Optimal. Leaf size=125 \[ -\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {a^2 x^2+1} C\left (\sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{2 a c \sqrt {a^2 c x^2+c}}+\frac {x \tan ^{-1}(a x)^{3/2}}{c \sqrt {a^2 c x^2+c}}+\frac {3 \sqrt {\tan ^{-1}(a x)}}{2 a c \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.14, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4898, 4905, 4904, 3304, 3352} \[ -\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {a^2 x^2+1} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{2 a c \sqrt {a^2 c x^2+c}}+\frac {x \tan ^{-1}(a x)^{3/2}}{c \sqrt {a^2 c x^2+c}}+\frac {3 \sqrt {\tan ^{-1}(a x)}}{2 a c \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3352
Rule 4898
Rule 4904
Rule 4905
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=\frac {3 \sqrt {\tan ^{-1}(a x)}}{2 a c \sqrt {c+a^2 c x^2}}+\frac {x \tan ^{-1}(a x)^{3/2}}{c \sqrt {c+a^2 c x^2}}-\frac {3}{4} \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx\\ &=\frac {3 \sqrt {\tan ^{-1}(a x)}}{2 a c \sqrt {c+a^2 c x^2}}+\frac {x \tan ^{-1}(a x)^{3/2}}{c \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \int \frac {1}{\left (1+a^2 x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx}{4 c \sqrt {c+a^2 c x^2}}\\ &=\frac {3 \sqrt {\tan ^{-1}(a x)}}{2 a c \sqrt {c+a^2 c x^2}}+\frac {x \tan ^{-1}(a x)^{3/2}}{c \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{4 a c \sqrt {c+a^2 c x^2}}\\ &=\frac {3 \sqrt {\tan ^{-1}(a x)}}{2 a c \sqrt {c+a^2 c x^2}}+\frac {x \tan ^{-1}(a x)^{3/2}}{c \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{2 a c \sqrt {c+a^2 c x^2}}\\ &=\frac {3 \sqrt {\tan ^{-1}(a x)}}{2 a c \sqrt {c+a^2 c x^2}}+\frac {x \tan ^{-1}(a x)^{3/2}}{c \sqrt {c+a^2 c x^2}}-\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {1+a^2 x^2} C\left (\sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{2 a c \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.17, size = 104, normalized size = 0.83 \[ \frac {\left (a^2 x^2+1\right )^{3/2} \sqrt {\tan ^{-1}(a x)} \left (\sqrt {i \tan ^{-1}(a x)} \Gamma \left (\frac {5}{2},-i \tan ^{-1}(a x)\right )+\sqrt {-i \tan ^{-1}(a x)} \Gamma \left (\frac {5}{2},i \tan ^{-1}(a x)\right )\right )}{2 a \left (c \left (a^2 x^2+1\right )\right )^{3/2} \sqrt {\tan ^{-1}(a x)^2}} \]
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] 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 = 1.42, size = 0, normalized size = 0.00 \[ \int \frac {\arctan \left (a x \right )^{\frac {3}{2}}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^{3/2}}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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