Optimal. Leaf size=132 \[ \frac {15 \text {Int}\left (\frac {\sqrt {\tan ^{-1}(a x)}}{\sqrt {a^2 c x^2+c}},x\right )}{8 a^2}-\frac {\text {Int}\left (\frac {\tan ^{-1}(a x)^{5/2}}{\sqrt {a^2 c x^2+c}},x\right )}{2 a^2}+\frac {x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}}{2 a^2 c}-\frac {5 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{4 a^3 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.25, 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 = {} \[ \int \frac {x^2 \tan ^{-1}(a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {x^2 \tan ^{-1}(a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx &=\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{2 a^2 c}-\frac {\int \frac {\tan ^{-1}(a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx}{2 a^2}-\frac {5 \int \frac {x \tan ^{-1}(a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx}{4 a}\\ &=-\frac {5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{4 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{2 a^2 c}-\frac {\int \frac {\tan ^{-1}(a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx}{2 a^2}+\frac {15 \int \frac {\sqrt {\tan ^{-1}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.60, size = 0, normalized size = 0.00 \[ \int \frac {x^2 \tan ^{-1}(a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [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]
________________________________________________________________________________________
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 = 8.69, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \arctan \left (a x \right )^{\frac {5}{2}}}{\sqrt {a^{2} c \,x^{2}+c}}\, dx \]
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 = -0.01 \[ \int \frac {x^2\,{\mathrm {atan}\left (a\,x\right )}^{5/2}}{\sqrt {c\,a^2\,x^2+c}} \,d x \]
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]
________________________________________________________________________________________