Optimal. Leaf size=64 \[ \frac{1}{4} x^4 \sqrt{\log \left (a x^n\right )}-\frac{1}{16} \sqrt{\pi } \sqrt{n} x^4 \left (a x^n\right )^{-4/n} \text{Erfi}\left (\frac{2 \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right ) \]
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Rubi [A] time = 0.0537024, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2305, 2310, 2180, 2204} \[ \frac{1}{4} x^4 \sqrt{\log \left (a x^n\right )}-\frac{1}{16} \sqrt{\pi } \sqrt{n} x^4 \left (a x^n\right )^{-4/n} \text{Erfi}\left (\frac{2 \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right ) \]
Antiderivative was successfully verified.
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Rule 2305
Rule 2310
Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int x^3 \sqrt{\log \left (a x^n\right )} \, dx &=\frac{1}{4} x^4 \sqrt{\log \left (a x^n\right )}-\frac{1}{8} n \int \frac{x^3}{\sqrt{\log \left (a x^n\right )}} \, dx\\ &=\frac{1}{4} x^4 \sqrt{\log \left (a x^n\right )}-\frac{1}{8} \left (x^4 \left (a x^n\right )^{-4/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{4 x}{n}}}{\sqrt{x}} \, dx,x,\log \left (a x^n\right )\right )\\ &=\frac{1}{4} x^4 \sqrt{\log \left (a x^n\right )}-\frac{1}{4} \left (x^4 \left (a x^n\right )^{-4/n}\right ) \operatorname{Subst}\left (\int e^{\frac{4 x^2}{n}} \, dx,x,\sqrt{\log \left (a x^n\right )}\right )\\ &=-\frac{1}{16} \sqrt{n} \sqrt{\pi } x^4 \left (a x^n\right )^{-4/n} \text{erfi}\left (\frac{2 \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )+\frac{1}{4} x^4 \sqrt{\log \left (a x^n\right )}\\ \end{align*}
Mathematica [A] time = 0.0221965, size = 61, normalized size = 0.95 \[ \frac{1}{16} x^4 \left (4 \sqrt{\log \left (a x^n\right )}-\sqrt{\pi } \sqrt{n} \left (a x^n\right )^{-4/n} \text{Erfi}\left (\frac{2 \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.293, size = 0, normalized size = 0. \begin{align*} \int{x}^{3}\sqrt{\ln \left ( a{x}^{n} \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \sqrt{\log \left (a x^{n}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \sqrt{\log{\left (a x^{n} \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \sqrt{\log \left (a x^{n}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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