Optimal. Leaf size=178 \[ \frac {15 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{16 a^3}-\frac {5 \sqrt {\frac {\pi }{6}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{144 a^3}+\frac {5 x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{18 a}-\frac {5 x \sqrt {\sin ^{-1}(a x)}}{6 a^2}+\frac {5 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{9 a^3}+\frac {1}{3} x^3 \sin ^{-1}(a x)^{5/2}-\frac {5}{36} x^3 \sqrt {\sin ^{-1}(a x)} \]
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Rubi [A] time = 0.47, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4629, 4707, 4677, 4619, 4723, 3305, 3351, 3312} \[ \frac {15 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{16 a^3}-\frac {5 \sqrt {\frac {\pi }{6}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{144 a^3}+\frac {5 x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{18 a}+\frac {5 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{9 a^3}-\frac {5 x \sqrt {\sin ^{-1}(a x)}}{6 a^2}+\frac {1}{3} x^3 \sin ^{-1}(a x)^{5/2}-\frac {5}{36} x^3 \sqrt {\sin ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 3305
Rule 3312
Rule 3351
Rule 4619
Rule 4629
Rule 4677
Rule 4707
Rule 4723
Rubi steps
\begin {align*} \int x^2 \sin ^{-1}(a x)^{5/2} \, dx &=\frac {1}{3} x^3 \sin ^{-1}(a x)^{5/2}-\frac {1}{6} (5 a) \int \frac {x^3 \sin ^{-1}(a x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {5 x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \sin ^{-1}(a x)^{5/2}-\frac {5}{12} \int x^2 \sqrt {\sin ^{-1}(a x)} \, dx-\frac {5 \int \frac {x \sin ^{-1}(a x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx}{9 a}\\ &=-\frac {5}{36} x^3 \sqrt {\sin ^{-1}(a x)}+\frac {5 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{9 a^3}+\frac {5 x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \sin ^{-1}(a x)^{5/2}-\frac {5 \int \sqrt {\sin ^{-1}(a x)} \, dx}{6 a^2}+\frac {1}{72} (5 a) \int \frac {x^3}{\sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}} \, dx\\ &=-\frac {5 x \sqrt {\sin ^{-1}(a x)}}{6 a^2}-\frac {5}{36} x^3 \sqrt {\sin ^{-1}(a x)}+\frac {5 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{9 a^3}+\frac {5 x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \sin ^{-1}(a x)^{5/2}+\frac {5 \operatorname {Subst}\left (\int \frac {\sin ^3(x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{72 a^3}+\frac {5 \int \frac {x}{\sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}} \, dx}{12 a}\\ &=-\frac {5 x \sqrt {\sin ^{-1}(a x)}}{6 a^2}-\frac {5}{36} x^3 \sqrt {\sin ^{-1}(a x)}+\frac {5 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{9 a^3}+\frac {5 x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \sin ^{-1}(a x)^{5/2}+\frac {5 \operatorname {Subst}\left (\int \left (\frac {3 \sin (x)}{4 \sqrt {x}}-\frac {\sin (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{72 a^3}+\frac {5 \operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{12 a^3}\\ &=-\frac {5 x \sqrt {\sin ^{-1}(a x)}}{6 a^2}-\frac {5}{36} x^3 \sqrt {\sin ^{-1}(a x)}+\frac {5 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{9 a^3}+\frac {5 x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \sin ^{-1}(a x)^{5/2}-\frac {5 \operatorname {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{288 a^3}+\frac {5 \operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{96 a^3}+\frac {5 \operatorname {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{6 a^3}\\ &=-\frac {5 x \sqrt {\sin ^{-1}(a x)}}{6 a^2}-\frac {5}{36} x^3 \sqrt {\sin ^{-1}(a x)}+\frac {5 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{9 a^3}+\frac {5 x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \sin ^{-1}(a x)^{5/2}+\frac {5 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{6 a^3}-\frac {5 \operatorname {Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{144 a^3}+\frac {5 \operatorname {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{48 