Optimal. Leaf size=125 \[ \frac{a^2 x^{2 m} \log \left (c x^n\right )}{2 m}-\frac{a^2 n x^{2 m}}{4 m^2}+\frac{12 a b n^2 x^m \log \left (c x^n\right )}{m^3}-\frac{6 a b n x^m \log ^2\left (c x^n\right )}{m^2}+\frac{2 a b x^m \log ^3\left (c x^n\right )}{m}-\frac{12 a b n^3 x^m}{m^4}+\frac{b^2 \log ^6\left (c x^n\right )}{6 n} \]
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Rubi [A] time = 0.165895, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{a^2 x^{2 m} \log \left (c x^n\right )}{2 m}-\frac{a^2 n x^{2 m}}{4 m^2}+\frac{12 a b n^2 x^m \log \left (c x^n\right )}{m^3}-\frac{6 a b n x^m \log ^2\left (c x^n\right )}{m^2}+\frac{2 a b x^m \log ^3\left (c x^n\right )}{m}-\frac{12 a b n^3 x^m}{m^4}+\frac{b^2 \log ^6\left (c x^n\right )}{6 n} \]
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 )^2}{x} \, dx &=\int \left (a^2 x^{-1+2 m} \log \left (c x^n\right )+2 a b x^{-1+m} \log ^3\left (c x^n\right )+\frac{b^2 \log ^5\left (c x^n\right )}{x}\right ) \, dx\\ &=a^2 \int x^{-1+2 m} \log \left (c x^n\right ) \, dx+(2 a b) \int x^{-1+m} \log ^3\left (c x^n\right ) \, dx+b^2 \int \frac{\log ^5\left (c x^n\right )}{x} \, dx\\ &=-\frac{a^2 n x^{2 m}}{4 m^2}+\frac{a^2 x^{2 m} \log \left (c x^n\right )}{2 m}+\frac{2 a b x^m \log ^3\left (c x^n\right )}{m}+\frac{b^2 \operatorname{Subst}\left (\int x^5 \, dx,x,\log \left (c x^n\right )\right )}{n}-\frac{(6 a b n) \int x^{-1+m} \log ^2\left (c x^n\right ) \, dx}{m}\\ &=-\frac{a^2 n x^{2 m}}{4 m^2}+\frac{a^2 x^{2 m} \log \left (c x^n\right )}{2 m}-\frac{6 a b n x^m \log ^2\left (c x^n\right )}{m^2}+\frac{2 a b x^m \log ^3\left (c x^n\right )}{m}+\frac{b^2 \log ^6\left (c x^n\right )}{6 n}+\frac{\left (12 a b n^2\right ) \int x^{-1+m} \log \left (c x^n\right ) \, dx}{m^2}\\ &=-\frac{12 a b n^3 x^m}{m^4}-\frac{a^2 n x^{2 m}}{4 m^2}+\frac{12 a b n^2 x^m \log \left (c x^n\right )}{m^3}+\frac{a^2 x^{2 m} \log \left (c x^n\right )}{2 m}-\frac{6 a b n x^m \log ^2\left (c x^n\right )}{m^2}+\frac{2 a b x^m \log ^3\left (c x^n\right )}{m}+\frac{b^2 \log ^6\left (c x^n\right )}{6 n}\\ \end{align*}
Mathematica [A] time = 0.104551, size = 115, normalized size = 0.92 \[ \frac{a x^m \log \left (c x^n\right ) \left (a m^2 x^m+24 b n^2\right )}{2 m^3}-\frac{6 a b n x^m \log ^2\left (c x^n\right )}{m^2}+\frac{2 a b x^m \log ^3\left (c x^n\right )}{m}-\frac{a n x^m \left (a m^2 x^m+48 b n^2\right )}{4 m^4}+\frac{b^2 \log ^6\left (c x^n\right )}{6 n} \]
Antiderivative was successfully verified.
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Maple [C] time = 1.375, size = 14983, normalized size = 119.9 \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 = 1.85593, size = 659, normalized size = 5.27 \begin{align*} \frac{2 \, b^{2} m^{4} n^{5} \log \left (x\right )^{6} + 12 \, b^{2} m^{4} n^{4} \log \left (c\right ) \log \left (x\right )^{5} + 30 \, b^{2} m^{4} n^{3} \log \left (c\right )^{2} \log \left (x\right )^{4} + 40 \, b^{2} m^{4} n^{2} \log \left (c\right )^{3} \log \left (x\right )^{3} + 30 \, b^{2} m^{4} n \log \left (c\right )^{4} \log \left (x\right )^{2} + 12 \, b^{2} m^{4} \log \left (c\right )^{5} \log \left (x\right ) + 3 \,{\left (2 \, a^{2} m^{3} n \log \left (x\right ) + 2 \, a^{2} m^{3} \log \left (c\right ) - a^{2} m^{2} n\right )} x^{2 \, m} + 24 \,{\left (a b m^{3} n^{3} \log \left (x\right )^{3} + a b m^{3} \log \left (c\right )^{3} - 3 \, a b m^{2} n \log \left (c\right )^{2} + 6 \, a b m n^{2} \log \left (c\right ) - 6 \, a b n^{3} + 3 \,{\left (a b m^{3} n^{2} \log \left (c\right ) - a b m^{2} n^{3}\right )} \log \left (x\right )^{2} + 3 \,{\left (a b m^{3} n \log \left (c\right )^{2} - 2 \, a b m^{2} n^{2} \log \left (c\right ) + 2 \, a b m n^{3}\right )} \log \left (x\right )\right )} x^{m}}{12 \, m^{4}} \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 = 1.65729, size = 386, normalized size = 3.09 \begin{align*} \frac{1}{6} \, b^{2} n^{5} \log \left (x\right )^{6} + b^{2} n^{4} \log \left (c\right ) \log \left (x\right )^{5} + \frac{5}{2} \, b^{2} n^{3} \log \left (c\right )^{2} \log \left (x\right )^{4} + \frac{10}{3} \, b^{2} n^{2} \log \left (c\right )^{3} \log \left (x\right )^{3} + \frac{5}{2} \, b^{2} n \log \left (c\right )^{4} \log \left (x\right )^{2} + b^{2} \log \left (c\right )^{5} \log \left (x\right ) + \frac{2 \, a b n^{3} x^{m} \log \left (x\right )^{3}}{m} + \frac{6 \, a b n^{2} x^{m} \log \left (c\right ) \log \left (x\right )^{2}}{m} + \frac{6 \, a b n x^{m} \log \left (c\right )^{2} \log \left (x\right )}{m} - \frac{6 \, a b n^{3} x^{m} \log \left (x\right )^{2}}{m^{2}} + \frac{2 \, a b x^{m} \log \left (c\right )^{3}}{m} - \frac{12 \, a b n^{2} x^{m} \log \left (c\right ) \log \left (x\right )}{m^{2}} - \frac{6 \, a b n x^{m} \log \left (c\right )^{2}}{m^{2}} + \frac{a^{2} n x^{2 \, m} \log \left (x\right )}{2 \, m} + \frac{12 \, a b n^{3} x^{m} \log \left (x\right )}{m^{3}} + \frac{a^{2} x^{2 \, m} \log \left (c\right )}{2 \, m} + \frac{12 \, a b n^{2} x^{m} \log \left (c\right )}{m^{3}} - \frac{a^{2} n x^{2 \, m}}{4 \, m^{2}} - \frac{12 \, a b n^{3} x^{m}}{m^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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