Optimal. Leaf size=229 \[ -\frac {2 x^3}{3 a c \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^{3/2}}-\frac {4 x^2}{a^2 c \sqrt {c+a^2 c x^2} \sqrt {\text {ArcTan}(a x)}}-\frac {8 x^4}{3 c \sqrt {c+a^2 c x^2} \sqrt {\text {ArcTan}(a x)}}+\frac {8 \sqrt {2 \pi } \sqrt {1+a^2 x^2} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {44}{3} \text {Int}\left (\frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\text {ArcTan}(a x)}},x\right )+8 a^2 \text {Int}\left (\frac {x^5}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\text {ArcTan}(a x)}},x\right ) \]
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Rubi [A]
time = 0.59, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)^{5/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx &=-\frac {2 x^3}{3 a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}+\frac {2 \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx}{a}+\frac {1}{3} (4 a) \int \frac {x^4}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac {2 x^3}{3 a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 c \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}-\frac {8 x^4}{3 c \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}+4 \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {32}{3} \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {8 \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx}{a^2}+\left (8 a^2\right ) \int \frac {x^5}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx\\ &=-\frac {2 x^3}{3 a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 c \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}-\frac {8 x^4}{3 c \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}+4 \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {32}{3} \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\left (8 a^2\right ) \int \frac {x^5}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {\left (8 \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}{a^2 c \sqrt {c+a^2 c x^2}}\\ &=-\frac {2 x^3}{3 a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 c \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}-\frac {8 x^4}{3 c \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}+4 \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {32}{3} \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\left (8 a^2\right ) \int \frac {x^5}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {\left (8 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c \sqrt {c+a^2 c x^2}}\\ &=-\frac {2 x^3}{3 a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 c \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}-\frac {8 x^4}{3 c \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}+4 \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {32}{3} \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\left (8 a^2\right ) \int \frac {x^5}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {\left (16 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}}\\ &=-\frac {2 x^3}{3 a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 c \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}-\frac {8 x^4}{3 c \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}+\frac {8 \sqrt {2 \pi } \sqrt {1+a^2 x^2} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}}+4 \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {32}{3} \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\left (8 a^2\right ) \int \frac {x^5}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx\\ \end {align*}
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Mathematica [A]
time = 1.89, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)^{5/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 6.42, size = 0, normalized size = 0.00 \[\int \frac {x^{3}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arctan \left (a x \right )^{\frac {5}{2}}}\, dx\]
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 \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 [A]
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 {3}{2}} \operatorname {atan}^{\frac {5}{2}}{\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 [A]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3}{{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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