Optimal. Leaf size=272 \[ \frac{9 a^2 b n^2 x^{2 m} \log \left (c x^n\right )}{4 m^3}-\frac{9 a^2 b n x^{2 m} \log ^2\left (c x^n\right )}{4 m^2}+\frac{3 a^2 b x^{2 m} \log ^3\left (c x^n\right )}{2 m}-\frac{9 a^2 b n^3 x^{2 m}}{8 m^4}+\frac{a^3 x^{3 m} \log \left (c x^n\right )}{3 m}-\frac{a^3 n x^{3 m}}{9 m^2}+\frac{60 a b^2 n^2 x^m \log ^3\left (c x^n\right )}{m^3}-\frac{180 a b^2 n^3 x^m \log ^2\left (c x^n\right )}{m^4}+\frac{360 a b^2 n^4 x^m \log \left (c x^n\right )}{m^5}-\frac{15 a b^2 n x^m \log ^4\left (c x^n\right )}{m^2}+\frac{3 a b^2 x^m \log ^5\left (c x^n\right )}{m}-\frac{360 a b^2 n^5 x^m}{m^6}+\frac{b^3 \log ^8\left (c x^n\right )}{8 n} \]
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Rubi [A] time = 0.305071, antiderivative size = 272, normalized size of antiderivative = 1., 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 2539
Rule 2304
Rule 2305
Rule 2302
Rule 30
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*}
Mathematica [A] time = 0.223594, size = 230, normalized size = 0.85 \[ \frac{a x^m \log \left (c x^n\right ) \left (4 a^2 m^4 x^{2 m}+27 a b m^2 n^2 x^m+4320 b^2 n^4\right )}{12 m^5}-\frac{a n x^m \left (8 a^2 m^4 x^{2 m}+81 a b m^2 n^2 x^m+25920 b^2 n^4\right )}{72 m^6}-\frac{15 a b^2 n x^m \log ^4\left (c x^n\right )}{m^2}+\frac{3 a b^2 x^m \log ^5\left (c x^n\right )}{m}+\frac{3 a b x^m \log ^3\left (c x^n\right ) \left (a m^2 x^m+40 b n^2\right )}{2 m^3}-\frac{9 a b n x^m \log ^2\left (c x^n\right ) \left (a m^2 x^m+80 b n^2\right )}{4 m^4}+\frac{b^3 \log ^8\left (c x^n\right )}{8 n} \]
Antiderivative was successfully verified.
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Maple [C] time = 5.05, size = 61910, normalized size = 227.6 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.16194, size = 1558, normalized size = 5.73 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.34126, size = 1034, normalized size = 3.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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