Optimal. Leaf size=119 \[ -\frac {5 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^{3/2}}{4 a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^{5/2}+\frac {15}{8} c \text {Int}\left (\frac {\sqrt {\text {ArcTan}(a x)}}{\sqrt {c+a^2 c x^2}},x\right )+\frac {1}{2} c \text {Int}\left (\frac {\text {ArcTan}(a x)^{5/2}}{\sqrt {c+a^2 c x^2}},x\right ) \]
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Rubi [A]
time = 0.07, 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 \sqrt {c+a^2 c x^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 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2} \, dx &=-\frac {5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{4 a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}+\frac {1}{2} c \int \frac {\tan ^{-1}(a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} (15 c) \int \frac {\sqrt {\tan ^{-1}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^{5/2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.56, size = 0, normalized size = 0.00 \[\int \arctan \left (a x \right )^{\frac {5}{2}} \sqrt {a^{2} c \,x^{2}+c}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.01 \begin {gather*} \int {\mathrm {atan}\left (a\,x\right )}^{5/2}\,\sqrt {c\,a^2\,x^2+c} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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