Integrand size = 14, antiderivative size = 112 \[ \int \frac {x^m}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\frac {4 (1+m)^{3/2} \sqrt {\pi } x^{1+m} \left (a x^n\right )^{-\frac {1+m}{n}} \text {erfi}\left (\frac {\sqrt {1+m} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{3 n^{5/2}}-\frac {2 x^{1+m}}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {4 (1+m) x^{1+m}}{3 n^2 \sqrt {\log \left (a x^n\right )}} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2343, 2347, 2211, 2235} \[ \int \frac {x^m}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\frac {4 \sqrt {\pi } (m+1)^{3/2} x^{m+1} \left (a x^n\right )^{-\frac {m+1}{n}} \text {erfi}\left (\frac {\sqrt {m+1} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{3 n^{5/2}}-\frac {4 (m+1) x^{m+1}}{3 n^2 \sqrt {\log \left (a x^n\right )}}-\frac {2 x^{m+1}}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )} \]
[In]
[Out]
Rule 2211
Rule 2235
Rule 2343
Rule 2347
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^{1+m}}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}+\frac {(2 (1+m)) \int \frac {x^m}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx}{3 n} \\ & = -\frac {2 x^{1+m}}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {4 (1+m) x^{1+m}}{3 n^2 \sqrt {\log \left (a x^n\right )}}+\frac {\left (4 (1+m)^2\right ) \int \frac {x^m}{\sqrt {\log \left (a x^n\right )}} \, dx}{3 n^2} \\ & = -\frac {2 x^{1+m}}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {4 (1+m) x^{1+m}}{3 n^2 \sqrt {\log \left (a x^n\right )}}+\frac {\left (4 (1+m)^2 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 )}{3 n^3} \\ & = -\frac {2 x^{1+m}}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {4 (1+m) x^{1+m}}{3 n^2 \sqrt {\log \left (a x^n\right )}}+\frac {\left (8 (1+m)^2 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 )}{3 n^3} \\ & = \frac {4 (1+m)^{3/2} \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 )}{3 n^{5/2}}-\frac {2 x^{1+m}}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {4 (1+m) x^{1+m}}{3 n^2 \sqrt {\log \left (a x^n\right )}} \\ \end{align*}
Time = 1.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.95 \[ \int \frac {x^m}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=-\frac {2 e^{-\frac {(1+m) \left (-n \log (x)+\log \left (a x^n\right )\right )}{n}} \left (2 n \Gamma \left (\frac {1}{2},-\frac {(1+m) \log \left (a x^n\right )}{n}\right ) \left (-\frac {(1+m) \log \left (a x^n\right )}{n}\right )^{3/2}+\left (a x^n\right )^{\frac {1+m}{n}} \left (n+2 (1+m) \log \left (a x^n\right )\right )\right )}{3 n^2 \log ^{\frac {3}{2}}\left (a x^n\right )} \]
[In]
[Out]
\[\int \frac {x^{m}}{\ln \left (a \,x^{n}\right )^{\frac {5}{2}}}d x\]
[In]
[Out]
\[ \int \frac {x^m}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\int { \frac {x^{m}}{\log \left (a x^{n}\right )^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^m}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\int \frac {x^{m}}{\log {\left (a x^{n} \right )}^{\frac {5}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {x^m}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\int { \frac {x^{m}}{\log \left (a x^{n}\right )^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^m}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\int { \frac {x^{m}}{\log \left (a x^{n}\right )^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x^m}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\int \frac {x^m}{{\ln \left (a\,x^n\right )}^{5/2}} \,d x \]
[In]
[Out]