Optimal. Leaf size=199 \[ \frac {5 \sqrt {c+a^2 c x^2} \sqrt {\text {ArcTan}(a x)}}{8 a^4 c}-\frac {5 x \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^{3/2}}{12 a^3 c}-\frac {2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^{5/2}}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^{5/2}}{3 a^2 c}-\frac {5 \text {Int}\left (\frac {1}{\sqrt {c+a^2 c x^2} \sqrt {\text {ArcTan}(a x)}},x\right )}{16 a^3}+\frac {25 \text {Int}\left (\frac {\text {ArcTan}(a x)^{3/2}}{\sqrt {c+a^2 c x^2}},x\right )}{12 a^3} \]
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Rubi [A]
time = 0.32, 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 \frac {x^3 \text {ArcTan}(a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^3 \tan ^{-1}(a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx &=\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{3 a^2 c}-\frac {2 \int \frac {x \tan ^{-1}(a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}-\frac {5 \int \frac {x^2 \tan ^{-1}(a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx}{6 a}\\ &=-\frac {5 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{12 a^3 c}-\frac {2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{3 a^2 c}+\frac {5 \int \frac {\tan ^{-1}(a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx}{12 a^3}+\frac {5 \int \frac {\tan ^{-1}(a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^3}+\frac {5 \int \frac {x \sqrt {\tan ^{-1}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}\\ &=\frac {5 \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}{8 a^4 c}-\frac {5 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{12 a^3 c}-\frac {2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{3 a^2 c}-\frac {5 \int \frac {1}{\sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}} \, dx}{16 a^3}+\frac {5 \int \frac {\tan ^{-1}(a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx}{12 a^3}+\frac {5 \int \frac {\tan ^{-1}(a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^3}\\ \end {align*}
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Mathematica [A]
time = 2.81, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3 \text {ArcTan}(a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 5.89, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \arctan \left (a x \right )^{\frac {5}{2}}}{\sqrt {a^{2} c \,x^{2}+c}}\, dx\]
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(-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 = -0.01 \begin {gather*} \int \frac {x^3\,{\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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