Integrand size = 14, antiderivative size = 83 \[ \int \frac {x^m}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=\frac {2 \sqrt {1+m} \sqrt {\pi } x^{1+m} \left (a x^n\right )^{-\frac {1+m}{n}} \text {erfi}\left (\frac {\sqrt {1+m} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{n^{3/2}}-\frac {2 x^{1+m}}{n \sqrt {\log \left (a x^n\right )}} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2343, 2347, 2211, 2235} \[ \int \frac {x^m}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=\frac {2 \sqrt {\pi } \sqrt {m+1} x^{m+1} \left (a x^n\right )^{-\frac {m+1}{n}} \text {erfi}\left (\frac {\sqrt {m+1} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{n^{3/2}}-\frac {2 x^{m+1}}{n \sqrt {\log \left (a x^n\right )}} \]
[In]
[Out]
Rule 2211
Rule 2235
Rule 2343
Rule 2347
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^{1+m}}{n \sqrt {\log \left (a x^n\right )}}+\frac {(2 (1+m)) \int \frac {x^m}{\sqrt {\log \left (a x^n\right )}} \, dx}{n} \\ & = -\frac {2 x^{1+m}}{n \sqrt {\log \left (a x^n\right )}}+\frac {\left (2 (1+m) x^{1+m} \left (a x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int \frac {e^{\frac {(1+m) x}{n}}}{\sqrt {x}} \, dx,x,\log \left (a x^n\right )\right )}{n^2} \\ & = -\frac {2 x^{1+m}}{n \sqrt {\log \left (a x^n\right )}}+\frac {\left (4 (1+m) x^{1+m} \left (a x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int e^{\frac {(1+m) x^2}{n}} \, dx,x,\sqrt {\log \left (a x^n\right )}\right )}{n^2} \\ & = \frac {2 \sqrt {1+m} \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 )}{n^{3/2}}-\frac {2 x^{1+m}}{n \sqrt {\log \left (a x^n\right )}} \\ \end{align*}
Time = 1.28 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.88 \[ \int \frac {x^m}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=\frac {2 x^{1+m} \left (-1+\left (a x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1}{2},-\frac {(1+m) \log \left (a x^n\right )}{n}\right ) \sqrt {-\frac {(1+m) \log \left (a x^n\right )}{n}}\right )}{n \sqrt {\log \left (a x^n\right )}} \]
[In]
[Out]
\[\int \frac {x^{m}}{\ln \left (a \,x^{n}\right )^{\frac {3}{2}}}d x\]
[In]
[Out]
\[ \int \frac {x^m}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=\int { \frac {x^{m}}{\log \left (a x^{n}\right )^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^m}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=\int \frac {x^{m}}{\log {\left (a x^{n} \right )}^{\frac {3}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {x^m}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=\int { \frac {x^{m}}{\log \left (a x^{n}\right )^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^m}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=\int { \frac {x^{m}}{\log \left (a x^{n}\right )^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x^m}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=\int \frac {x^m}{{\ln \left (a\,x^n\right )}^{3/2}} \,d x \]
[In]
[Out]