Optimal. Leaf size=112 \[ \frac {4 \sqrt {\pi } (m+1)^{3/2} x^{m+1} \left (a x^n\right )^{-\frac {m+1}{n}} \text {erfi}\left (\frac {\sqrt {m+1} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{3 n^{5/2}}-\frac {4 (m+1) x^{m+1}}{3 n^2 \sqrt {\log \left (a x^n\right )}}-\frac {2 x^{m+1}}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2306, 2310, 2180, 2204} \[ \frac {4 \sqrt {\pi } (m+1)^{3/2} x^{m+1} \left (a x^n\right )^{-\frac {m+1}{n}} \text {Erfi}\left (\frac {\sqrt {m+1} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{3 n^{5/2}}-\frac {4 (m+1) x^{m+1}}{3 n^2 \sqrt {\log \left (a x^n\right )}}-\frac {2 x^{m+1}}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2180
Rule 2204
Rule 2306
Rule 2310
Rubi steps
\begin {align*} \int \frac {x^m}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx &=-\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 ) \operatorname {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 ) \operatorname {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*}
________________________________________________________________________________________
Mathematica [A] time = 0.42, size = 103, normalized size = 0.92 \[ \frac {2 \left (\frac {2 \sqrt {\pi } (m+1)^{3/2} e^{\frac {(m+1) \left (n \log (x)-\log \left (a x^n\right )\right )}{n}} \text {erfi}\left (\frac {\sqrt {m+1} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{\sqrt {n}}-\frac {x^{m+1} \left (2 (m+1) \log \left (a x^n\right )+n\right )}{\log ^{\frac {3}{2}}\left (a x^n\right )}\right )}{3 n^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m}}{\log \left (a x^{n}\right )^{\frac {5}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\log \left (a x^{n}\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\ln \left (a \,x^{n}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\log \left (a x^{n}\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^m}{{\ln \left (a\,x^n\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\log {\left (a x^{n} \right )}^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________