Optimal. Leaf size=272 \[ \frac {a^3 x^{3 m} \log \left (c x^n\right )}{3 m}-\frac {a^3 n x^{3 m}}{9 m^2}+\frac {9 a^2 b n^2 x^{2 m} \log \left (c x^n\right )}{4 m^3}-\frac {9 a^2 b n x^{2 m} \log ^2\left (c x^n\right )}{4 m^2}+\frac {3 a^2 b x^{2 m} \log ^3\left (c x^n\right )}{2 m}-\frac {9 a^2 b n^3 x^{2 m}}{8 m^4}+\frac {360 a b^2 n^4 x^m \log \left (c x^n\right )}{m^5}-\frac {180 a b^2 n^3 x^m \log ^2\left (c x^n\right )}{m^4}+\frac {60 a b^2 n^2 x^m \log ^3\left (c x^n\right )}{m^3}-\frac {15 a b^2 n x^m \log ^4\left (c x^n\right )}{m^2}+\frac {3 a b^2 x^m \log ^5\left (c x^n\right )}{m}-\frac {360 a b^2 n^5 x^m}{m^6}+\frac {b^3 \log ^8\left (c x^n\right )}{8 n} \]
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Rubi [A] time = 0.31, antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2539, 2304, 2305, 2302, 30} \[ \frac {9 a^2 b n^2 x^{2 m} \log \left (c x^n\right )}{4 m^3}-\frac {9 a^2 b n x^{2 m} \log ^2\left (c x^n\right )}{4 m^2}+\frac {3 a^2 b x^{2 m} \log ^3\left (c x^n\right )}{2 m}-\frac {9 a^2 b n^3 x^{2 m}}{8 m^4}+\frac {a^3 x^{3 m} \log \left (c x^n\right )}{3 m}-\frac {a^3 n x^{3 m}}{9 m^2}+\frac {60 a b^2 n^2 x^m \log ^3\left (c x^n\right )}{m^3}-\frac {180 a b^2 n^3 x^m \log ^2\left (c x^n\right )}{m^4}+\frac {360 a b^2 n^4 x^m \log \left (c x^n\right )}{m^5}-\frac {15 a b^2 n x^m \log ^4\left (c x^n\right )}{m^2}+\frac {3 a b^2 x^m \log ^5\left (c x^n\right )}{m}-\frac {360 a b^2 n^5 x^m}{m^6}+\frac {b^3 \log ^8\left (c x^n\right )}{8 n} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2302
Rule 2304
Rule 2305
Rule 2539
Rubi steps
\begin {align*} \int \frac {\log \left (c x^n\right ) \left (a x^m+b \log ^2\left (c x^n\right )\right )^3}{x} \, dx &=\int \left (a^3 x^{-1+3 m} \log \left (c x^n\right )+3 a^2 b x^{-1+2 m} \log ^3\left (c x^n\right )+3 a b^2 x^{-1+m} \log ^5\left (c x^n\right )+\frac {b^3 \log ^7\left (c x^n\right )}{x}\right ) \, dx\\ &=a^3 \int x^{-1+3 m} \log \left (c x^n\right ) \, dx+\left (3 a^2 b\right ) \int x^{-1+2 m} \log ^3\left (c x^n\right ) \, dx+\left (3 a b^2\right ) \int x^{-1+m} \log ^5\left (c x^n\right ) \, dx+b^3 \int \frac {\log ^7\left (c x^n\right )}{x} \, dx\\ &=-\frac {a^3 n x^{3 m}}{9 m^2}+\frac {a^3 x^{3 m} \log \left (c x^n\right )}{3 m}+\frac {3 a^2 b x^{2 m} \log ^3\left (c x^n\right )}{2 m}+\frac {3 a b^2 x^m \log ^5\left (c x^n\right )}{m}+\frac {b^3 \operatorname {Subst}\left (\int x^7 \, dx,x,\log \left (c x^n\right )\right )}{n}-\frac {\left (9 a^2 b n\right ) \int x^{-1+2 m} \log ^2\left (c x^n\right ) \, dx}{2 m}-\frac {\left (15 a b^2 n\right ) \int x^{-1+m} \log ^4\left (c x^n\right ) \, dx}{m}\\ &=-\frac {a^3 n x^{3 m}}{9 m^2}+\frac {a^3 x^{3 m} \log \left (c x^n\right )}{3 m}-\frac {9 a^2 b n x^{2 m} \log ^2\left (c x^n\right )}{4 m^2}+\frac {3 a^2 b x^{2 m} \log ^3\left (c x^n\right )}{2 m}-\frac {15 a b^2 n x^m \log ^4\left (c x^n\right )}{m^2}+\frac {3 a b^2 x^m \log ^5\left (c x^n\right )}{m}+\frac {b^3 \log ^8\left (c x^n\right )}{8 n}+\frac {\left (9 a^2 b n^2\right ) \int x^{-1+2 m} \log \left (c x^n\right ) \, dx}{2 m^2}+\frac {\left (60 a b^2 n^2\right ) \int x^{-1+m} \log ^3\left (c x^n\right ) \, dx}{m^2}\\ &=-\frac {9 a^2 b n^3 x^{2 m}}{8 m^4}-\frac {a^3 n x^{3 m}}{9 m^2}+\frac {9 a^2 b n^2 x^{2 m} \log \left (c x^n\right )}{4 m^3}+\frac {a^3 x^{3 m} \log \left (c x^n\right )}{3 m}-\frac {9 a^2 b n x^{2 m} \log ^2\left (c x^n\right )}{4 m^2}+\frac {60 a b^2 n^2 x^m \log ^3\left (c x^n\right )}{m^3}+\frac {3 a^2 b x^{2 m} \log ^3\left (c x^n\right )}{2 m}-\frac {15 a b^2 n x^m \log ^4\left (c x^n\right )}{m^2}+\frac {3 a b^2 x^m \log ^5\left (c x^n\right )}{m}+\frac {b^3 \log ^8\left (c x^n\right )}{8 n}-\frac {\left (180 a b^2 n^3\right ) \int x^{-1+m} \log ^2\left (c x^n\right ) \, dx}{m^3}\\ &=-\frac {9 a^2 b n^3 x^{2 m}}{8 m^4}-\frac {a^3 n x^{3 m}}{9 m^2}+\frac {9 a^2 b n^2 x^{2 m} \log \left (c x^n\right )}{4 m^3}+\frac {a^3 x^{3 m} \log \left (c x^n\right )}{3 m}-\frac {180 a b^2 n^3 x^m \log ^2\left (c x^n\right )}{m^4}-\frac {9 a^2 b n x^{2 m} \log ^2\left (c x^n\right )}{4 m^2}+\frac {60 a b^2 n^2 x^m \log ^3\left (c x^n\right )}{m^3}+\frac {3 a^2 b x^{2 m} \log ^3\left (c x^n\right )}{2 m}-\frac {15 a b^2 n x^m \log ^4\left (c x^n\right )}{m^2}+\frac {3 a b^2 x^m \log ^5\left (c x^n\right )}{m}+\frac {b^3 \log ^8\left (c x^n\right )}{8 n}+\frac {\left (360 a b^2 n^4\right ) \int x^{-1+m} \log \left (c x^n\right ) \, dx}{m^4}\\ &=-\frac {360 a b^2 n^5 x^m}{m^6}-\frac {9 a^2 b n^3 x^{2 m}}{8 m^4}-\frac {a^3 n x^{3 m}}{9 m^2}+\frac {360 a b^2 n^4 x^m \log \left (c x^n\right )}{m^5}+\frac {9 a^2 b n^2 x^{2 m} \log \left (c x^n\right )}{4 m^3}+\frac {a^3 x^{3 m} \log \left (c x^n\right )}{3 m}-\frac {180 a b^2 n^3 x^m \log ^2\left (c x^n\right )}{m^4}-\frac {9 a^2 b n x^{2 m} \log ^2\left (c x^n\right )}{4 m^2}+\frac {60 a b^2 n^2 x^m \log ^3\left (c x^n\right )}{m^3}+\frac {3 a^2 b x^{2 m} \log ^3\left (c x^n\right )}{2 m}-\frac {15 a b^2 n x^m \log ^4\left (c x^n\right )}{m^2}+\frac {3 a b^2 x^m \log ^5\left (c x^n\right )}{m}+\frac {b^3 \log ^8\left (c x^n\right )}{8 n}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 230, normalized size = 0.85 \[ \frac {a x^m \log \left (c x^n\right ) \left (4 a^2 m^4 x^{2 m}+27 a b m^2 n^2 x^m+4320 b^2 n^4\right )}{12 m^5}-\frac {a n x^m \left (8 a^2 m^4 x^{2 m}+81 a b m^2 n^2 x^m+25920 b^2 n^4\right )}{72 m^6}-\frac {15 a b^2 n x^m \log ^4\left (c x^n\right )}{m^2}+\frac {3 a b^2 x^m \log ^5\left (c x^n\right )}{m}-\frac {9 a b n x^m \log ^2\left (c x^n\right ) \left (a m^2 x^m+80 b n^2\right )}{4 m^4}+\frac {3 a b x^m \log ^3\left (c x^n\right ) \left (a m^2 x^m+40 b n^2\right )}{2 m^3}+\frac {b^3 \log ^8\left (c x^n\right )}{8 n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 655, normalized size = 2.41 \[ \frac {9 \, b^{3} m^{6} n^{7} \log \relax (x)^{8} + 72 \, b^{3} m^{6} n^{6} \log \relax (c) \log \relax (x)^{7} + 252 \, b^{3} m^{6} n^{5} \log \relax (c)^{2} \log \relax (x)^{6} + 504 \, b^{3} m^{6} n^{4} \log \relax (c)^{3} \log \relax (x)^{5} + 630 \, b^{3} m^{6} n^{3} \log \relax (c)^{4} \log \relax (x)^{4} + 504 \, b^{3} m^{6} n^{2} \log \relax (c)^{5} \log \relax (x)^{3} + 252 \, b^{3} m^{6} n \log \relax (c)^{6} \log \relax (x)^{2} + 72 \, b^{3} m^{6} \log \relax (c)^{7} \log \relax (x) + 8 \, {\left (3 \, a^{3} m^{5} n \log \relax (x) + 3 \, a^{3} m^{5} \log \relax (c) - a^{3} m^{4} n\right )} x^{3 \, m} + 27 \, {\left (4 \, a^{2} b m^{5} n^{3} \log \relax (x)^{3} + 4 \, a^{2} b m^{5} \log \relax (c)^{3} - 6 \, a^{2} b m^{4} n \log \relax (c)^{2} + 6 \, a^{2} b m^{3} n^{2} \log \relax (c) - 3 \, a^{2} b m^{2} n^{3} + 6 \, {\left (2 \, a^{2} b m^{5} n^{2} \log \relax (c) - a^{2} b m^{4} n^{3}\right )} \log \relax (x)^{2} + 6 \, {\left (2 \, a^{2} b m^{5} n \log \relax (c)^{2} - 2 \, a^{2} b m^{4} n^{2} \log \relax (c) + a^{2} b m^{3} n^{3}\right )} \log \relax (x)\right )} x^{2 \, m} + 216 \, {\left (a b^{2} m^{5} n^{5} \log \relax (x)^{5} + a b^{2} m^{5} \log \relax (c)^{5} - 5 \, a b^{2} m^{4} n \log \relax (c)^{4} + 20 \, a b^{2} m^{3} n^{2} \log \relax (c)^{3} - 60 \, a b^{2} m^{2} n^{3} \log \relax (c)^{2} + 120 \, a b^{2} m n^{4} \log \relax (c) - 120 \, a b^{2} n^{5} + 5 \, {\left (a b^{2} m^{5} n^{4} \log \relax (c) - a b^{2} m^{4} n^{5}\right )} \log \relax (x)^{4} + 10 \, {\left (a b^{2} m^{5} n^{3} \log \relax (c)^{2} - 2 \, a b^{2} m^{4} n^{4} \log \relax (c) + 2 \, a b^{2} m^{3} n^{5}\right )} \log \relax (x)^{3} + 10 \, {\left (a b^{2} m^{5} n^{2} \log \relax (c)^{3} - 3 \, a b^{2} m^{4} n^{3} \log \relax (c)^{2} + 6 \, a b^{2} m^{3} n^{4} \log \relax (c) - 6 \, a b^{2} m^{2} n^{5}\right )} \log \relax (x)^{2} + 5 \, {\left (a b^{2} m^{5} n \log \relax (c)^{4} - 4 \, a b^{2} m^{4} n^{2} \log \relax (c)^{3} + 12 \, a b^{2} m^{3} n^{3} \log \relax (c)^{2} - 24 \, a b^{2} m^{2} n^{4} \log \relax (c) + 24 \, a b^{2} m n^{5}\right )} \log \relax (x)\right )} x^{m}}{72 \, m^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 766, normalized size = 2.82 \[ \frac {1}{8} \, b^{3} n^{7} \log \relax (x)^{8} + b^{3} n^{6} \log \relax (c) \log \relax (x)^{7} + \frac {7}{2} \, b^{3} n^{5} \log \relax (c)^{2} \log \relax (x)^{6} + 7 \, b^{3} n^{4} \log \relax (c)^{3} \log \relax (x)^{5} + \frac {35}{4} \, b^{3} n^{3} \log \relax (c)^{4} \log \relax (x)^{4} + 7 \, b^{3} n^{2} \log \relax (c)^{5} \log \relax (x)^{3} + \frac {3 \, a b^{2} n^{5} x^{m} \log \relax (x)^{5}}{m} + \frac {7}{2} \, b^{3} n \log \relax (c)^{6} \log \relax (x)^{2} + \frac {15 \, a b^{2} n^{4} x^{m} \log \relax (c) \log \relax (x)^{4}}{m} + b^{3} \log \relax (c)^{7} \log \relax (x) + \frac {30 \, a b^{2} n^{3} x^{m} \log \relax (c)^{2} \log \relax (x)^{3}}{m} - \frac {15 \, a b^{2} n^{5} x^{m} \log \relax (x)^{4}}{m^{2}} + \frac {30 \, a b^{2} n^{2} x^{m} \log \relax (c)^{3} \log \relax (x)^{2}}{m} - \frac {60 \, a b^{2} n^{4} x^{m} \log \relax (c) \log \relax (x)^{3}}{m^{2}} + \frac {15 \, a b^{2} n x^{m} \log \relax (c)^{4} \log \relax (x)}{m} - \frac {90 \, a b^{2} n^{3} x^{m} \log \relax (c)^{2} \log \relax (x)^{2}}{m^{2}} + \frac {3 \, a^{2} b n^{3} x^{2 \, m} \log \relax (x)^{3}}{2 \, m} + \frac {60 \, a b^{2} n^{5} x^{m} \log \relax (x)^{3}}{m^{3}} + \frac {3 \, a b^{2} x^{m} \log \relax (c)^{5}}{m} - \frac {60 \, a b^{2} n^{2} x^{m} \log \relax (c)^{3} \log \relax (x)}{m^{2}} + \frac {9 \, a^{2} b n^{2} x^{2 \, m} \log \relax (c) \log \relax (x)^{2}}{2 \, m} + \frac {180 \, a b^{2} n^{4} x^{m} \log \relax (c) \log \relax (x)^{2}}{m^{3}} - \frac {15 \, a b^{2} n x^{m} \log \relax (c)^{4}}{m^{2}} + \frac {9 \, a^{2} b n x^{2 \, m} \log \relax (c)^{2} \log \relax (x)}{2 \, m} + \frac {180 \, a b^{2} n^{3} x^{m} \log \relax (c)^{2} \log \relax (x)}{m^{3}} - \frac {9 \, a^{2} b n^{3} x^{2 \, m} \log \relax (x)^{2}}{4 \, m^{2}} - \frac {180 \, a b^{2} n^{5} x^{m} \log \relax (x)^{2}}{m^{4}} + \frac {3 \, a^{2} b x^{2 \, m} \log \relax (c)^{3}}{2 \, m} + \frac {60 \, a b^{2} n^{2} x^{m} \log \relax (c)^{3}}{m^{3}} - \frac {9 \, a^{2} b n^{2} x^{2 \, m} \log \relax (c) \log \relax (x)}{2 \, m^{2}} - \frac {360 \, a b^{2} n^{4} x^{m} \log \relax (c) \log \relax (x)}{m^{4}} - \frac {9 \, a^{2} b n x^{2 \, m} \log \relax (c)^{2}}{4 \, m^{2}} - \frac {180 \, a b^{2} n^{3} x^{m} \log \relax (c)^{2}}{m^{4}} + \frac {a^{3} n x^{3 \, m} \log \relax (x)}{3 \, m} + \frac {9 \, a^{2} b n^{3} x^{2 \, m} \log \relax (x)}{4 \, m^{3}} + \frac {360 \, a b^{2} n^{5} x^{m} \log \relax (x)}{m^{5}} + \frac {a^{3} x^{3 \, m} \log \relax (c)}{3 \, m} + \frac {9 \, a^{2} b n^{2} x^{2 \, m} \log \relax (c)}{4 \, m^{3}} + \frac {360 \, a b^{2} n^{4} x^{m} \log \relax (c)}{m^{5}} - \frac {a^{3} n x^{3 \, m}}{9 \, m^{2}} - \frac {9 \, a^{2} b n^{3} x^{2 \, m}}{8 \, m^{4}} - \frac {360 \, a b^{2} n^{5} x^{m}}{m^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 9.06, size = 61910, normalized size = 227.61 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.94, size = 1115, normalized size = 4.10 \[ \text {result too large to display} \]
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 \[ \int \frac {\ln \left (c\,x^n\right )\,{\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^2\right )}^3}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 103.52, size = 762, normalized size = 2.80 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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