Optimal. Leaf size=213 \[ \frac {3 x \sqrt {\text {ArcTan}(a x)}}{4 c^2 \sqrt {c+a^2 c x^2}}-\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {1+a^2 x^2} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{4 a c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {1+a^2 x^2} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{12 a c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \sqrt {\text {ArcTan}(a x)} \sin (3 \text {ArcTan}(a x))}{12 a c^2 \sqrt {c+a^2 c x^2}} \]
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Rubi [A]
time = 0.14, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5025, 5024,
3393, 3377, 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 )}{4 a 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 )}{12 a c^2 \sqrt {a^2 c x^2+c}}+\frac {3 x \sqrt {\text {ArcTan}(a x)}}{4 c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 x^2+1} \sqrt {\text {ArcTan}(a x)} \sin (3 \text {ArcTan}(a x))}{12 a c^2 \sqrt {a^2 c x^2+c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3377
Rule 3386
Rule 3393
Rule 3432
Rule 5024
Rule 5025
Rubi steps
\begin {align*} \int \frac {\sqrt {\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=\frac {\sqrt {1+a^2 x^2} \int \frac {\sqrt {\tan ^{-1}(a x)}}{\left (1+a^2 x^2\right )^{5/2}} \, dx}{c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \sqrt {x} \cos ^3(x) \, dx,x,\tan ^{-1}(a x)\right )}{a c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \left (\frac {3}{4} \sqrt {x} \cos (x)+\frac {1}{4} \sqrt {x} \cos (3 x)\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \sqrt {x} \cos (3 x) \, dx,x,\tan ^{-1}(a x)\right )}{4 a c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \sqrt {x} \cos (x) \, dx,x,\tan ^{-1}(a x)\right )}{4 a c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {3 x \sqrt {\tan ^{-1}(a x)}}{4 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \sqrt {\tan ^{-1}(a x)} \sin \left (3 \tan ^{-1}(a x)\right )}{12 a 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 )}{24 a 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 )}{8 a c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {3 x \sqrt {\tan ^{-1}(a x)}}{4 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \sqrt {\tan ^{-1}(a x)} \sin \left (3 \tan ^{-1}(a x)\right )}{12 a 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 )}{12 a 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 )}{4 a c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {3 x \sqrt {\tan ^{-1}(a x)}}{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 )}{4 a 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 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \sqrt {\tan ^{-1}(a x)} \sin \left (3 \tan ^{-1}(a x)\right )}{12 a c^2 \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 137, normalized size = 0.64 \begin {gather*} \frac {24 a x \left (3+2 a^2 x^2\right ) \sqrt {\text {ArcTan}(a x)}-27 \sqrt {2 \pi } \left (1+a^2 x^2\right )^{3/2} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )-\sqrt {6 \pi } \left (1+a^2 x^2\right )^{3/2} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{72 c^2 \left (a+a^3 x^2\right ) \sqrt {c+a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.51, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {\arctan \left (a x \right )}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}\, dx\]
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 \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 {\sqrt {\operatorname {atan}{\left (a x \right )}}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 = -0.00 \begin {gather*} \int \frac {\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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