Optimal. Leaf size=212 \[ -\frac {15 c \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^{3/2}}{16 a}-\frac {5 \left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)^{3/2}}{24 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^{5/2}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)^{5/2}+\frac {45}{32} c^2 \text {Int}\left (\frac {\sqrt {\text {ArcTan}(a x)}}{\sqrt {c+a^2 c x^2}},x\right )+\frac {5}{16} c \text {Int}\left (\sqrt {c+a^2 c x^2} \sqrt {\text {ArcTan}(a x)},x\right )+\frac {3}{8} c^2 \text {Int}\left (\frac {\text {ArcTan}(a x)^{5/2}}{\sqrt {c+a^2 c x^2}},x\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.12, 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 = {}
\begin {gather*} \int \left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)^{5/2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2} \, dx &=-\frac {5 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}{24 a}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}+\frac {1}{16} (5 c) \int \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)} \, dx+\frac {1}{4} (3 c) \int \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2} \, dx\\ &=-\frac {15 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{16 a}-\frac {5 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}{24 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}+\frac {1}{16} (5 c) \int \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)} \, dx+\frac {1}{8} \left (3 c^2\right ) \int \frac {\tan ^{-1}(a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{32} \left (45 c^2\right ) \int \frac {\sqrt {\tan ^{-1}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.10, size = 0, normalized size = 0.00 \begin {gather*} \int \left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)^{5/2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.53, size = 0, normalized size = 0.00 \[\int \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arctan \left (a x \right )^{\frac {5}{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [A]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________