Optimal. Leaf size=163 \[ \frac {x^3 \sqrt {\tan ^{-1}(a x)}}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {a^2 x^2+1} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{4 a^3 c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {a^2 x^2+1} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{12 a^3 c^2 \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.42, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {4944, 4971, 4970, 3312, 3305, 3351} \[ -\frac {\sqrt {\frac {\pi }{2}} \sqrt {a^2 x^2+1} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{4 a^3 c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {a^2 x^2+1} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{12 a^3 c^2 \sqrt {a^2 c x^2+c}}+\frac {x^3 \sqrt {\tan ^{-1}(a x)}}{3 c \left (a^2 c x^2+c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3305
Rule 3312
Rule 3351
Rule 4944
Rule 4970
Rule 4971
Rubi steps
\begin {align*} \int \frac {x^2 \sqrt {\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=\frac {x^3 \sqrt {\tan ^{-1}(a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {1}{6} a \int \frac {x^3}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)}} \, dx\\ &=\frac {x^3 \sqrt {\tan ^{-1}(a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {\left (a \sqrt {1+a^2 x^2}\right ) \int \frac {x^3}{\left (1+a^2 x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)}} \, dx}{6 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {x^3 \sqrt {\tan ^{-1}(a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {\sqrt {1+a^2 x^2} \operatorname {Subst}\left (\int \frac {\sin ^3(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{6 a^3 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {x^3 \sqrt {\tan ^{-1}(a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {\sqrt {1+a^2 x^2} \operatorname {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 )}{6 a^3 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {x^3 \sqrt {\tan ^{-1}(a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\sqrt {1+a^2 x^2} \operatorname {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{24 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {1+a^2 x^2} \operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {x^3 \sqrt {\tan ^{-1}(a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\sqrt {1+a^2 x^2} \operatorname {Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{12 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {1+a^2 x^2} \operatorname {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{4 a^3 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {x^3 \sqrt {\tan ^{-1}(a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {1+a^2 x^2} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{4 a^3 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 )}{12 a^3 c^2 \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 133, normalized size = 0.82 \[ \frac {24 a^3 x^3 \sqrt {\tan ^{-1}(a x)}-9 \sqrt {2 \pi } \left (a^2 x^2+1\right )^{3/2} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )+\sqrt {6 \pi } \left (a^2 x^2+1\right )^{3/2} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{72 a^3 c^2 \left (a^2 x^2+1\right ) \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 9.89, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \sqrt {\arctan \left (a x \right )}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}\, dx \]
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 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {x^2\,\sqrt {\mathrm {atan}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
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 \[ \int \frac {x^{2} \sqrt {\operatorname {atan}{\left (a x \right )}}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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