Optimal. Leaf size=131 \[ \frac {3 \sqrt {\frac {\pi }{2}} \sqrt {1+a^2 x^2} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{2 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {1+a^2 x^2} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{2 a^4 c^2 \sqrt {c+a^2 c x^2}} \]
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Rubi [A]
time = 0.20, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {5091, 5090,
3393, 3386, 3432} \begin {gather*} \frac {3 \sqrt {\frac {\pi }{2}} \sqrt {a^2 x^2+1} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{2 a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {a^2 x^2+1} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{2 a^4 c^2 \sqrt {a^2 c x^2+c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3386
Rule 3393
Rule 3432
Rule 5090
Rule 5091
Rubi steps
\begin {align*} \int \frac {x^3}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)}} \, dx &=\frac {\sqrt {1+a^2 x^2} \int \frac {x^3}{\left (1+a^2 x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)}} \, dx}{c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\sin ^3(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \left (\frac {3 \sin (x)}{4 \sqrt {x}}-\frac {\sin (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^4 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^4 c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{2 a^4 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{2 a^4 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {1+a^2 x^2} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{2 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {1+a^2 x^2} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{2 a^4 c^2 \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 95, normalized size = 0.73 \begin {gather*} \frac {\sqrt {\frac {\pi }{6}} \left (1+a^2 x^2\right )^{3/2} \left (3 \sqrt {3} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )-S\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )\right )}{2 a^4 c \left (c \left (1+a^2 x^2\right )\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 6.24, size = 0, normalized size = 0.00 \[\int \frac {x^{3}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \sqrt {\arctan \left (a x \right )}}\, dx\]
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^{3}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \sqrt {\operatorname {atan}{\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^3}{\sqrt {\mathrm {atan}\left (a\,x\right )}\,{\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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