Optimal. Leaf size=52 \[ \frac{1}{3} x^3 \log ^2\left (c \left (b x^n\right )^p\right )-\frac{2}{9} n p x^3 \log \left (c \left (b x^n\right )^p\right )+\frac{2}{27} n^2 p^2 x^3 \]
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Rubi [A] time = 0.0708018, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2305, 2304, 2445} \[ \frac{1}{3} x^3 \log ^2\left (c \left (b x^n\right )^p\right )-\frac{2}{9} n p x^3 \log \left (c \left (b x^n\right )^p\right )+\frac{2}{27} n^2 p^2 x^3 \]
Antiderivative was successfully verified.
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Rule 2305
Rule 2304
Rule 2445
Rubi steps
\begin{align*} \int x^2 \log ^2\left (c \left (b x^n\right )^p\right ) \, dx &=\operatorname{Subst}\left (\int x^2 \log ^2\left (b^p c x^{n p}\right ) \, dx,b^p c x^{n p},c \left (b x^n\right )^p\right )\\ &=\frac{1}{3} x^3 \log ^2\left (c \left (b x^n\right )^p\right )-\operatorname{Subst}\left (\frac{1}{3} (2 n p) \int x^2 \log \left (b^p c x^{n p}\right ) \, dx,b^p c x^{n p},c \left (b x^n\right )^p\right )\\ &=\frac{2}{27} n^2 p^2 x^3-\frac{2}{9} n p x^3 \log \left (c \left (b x^n\right )^p\right )+\frac{1}{3} x^3 \log ^2\left (c \left (b x^n\right )^p\right )\\ \end{align*}
Mathematica [A] time = 0.0027718, size = 52, normalized size = 1. \[ \frac{1}{3} x^3 \log ^2\left (c \left (b x^n\right )^p\right )-\frac{2}{9} n p x^3 \log \left (c \left (b x^n\right )^p\right )+\frac{2}{27} n^2 p^2 x^3 \]
Antiderivative was successfully verified.
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Maple [F] time = 0.028, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( \ln \left ( c \left ( b{x}^{n} \right ) ^{p} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16307, size = 62, normalized size = 1.19 \begin{align*} \frac{2}{27} \, n^{2} p^{2} x^{3} - \frac{2}{9} \, n p x^{3} \log \left (\left (b x^{n}\right )^{p} c\right ) + \frac{1}{3} \, x^{3} \log \left (\left (b x^{n}\right )^{p} c\right )^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.734404, size = 293, normalized size = 5.63 \begin{align*} \frac{1}{3} \, n^{2} p^{2} x^{3} \log \left (x\right )^{2} + \frac{2}{27} \, n^{2} p^{2} x^{3} - \frac{2}{9} \, n p^{2} x^{3} \log \left (b\right ) + \frac{1}{3} \, p^{2} x^{3} \log \left (b\right )^{2} + \frac{1}{3} \, x^{3} \log \left (c\right )^{2} - \frac{2}{9} \,{\left (n p x^{3} - 3 \, p x^{3} \log \left (b\right )\right )} \log \left (c\right ) - \frac{2}{9} \,{\left (n^{2} p^{2} x^{3} - 3 \, n p^{2} x^{3} \log \left (b\right ) - 3 \, n p x^{3} \log \left (c\right )\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 8.06876, size = 150, normalized size = 2.88 \begin{align*} \frac{n^{2} p^{2} x^{3} \log{\left (x \right )}^{2}}{3} - \frac{2 n^{2} p^{2} x^{3} \log{\left (x \right )}}{9} + \frac{2 n^{2} p^{2} x^{3}}{27} + \frac{2 n p^{2} x^{3} \log{\left (b \right )} \log{\left (x \right )}}{3} - \frac{2 n p^{2} x^{3} \log{\left (b \right )}}{9} + \frac{2 n p x^{3} \log{\left (c \right )} \log{\left (x \right )}}{3} - \frac{2 n p x^{3} \log{\left (c \right )}}{9} + \frac{p^{2} x^{3} \log{\left (b \right )}^{2}}{3} + \frac{2 p x^{3} \log{\left (b \right )} \log{\left (c \right )}}{3} + \frac{x^{3} \log{\left (c \right )}^{2}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.30326, size = 155, normalized size = 2.98 \begin{align*} \frac{1}{3} \, n^{2} p^{2} x^{3} \log \left (x\right )^{2} - \frac{2}{9} \, n^{2} p^{2} x^{3} \log \left (x\right ) + \frac{2}{3} \, n p^{2} x^{3} \log \left (b\right ) \log \left (x\right ) + \frac{2}{27} \, n^{2} p^{2} x^{3} - \frac{2}{9} \, n p^{2} x^{3} \log \left (b\right ) + \frac{1}{3} \, p^{2} x^{3} \log \left (b\right )^{2} + \frac{2}{3} \, n p x^{3} \log \left (c\right ) \log \left (x\right ) - \frac{2}{9} \, n p x^{3} \log \left (c\right ) + \frac{2}{3} \, p x^{3} \log \left (b\right ) \log \left (c\right ) + \frac{1}{3} \, x^{3} \log \left (c\right )^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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