Optimal. Leaf size=36 \[ \frac {x^{m+1} F_1\left (m+1;\frac {5}{4},-\frac {5}{4};m+2;i a x,-i a x\right )}{m+1} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5062, 133} \[ \frac {x^{m+1} F_1\left (m+1;\frac {5}{4},-\frac {5}{4};m+2;i a x,-i a x\right )}{m+1} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 133
Rule 5062
Rubi steps
\begin {align*} \int e^{\frac {5}{2} i \tan ^{-1}(a x)} x^m \, dx &=\int \frac {x^m (1+i a x)^{5/4}}{(1-i a x)^{5/4}} \, dx\\ &=\frac {x^{1+m} F_1\left (1+m;\frac {5}{4},-\frac {5}{4};2+m;i a x,-i a x\right )}{1+m}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int e^{\frac {5}{2} i \tan ^{-1}(a x)} x^m \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a x - i\right )} x^{m} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{a x + i}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [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.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )^{\frac {5}{2}} x^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int x^m\,{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________