Optimal. Leaf size=243 \[ \frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {e^3 m \log (e+f x) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}-\frac {e^2 m x \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac {1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b n x^3 \log \left (d (e+f x)^m\right )-\frac {b e^3 m n \text {Li}_2\left (\frac {f x}{e}+1\right )}{3 f^3}-\frac {b e^3 m n \log (e+f x)}{9 f^3}-\frac {b e^3 m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{3 f^3}+\frac {4 b e^2 m n x}{9 f^2}-\frac {5 b e m n x^2}{36 f}+\frac {2}{27} b m n x^3 \]
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Rubi [A] time = 0.17, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2395, 43, 2376, 2394, 2315} \[ -\frac {b e^3 m n \text {PolyLog}\left (2,\frac {f x}{e}+1\right )}{3 f^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {e^3 m \log (e+f x) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}-\frac {e^2 m x \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac {1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b n x^3 \log \left (d (e+f x)^m\right )+\frac {4 b e^2 m n x}{9 f^2}-\frac {b e^3 m n \log (e+f x)}{9 f^3}-\frac {b e^3 m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{3 f^3}-\frac {5 b e m n x^2}{36 f}+\frac {2}{27} b m n x^3 \]
Antiderivative was successfully verified.
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Rule 43
Rule 2315
Rule 2376
Rule 2394
Rule 2395
Rubi steps
\begin {align*} \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right ) \, dx &=-\frac {e^2 m x \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac {1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {e^3 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{3 f^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-(b n) \int \left (-\frac {e^2 m}{3 f^2}+\frac {e m x}{6 f}-\frac {m x^2}{9}+\frac {e^3 m \log (e+f x)}{3 f^3 x}+\frac {1}{3} x^2 \log \left (d (e+f x)^m\right )\right ) \, dx\\ &=\frac {b e^2 m n x}{3 f^2}-\frac {b e m n x^2}{12 f}+\frac {1}{27} b m n x^3-\frac {e^2 m x \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac {1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {e^3 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{3 f^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac {1}{3} (b n) \int x^2 \log \left (d (e+f x)^m\right ) \, dx-\frac {\left (b e^3 m n\right ) \int \frac {\log (e+f x)}{x} \, dx}{3 f^3}\\ &=\frac {b e^2 m n x}{3 f^2}-\frac {b e m n x^2}{12 f}+\frac {1}{27} b m n x^3-\frac {e^2 m x \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac {1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {b e^3 m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{3 f^3}+\frac {e^3 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{3 f^3}-\frac {1}{9} b n x^3 \log \left (d (e+f x)^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {\left (b e^3 m n\right ) \int \frac {\log \left (-\frac {f x}{e}\right )}{e+f x} \, dx}{3 f^2}+\frac {1}{9} (b f m n) \int \frac {x^3}{e+f x} \, dx\\ &=\frac {b e^2 m n x}{3 f^2}-\frac {b e m n x^2}{12 f}+\frac {1}{27} b m n x^3-\frac {e^2 m x \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac {1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {b e^3 m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{3 f^3}+\frac {e^3 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{3 f^3}-\frac {1}{9} b n x^3 \log \left (d (e+f x)^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac {b e^3 m n \text {Li}_2\left (1+\frac {f x}{e}\right )}{3 f^3}+\frac {1}{9} (b f m n) \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 {4 b e^2 m n x}{9 f^2}-\frac {5 b e m n x^2}{36 f}+\frac {2}{27} b m n x^3-\frac {e^2 m x \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac {1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {b e^3 m n \log (e+f x)}{9 f^3}-\frac {b e^3 m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{3 f^3}+\frac {e^3 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{3 f^3}-\frac {1}{9} b n x^3 \log \left (d (e+f x)^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac {b e^3 m n \text {Li}_2\left (1+\frac {f x}{e}\right )}{3 f^3}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 252, normalized size = 1.04 \[ \frac {36 a f^3 x^3 \log \left (d (e+f x)^m\right )+36 a e^3 m \log (e+f x)-36 a e^2 f m x+18 a e f^2 m x^2-12 a f^3 m x^3-6 b \log \left (c x^n\right ) \left (-6 f^3 x^3 \log \left (d (e+f x)^m\right )-6 e^3 m \log (e+f x)+f m x \left (6 e^2-3 e f x+2 f^2 x^2\right )\right )-12 b f^3 n x^3 \log \left (d (e+f x)^m\right )+36 b e^3 m n \text {Li}_2\left (-\frac {f x}{e}\right )-12 b e^3 m n \log (e+f x)-36 b e^3 m n \log (x) \log (e+f x)+36 b e^3 m n \log (x) \log \left (\frac {f x}{e}+1\right )+48 b e^2 f m n x-15 b e f^2 m n x^2+8 b f^3 m n x^3}{108 f^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{2} \log \left (c x^{n}\right ) + a x^{2}\right )} \log \left ({\left (f x + e\right )}^{m} d\right ), x\right ) \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )} x^{2} \log \left ({\left (f x + e\right )}^{m} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.75, size = 2222, normalized size = 9.14 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.39, size = 328, normalized size = 1.35 \[ \frac {{\left (\log \left (\frac {f x}{e} + 1\right ) \log \relax (x) + {\rm Li}_2\left (-\frac {f x}{e}\right )\right )} b e^{3} m n}{3 \, f^{3}} + \frac {{\left (3 \, a e^{3} m - {\left (e^{3} m n - 3 \, e^{3} m \log \relax (c)\right )} b\right )} \log \left (f x + e\right )}{9 \, f^{3}} - \frac {36 \, b e^{3} m n \log \left (f x + e\right ) \log \relax (x) + 4 \, {\left (3 \, {\left (f^{3} m - 3 \, f^{3} \log \relax (d)\right )} a - {\left (2 \, f^{3} m n - 3 \, f^{3} n \log \relax (d) - 3 \, {\left (f^{3} m - 3 \, f^{3} \log \relax (d)\right )} \log \relax (c)\right )} b\right )} x^{3} - 3 \, {\left (6 \, a e f^{2} m - {\left (5 \, e f^{2} m n - 6 \, e f^{2} m \log \relax (c)\right )} b\right )} x^{2} + 12 \, {\left (3 \, a e^{2} f m - {\left (4 \, e^{2} f m n - 3 \, e^{2} f m \log \relax (c)\right )} b\right )} x - 12 \, {\left (3 \, b f^{3} x^{3} \log \left (x^{n}\right ) + {\left (3 \, a f^{3} - {\left (f^{3} n - 3 \, f^{3} \log \relax (c)\right )} b\right )} x^{3}\right )} \log \left ({\left (f x + e\right )}^{m}\right ) - 6 \, {\left (3 \, b e f^{2} m x^{2} - 6 \, b e^{2} f m x + 6 \, b e^{3} m \log \left (f x + e\right ) - 2 \, {\left (f^{3} m - 3 \, f^{3} \log \relax (d)\right )} b x^{3}\right )} \log \left (x^{n}\right )}{108 \, f^{3}} \]
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 \[ \int x^2\,\ln \left (d\,{\left (e+f\,x\right )}^m\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]
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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