Optimal. Leaf size=129 \[ -\frac {4 \sqrt {2 \pi } \sqrt {a^2 x^2+1} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{3 a^2 c \sqrt {a^2 c x^2+c}}-\frac {2 x}{3 a c \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}-\frac {4}{3 a^2 c \sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}} \]
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Rubi [A] time = 0.30, 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 = {4942, 4902, 4971, 4970, 3305, 3351} \[ -\frac {4 \sqrt {2 \pi } \sqrt {a^2 x^2+1} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{3 a^2 c \sqrt {a^2 c x^2+c}}-\frac {2 x}{3 a c \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}-\frac {4}{3 a^2 c \sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 3305
Rule 3351
Rule 4902
Rule 4942
Rule 4970
Rule 4971
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 ) \operatorname {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 ) \operatorname {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] time = 0.19, size = 124, normalized size = 0.96 \[ -\frac {2 \left (-i \sqrt {a^2 x^2+1} \left (-i \tan ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-i \tan ^{-1}(a x)\right )+i \sqrt {a^2 x^2+1} \left (i \tan ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},i \tan ^{-1}(a x)\right )+a x+2 \tan ^{-1}(a x)\right )}{3 a^2 c \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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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(-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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maple [F] time = 3.12, 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 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {x}{{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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