Optimal. Leaf size=69 \[ \frac {2 \sqrt {3 \pi } x^3 \left (a x^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{n^{3/2}}-\frac {2 x^3}{n \sqrt {\log \left (a x^n\right )}} \]
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Rubi [A] time = 0.06, antiderivative size = 69, 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 = {2306, 2310, 2180, 2204} \[ \frac {2 \sqrt {3 \pi } x^3 \left (a x^n\right )^{-3/n} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{n^{3/2}}-\frac {2 x^3}{n \sqrt {\log \left (a x^n\right )}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2306
Rule 2310
Rubi steps
\begin {align*} \int \frac {x^2}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx &=-\frac {2 x^3}{n \sqrt {\log \left (a x^n\right )}}+\frac {6 \int \frac {x^2}{\sqrt {\log \left (a x^n\right )}} \, dx}{n}\\ &=-\frac {2 x^3}{n \sqrt {\log \left (a x^n\right )}}+\frac {\left (6 x^3 \left (a x^n\right )^{-3/n}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {3 x}{n}}}{\sqrt {x}} \, dx,x,\log \left (a x^n\right )\right )}{n^2}\\ &=-\frac {2 x^3}{n \sqrt {\log \left (a x^n\right )}}+\frac {\left (12 x^3 \left (a x^n\right )^{-3/n}\right ) \operatorname {Subst}\left (\int e^{\frac {3 x^2}{n}} \, dx,x,\sqrt {\log \left (a x^n\right )}\right )}{n^2}\\ &=\frac {2 \sqrt {3 \pi } x^3 \left (a x^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{n^{3/2}}-\frac {2 x^3}{n \sqrt {\log \left (a x^n\right )}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 78, normalized size = 1.13 \[ -\frac {2 x^3 \left (a x^n\right )^{-3/n} \left (\left (a x^n\right )^{3/n}-\sqrt {3} \sqrt {-\frac {\log \left (a x^n\right )}{n}} \Gamma \left (\frac {1}{2},-\frac {3 \log \left (a x^n\right )}{n}\right )\right )}{n \sqrt {\log \left (a x^n\right )}} \]
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 \[ \int \frac {x^{2}}{\log \left (a x^{n}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.30, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\ln \left (a \,x^{n}\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {x^{2}}{\log \left (a x^{n}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2}{{\ln \left (a\,x^n\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{2}}{\log {\left (a x^{n} \right )}^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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