Optimal. Leaf size=307 \[ -\frac {25 c^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^{3/2}}{32 a}-\frac {25 c \left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)^{3/2}}{144 a}-\frac {\left (c+a^2 c x^2\right )^{5/2} \text {ArcTan}(a x)^{3/2}}{12 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^{5/2}+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)^{5/2}+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \text {ArcTan}(a x)^{5/2}+\frac {75}{64} c^3 \text {Int}\left (\frac {\sqrt {\text {ArcTan}(a x)}}{\sqrt {c+a^2 c x^2}},x\right )+\frac {25}{96} c^2 \text {Int}\left (\sqrt {c+a^2 c x^2} \sqrt {\text {ArcTan}(a x)},x\right )+\frac {1}{8} c \text {Int}\left (\left (c+a^2 c x^2\right )^{3/2} \sqrt {\text {ArcTan}(a x)},x\right )+\frac {5}{16} c^3 \text {Int}\left (\frac {\text {ArcTan}(a x)^{5/2}}{\sqrt {c+a^2 c x^2}},x\right ) \]
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Rubi [A]
time = 0.18, 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 )^{5/2} \text {ArcTan}(a x)^{5/2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{5/2} \, dx &=-\frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2}}{12 a}+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{5/2}+\frac {1}{8} c \int \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)} \, dx+\frac {1}{6} (5 c) \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2} \, dx\\ &=-\frac {25 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}{144 a}-\frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2}}{12 a}+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{5/2}+\frac {1}{8} c \int \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)} \, dx+\frac {1}{96} \left (25 c^2\right ) \int \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)} \, dx+\frac {1}{8} \left (5 c^2\right ) \int \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2} \, dx\\ &=-\frac {25 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{32 a}-\frac {25 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}{144 a}-\frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2}}{12 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{5/2}+\frac {1}{8} c \int \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)} \, dx+\frac {1}{96} \left (25 c^2\right ) \int \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)} \, dx+\frac {1}{16} \left (5 c^3\right ) \int \frac {\tan ^{-1}(a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{64} \left (75 c^3\right ) \int \frac {\sqrt {\tan ^{-1}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 0, normalized size = 0.00 \begin {gather*} \int \left (c+a^2 c x^2\right )^{5/2} \text {ArcTan}(a x)^{5/2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.66, size = 0, normalized size = 0.00 \[\int \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{\frac {5}{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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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.
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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.
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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 )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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