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 = 0.05, antiderivative size = 64, normalized size of antiderivative = 1.00, 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 2180
Rule 2204
Rule 2305
Rule 2310
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*}
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Mathematica [A] time = 0.02, 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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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \sqrt {\log \left (a x^{n}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.65, size = 0, normalized size = 0.00 \[ \int x^{3} \sqrt {\ln \left (a \,x^{n}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \sqrt {\log \left (a x^{n}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^3\,\sqrt {\ln \left (a\,x^n\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \sqrt {\log {\left (a x^{n} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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