Optimal. Leaf size=129 \[ -\frac {2 x}{3 a c \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^{3/2}}-\frac {4}{3 a^2 c \sqrt {c+a^2 c x^2} \sqrt {\text {ArcTan}(a x)}}-\frac {4 \sqrt {2 \pi } \sqrt {1+a^2 x^2} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{3 a^2 c \sqrt {c+a^2 c x^2}} \]
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Rubi [A]
time = 0.20, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5062, 5022,
5091, 5090, 3386, 3432} \begin {gather*} -\frac {4 \sqrt {2 \pi } \sqrt {a^2 x^2+1} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{3 a^2 c \sqrt {a^2 c x^2+c}}-\frac {2 x}{3 a c \text {ArcTan}(a x)^{3/2} \sqrt {a^2 c x^2+c}}-\frac {4}{3 a^2 c \sqrt {\text {ArcTan}(a x)} \sqrt {a^2 c x^2+c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3386
Rule 3432
Rule 5022
Rule 5062
Rule 5090
Rule 5091
Rubi steps
\begin {align*} \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx &=-\frac {2 x}{3 a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}+\frac {2 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx}{3 a}\\ &=-\frac {2 x}{3 a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}-\frac {4}{3 a^2 c \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}-\frac {4}{3} \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx\\ &=-\frac {2 x}{3 a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}-\frac {4}{3 a^2 c \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}-\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \int \frac {x}{\left (1+a^2 x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx}{3 c \sqrt {c+a^2 c x^2}}\\ &=-\frac {2 x}{3 a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}-\frac {4}{3 a^2 c \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}-\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{3 a^2 c \sqrt {c+a^2 c x^2}}\\ &=-\frac {2 x}{3 a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}-\frac {4}{3 a^2 c \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}-\frac {\left (8 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{3 a^2 c \sqrt {c+a^2 c x^2}}\\ &=-\frac {2 x}{3 a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}-\frac {4}{3 a^2 c \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}-\frac {4 \sqrt {2 \pi } \sqrt {1+a^2 x^2} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{3 a^2 c \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.13, size = 124, normalized size = 0.96 \begin {gather*} -\frac {2 \left (a x+2 \text {ArcTan}(a x)-i \sqrt {1+a^2 x^2} (-i \text {ArcTan}(a x))^{3/2} \text {Gamma}\left (\frac {1}{2},-i \text {ArcTan}(a x)\right )+i \sqrt {1+a^2 x^2} (i \text {ArcTan}(a x))^{3/2} \text {Gamma}\left (\frac {1}{2},i \text {ArcTan}(a x)\right )\right )}{3 a^2 c \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.78, size = 0, normalized size = 0.00 \[\int \frac {x}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arctan \left (a x \right )^{\frac {5}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x}{{\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.
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