a^3}\\ &=-\frac {5 x \sqrt {\sin ^{-1}(a x)}}{6 a^2}-\frac {5}{36} x^3 \sqrt {\sin ^{-1}(a x)}+\frac {5 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{9 a^3}+\frac {5 x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \sin ^{-1}(a x)^{5/2}+\frac {15 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{16 a^3}-\frac {5 \sqrt {\frac {\pi }{6}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{144 a^3}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 125, normalized size = 0.70 \[ \frac {-81 \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {7}{2},-i \sin ^{-1}(a x)\right )-81 \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {7}{2},i \sin ^{-1}(a x)\right )+\sqrt {3} \left (\sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {7}{2},-3 i \sin ^{-1}(a x)\right )+\sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {7}{2},3 i \sin ^{-1}(a x)\right )\right )}{648 a^3 \sqrt {\sin ^{-1}(a x)}} \]
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 [C] time = 0.54, size = 309, normalized size = 1.74 \[ \frac {i \, \arcsin \left (a x\right )^{\frac {5}{2}} e^{\left (3 i \, \arcsin \left (a x\right )\right )}}{24 \, a^{3}} - \frac {i \, \arcsin \left (a x\right )^{\frac {5}{2}} e^{\left (i \, \arcsin \left (a x\right )\right )}}{8 \, a^{3}} + \frac {i \, \arcsin \left (a x\right )^{\frac {5}{2}} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{8 \, a^{3}} - \frac {i \, \arcsin \left (a x\right )^{\frac {5}{2}} e^{\left (-3 i \, \arcsin \left (a x\right )\right )}}{24 \, a^{3}} - \frac {5 \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (3 i \, \arcsin \left (a x\right )\right )}}{144 \, a^{3}} + \frac {5 \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (i \, \arcsin \left (a x\right )\right )}}{16 \, a^{3}} + \frac {5 \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{16 \, a^{3}} - \frac {5 \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (-3 i \, \arcsin \left (a x\right )\right )}}{144 \, a^{3}} - \frac {\left (5 i - 5\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arcsin \left (a x\right )}\right )}{3456 \, a^{3}} + \frac {\left (5 i + 5\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arcsin \left (a x\right )}\right )}{3456 \, a^{3}} + \frac {\left (15 i - 15\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{128 \, a^{3}} - \frac {\left (15 i + 15\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{128 \, a^{3}} - \frac {5 i \, \sqrt {\arcsin \left (a x\right )} e^{\left (3 i \, \arcsin \left (a x\right )\right )}}{288 \, a^{3}} + \frac {15 i \, \sqrt {\arcsin \left (a x\right )} e^{\left (i \, \arcsin \left (a x\right )\right )}}{32 \, a^{3}} - \frac {15 i \, \sqrt {\arcsin \left (a x\right )} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{32 \, a^{3}} + \frac {5 i \, \sqrt {\arcsin \left (a x\right )} e^{\left (-3 i \, \arcsin \left (a x\right )\right )}}{288 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 156, normalized size = 0.88 \[ -\frac {-216 a x \arcsin \left (a x \right )^{3}+72 \arcsin \left (a x \right )^{3} \sin \left (3 \arcsin \left (a x \right )\right )+5 \sqrt {3}\, \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )+60 \arcsin \left (a x \right )^{2} \cos \left (3 \arcsin \left (a x \right )\right )-540 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}-405 \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )+810 a x \arcsin \left (a x \right )-30 \arcsin \left (a x \right ) \sin \left (3 \arcsin \left (a x \right )\right )}{864 a^{3} \sqrt {\arcsin \left (a x \right )}} \]
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 \[ \int x^2\,{\mathrm {asin}\left (a\,x\right )}^{5/2} \,d x \]
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 \[ \int x^{2} \operatorname {asin}^{\frac {5}{2}}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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