x \log \left (c x^n\right )+18 b^2 e f^2 m x^2 \log ^2\left (c x^n\right )-30 b^2 e f^2 m n x^2 \log \left (c x^n\right )-12 b^2 f^3 m x^3 \log ^2\left (c x^n\right )+16 b^2 f^3 m n x^3 \log \left (c x^n\right )+8 b^2 f^3 n^2 x^3 \log \left (d (e+f x)^m\right )-72 b^2 e^3 m n^2 \text {Li}_3\left (-\frac {f x}{e}\right )+36 b^2 e^3 m n^2 \log ^2(x) \log (e+f x)-36 b^2 e^3 m n^2 \log ^2(x) \log \left (\frac {f x}{e}+1\right )+8 b^2 e^3 m n^2 \log (e+f x)+24 b^2 e^3 m n^2 \log (x) \log (e+f x)-24 b^2 e^3 m n^2 \log (x) \log \left (\frac {f x}{e}+1\right )-104 b^2 e^2 f m n^2 x+19 b^2 e f^2 m n^2 x^2-8 b^2 f^3 m n^2 x^3}{108 f^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} x^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b x^{2} \log \left (c x^{n}\right ) + a^{2} x^{2}\right )} \log \left ({\left (f x + e\right )}^{m} d\right ), x\right ) \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{2} \log \left ({\left (f x + e\right )}^{m} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.81, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right )^{2} x^{2} \ln \left (d \left (f x +e \right )^{m}\right )\, dx \]
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 \[ \frac {3 \, {\left (3 \, b^{2} e f^{2} m x^{2} - 6 \, b^{2} e^{2} f m x + 6 \, b^{2} e^{3} m \log \left (f x + e\right ) - 2 \, {\left (f^{3} m - 3 \, f^{3} \log \relax (d)\right )} b^{2} x^{3}\right )} \log \left (x^{n}\right )^{2} + 2 \, {\left (9 \, b^{2} f^{3} x^{3} \log \left (x^{n}\right )^{2} + 6 \, {\left (3 \, a b f^{3} - {\left (f^{3} n - 3 \, f^{3} \log \relax (c)\right )} b^{2}\right )} x^{3} \log \left (x^{n}\right ) + {\left (9 \, a^{2} f^{3} - 6 \, {\left (f^{3} n - 3 \, f^{3} \log \relax (c)\right )} a b + {\left (2 \, f^{3} n^{2} - 6 \, f^{3} n \log \relax (c) + 9 \, f^{3} \log \relax (c)^{2}\right )} b^{2}\right )} x^{3}\right )} \log \left ({\left (f x + e\right )}^{m}\right )}{54 \, f^{3}} - \int \frac {{\left (9 \, {\left (f^{4} m - 3 \, f^{4} \log \relax (d)\right )} a^{2} - 6 \, {\left (f^{4} m n - 3 \, {\left (f^{4} m - 3 \, f^{4} \log \relax (d)\right )} \log \relax (c)\right )} a b + {\left (2 \, f^{4} m n^{2} - 6 \, f^{4} m n \log \relax (c) + 9 \, {\left (f^{4} m - 3 \, f^{4} \log \relax (d)\right )} \log \relax (c)^{2}\right )} b^{2}\right )} x^{4} - 27 \, {\left (b^{2} e f^{3} \log \relax (c)^{2} \log \relax (d) + 2 \, a b e f^{3} \log \relax (c) \log \relax (d) + a^{2} e f^{3} \log \relax (d)\right )} x^{3} - 3 \, {\left (3 \, b^{2} e^{2} f^{2} m n x^{2} + 6 \, b^{2} e^{3} f m n x - 2 \, {\left (3 \, {\left (f^{4} m - 3 \, f^{4} \log \relax (d)\right )} a b - {\left (2 \, f^{4} m n - 3 \, f^{4} n \log \relax (d) - 3 \, {\left (f^{4} m - 3 \, f^{4} \log \relax (d)\right )} \log \relax (c)\right )} b^{2}\right )} x^{4} + {\left (18 \, a b e f^{3} \log \relax (d) - {\left (e f^{3} m n + 6 \, e f^{3} n \log \relax (d) - 18 \, e f^{3} \log \relax (c) \log \relax (d)\right )} b^{2}\right )} x^{3} - 6 \, {\left (b^{2} e^{3} f m n x + b^{2} e^{4} m n\right )} \log \left (f x + e\right )\right )} \log \left (x^{n}\right )}{27 \, {\left (f^{4} x^{2} + e f^{3} x\right )}}\,{d x} \]
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 \[ \int x^2\,\ln \left (d\,{\left (e+f\,x\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]
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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