Optimal. Leaf size=46 \[ \frac {2 m \text {Int}\left (\frac {x^{m-1}}{\sqrt {\tan ^{-1}(a x)}},x\right )}{a c}-\frac {2 x^m}{a c \sqrt {\tan ^{-1}(a x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {x^m}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {x^m}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2}} \, dx &=-\frac {2 x^m}{a c \sqrt {\tan ^{-1}(a x)}}+\frac {(2 m) \int \frac {x^{-1+m}}{\sqrt {\tan ^{-1}(a x)}} \, dx}{a c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.75, size = 0, normalized size = 0.00 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 2.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m}}{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{\frac {3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.33, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [A] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^m}{{\mathrm {atan}\left (a\,x\right )}^{3/2}\,\left (c\,a^2\,x^2+c\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {x^{m}}{a^{2} x^{2} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________