Optimal. Leaf size=313 \[ -\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {\sqrt {\frac {\pi }{6}} b^{3/2} \cos \left (\frac {3 a}{b}\right ) C\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {\sqrt {\frac {\pi }{6}} b^{3/2} \sin \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{24 c^3}+\frac {b x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{6 c}+\frac {b \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{3 c^3}+\frac {1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2} \]
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Rubi [A] time = 1.05, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 11, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {4629, 4707, 4677, 4623, 3306, 3305, 3351, 3304, 3352, 4635, 4406} \[ -\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {\sqrt {\frac {\pi }{6}} b^{3/2} \cos \left (\frac {3 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {\sqrt {\frac {\pi }{6}} b^{3/2} \sin \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{24 c^3}+\frac {b x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{6 c}+\frac {b \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{3 c^3}+\frac {1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3305
Rule 3306
Rule 3351
Rule 3352
Rule 4406
Rule 4623
Rule 4629
Rule 4635
Rule 4677
Rule 4707
Rubi steps
\begin {align*} \int x^2 \left (a+b \sin ^{-1}(c x)\right )^{3/2} \, dx &=\frac {1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac {1}{2} (b c) \int \frac {x^3 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {b x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{6 c}+\frac {1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac {1}{12} b^2 \int \frac {x^2}{\sqrt {a+b \sin ^{-1}(c x)}} \, dx-\frac {b \int \frac {x \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {1-c^2 x^2}} \, dx}{3 c}\\ &=\frac {b \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{3 c^3}+\frac {b x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{6 c}+\frac {1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac {b^2 \operatorname {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{12 c^3}-\frac {b^2 \int \frac {1}{\sqrt {a+b \sin ^{-1}(c x)}} \, dx}{6 c^2}\\ &=\frac {b \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{3 c^3}+\frac {b x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{6 c}+\frac {1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac {b \operatorname {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c x)\right )}{6 c^3}-\frac {b^2 \operatorname {Subst}\left (\int \left (\frac {\cos (x)}{4 \sqrt {a+b x}}-\frac {\cos (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{12 c^3}\\ &=\frac {b \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{3 c^3}+\frac {b x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{6 c}+\frac {1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac {b^2 \operatorname {Subst}\left (\int \frac {\cos (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{48 c^3}+\frac {b^2 \operatorname {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{48 c^3}-\frac {\left (b \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c x)\right )}{6 c^3}-\frac {\left (b \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c x)\right )}{6 c^3}\\ &=\frac {b \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{3 c^3}+\frac {b x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{6 c}+\frac {1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac {\left (b \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{3 c^3}-\frac {\left (b^2 \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{48 c^3}+\frac {\left (b^2 \cos \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{48 c^3}-\frac {\left (b \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{3 c^3}-\frac {\left (b^2 \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{48 c^3}+\frac {\left (b^2 \sin \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{48 c^3}\\ &=\frac {b \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{3 c^3}+\frac {b x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{6 c}+\frac {1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac {b^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{3 c^3}-\frac {b^{3/2} \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{3 c^3}-\frac {\left (b \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{24 c^3}+\frac {\left (b \cos \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{24 c^3}-\frac {\left (b \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{24 c^3}+\frac {\left (b \sin \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{24 c^3}\\ &=\frac {b \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{3 c^3}+\frac {b x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{6 c}+\frac {1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {b^{3/2} \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) C\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{8 c^3}+\frac {b^{3/2} \sqrt {\frac {\pi }{6}} S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{24 c^3}\\ \end {align*}
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Mathematica [C] time = 0.31, size = 245, normalized size = 0.78 \[ \frac {b e^{-\frac {3 i a}{b}} \sqrt {a+b \sin ^{-1}(c x)} \left (27 e^{\frac {2 i a}{b}} \sqrt {\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {5}{2},-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+27 e^{\frac {4 i a}{b}} \sqrt {-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {5}{2},\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-\sqrt {3} \left (\sqrt {\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {5}{2},-\frac {3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+e^{\frac {6 i a}{b}} \sqrt {-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {5}{2},\frac {3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )\right )}{216 c^3 \sqrt {\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}}} \]
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 [C] time = 4.17, size = 1967, normalized size = 6.28 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.17, size = 542, normalized size = 1.73 \[ -\frac {-\sqrt {\pi }\, \sqrt {2}\, \sqrt {3}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {3 a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {\frac {1}{b}}\, b^{2}-\sqrt {\pi }\, \sqrt {2}\, \sqrt {3}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {3 a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {\frac {1}{b}}\, b^{2}+27 \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \cos \left (\frac {a}{b}\right ) \sqrt {\frac {1}{b}}\, b^{2}+27 \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {\frac {1}{b}}\, b^{2}+12 \arcsin \left (c x \right )^{2} \sin \left (\frac {3 a +3 b \arcsin \left (c x \right )}{b}-\frac {3 a}{b}\right ) b^{2}-36 \arcsin \left (c x \right )^{2} \sin \left (\frac {a +b \arcsin \left (c x \right )}{b}-\frac {a}{b}\right ) b^{2}-54 \arcsin \left (c x \right ) \cos \left (\frac {a +b \arcsin \left (c x \right )}{b}-\frac {a}{b}\right ) b^{2}+24 \arcsin \left (c x \right ) \sin \left (\frac {3 a +3 b \arcsin \left (c x \right )}{b}-\frac {3 a}{b}\right ) a b +6 \arcsin \left (c x \right ) \cos \left (\frac {3 a +3 b \arcsin \left (c x \right )}{b}-\frac {3 a}{b}\right ) b^{2}-72 \arcsin \left (c x \right ) \sin \left (\frac {a +b \arcsin \left (c x \right )}{b}-\frac {a}{b}\right ) a b -54 \cos \left (\frac {a +b \arcsin \left (c x \right )}{b}-\frac {a}{b}\right ) a b +12 \sin \left (\frac {3 a +3 b \arcsin \left (c x \right )}{b}-\frac {3 a}{b}\right ) a^{2}+6 \cos \left (\frac {3 a +3 b \arcsin \left (c x \right )}{b}-\frac {3 a}{b}\right ) a b -36 \sin \left (\frac {a +b \arcsin \left (c x \right )}{b}-\frac {a}{b}\right ) a^{2}}{144 c^{3} \sqrt {a +b \arcsin \left (c x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}} x^{2}\,{d x} \]
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 \[ \int x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{3/2} \,d x \]
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 \[ \int x^{2} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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