Optimal. Leaf size=52 \[ \frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{32} b^2 n^2 x^4 \]
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Rubi [A] time = 0.0362843, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2305, 2304} \[ \frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{32} b^2 n^2 x^4 \]
Antiderivative was successfully verified.
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Rule 2305
Rule 2304
Rubi steps
\begin{align*} \int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx &=\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{2} (b n) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=\frac{1}{32} b^2 n^2 x^4-\frac{1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2\\ \end{align*}
Mathematica [A] time = 0.0161885, size = 43, normalized size = 0.83 \[ \frac{1}{32} x^4 \left (-4 b n \left (a+b \log \left (c x^n\right )\right )+8 \left (a+b \log \left (c x^n\right )\right )^2+b^2 n^2\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.191, size = 691, normalized size = 13.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12839, size = 96, normalized size = 1.85 \begin{align*} \frac{1}{4} \, b^{2} x^{4} \log \left (c x^{n}\right )^{2} - \frac{1}{8} \, a b n x^{4} + \frac{1}{2} \, a b x^{4} \log \left (c x^{n}\right ) + \frac{1}{4} \, a^{2} x^{4} + \frac{1}{32} \,{\left (n^{2} x^{4} - 4 \, n x^{4} \log \left (c x^{n}\right )\right )} b^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.890636, size = 244, normalized size = 4.69 \begin{align*} \frac{1}{4} \, b^{2} n^{2} x^{4} \log \left (x\right )^{2} + \frac{1}{4} \, b^{2} x^{4} \log \left (c\right )^{2} - \frac{1}{8} \,{\left (b^{2} n - 4 \, a b\right )} x^{4} \log \left (c\right ) + \frac{1}{32} \,{\left (b^{2} n^{2} - 4 \, a b n + 8 \, a^{2}\right )} x^{4} + \frac{1}{8} \,{\left (4 \, b^{2} n x^{4} \log \left (c\right ) -{\left (b^{2} n^{2} - 4 \, a b n\right )} x^{4}\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.9114, size = 131, normalized size = 2.52 \begin{align*} \frac{a^{2} x^{4}}{4} + \frac{a b n x^{4} \log{\left (x \right )}}{2} - \frac{a b n x^{4}}{8} + \frac{a b x^{4} \log{\left (c \right )}}{2} + \frac{b^{2} n^{2} x^{4} \log{\left (x \right )}^{2}}{4} - \frac{b^{2} n^{2} x^{4} \log{\left (x \right )}}{8} + \frac{b^{2} n^{2} x^{4}}{32} + \frac{b^{2} n x^{4} \log{\left (c \right )} \log{\left (x \right )}}{2} - \frac{b^{2} n x^{4} \log{\left (c \right )}}{8} + \frac{b^{2} x^{4} \log{\left (c \right )}^{2}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15678, size = 150, normalized size = 2.88 \begin{align*} \frac{1}{4} \, b^{2} n^{2} x^{4} \log \left (x\right )^{2} - \frac{1}{8} \, b^{2} n^{2} x^{4} \log \left (x\right ) + \frac{1}{2} \, b^{2} n x^{4} \log \left (c\right ) \log \left (x\right ) + \frac{1}{32} \, b^{2} n^{2} x^{4} - \frac{1}{8} \, b^{2} n x^{4} \log \left (c\right ) + \frac{1}{4} \, b^{2} x^{4} \log \left (c\right )^{2} + \frac{1}{2} \, a b n x^{4} \log \left (x\right ) - \frac{1}{8} \, a b n x^{4} + \frac{1}{2} \, a b x^{4} \log \left (c\right ) + \frac{1}{4} \, a^{2} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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