Optimal. Leaf size=61 \[ \frac{\sqrt{\pi } x^{m+1} \left (a x^n\right )^{-\frac{m+1}{n}} \text{Erfi}\left (\frac{\sqrt{m+1} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{\sqrt{m+1} \sqrt{n}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0472519, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2310, 2180, 2204} \[ \frac{\sqrt{\pi } x^{m+1} \left (a x^n\right )^{-\frac{m+1}{n}} \text{Erfi}\left (\frac{\sqrt{m+1} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{\sqrt{m+1} \sqrt{n}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2310
Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int \frac{x^m}{\sqrt{\log \left (a x^n\right )}} \, dx &=\frac{\left (x^{1+m} \left (a x^n\right )^{-\frac{1+m}{n}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{(1+m) x}{n}}}{\sqrt{x}} \, dx,x,\log \left (a x^n\right )\right )}{n}\\ &=\frac{\left (2 x^{1+m} \left (a x^n\right )^{-\frac{1+m}{n}}\right ) \operatorname{Subst}\left (\int e^{\frac{(1+m) x^2}{n}} \, dx,x,\sqrt{\log \left (a x^n\right )}\right )}{n}\\ &=\frac{\sqrt{\pi } x^{1+m} \left (a x^n\right )^{-\frac{1+m}{n}} \text{erfi}\left (\frac{\sqrt{1+m} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{\sqrt{1+m} \sqrt{n}}\\ \end{align*}
Mathematica [A] time = 0.0105832, size = 61, normalized size = 1. \[ \frac{\sqrt{\pi } x^{m+1} \left (a x^n\right )^{-\frac{m+1}{n}} \text{Erfi}\left (\frac{\sqrt{m+1} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{\sqrt{m+1} \sqrt{n}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.201, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m}{\frac{1}{\sqrt{\ln \left ( a{x}^{n} \right ) }}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{\sqrt{\log \left (a x^{n}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{m}}{\sqrt{\log \left (a x^{n}\right )}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{\sqrt{\log{\left (a x^{n} \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{\sqrt{\log \left (a x^{n}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]