Optimal. Leaf size=365 \[ \frac {51 a c x^2 \sqrt {c-a^2 c x^2}}{128 \sqrt {1-a^2 x^2}}-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{16 \sqrt {1-a^2 x^2}}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^3+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^3-\frac {45}{64} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)-\frac {3}{32} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^4}{32 a \sqrt {1-a^2 x^2}}+\frac {3 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{16 a}+\frac {27 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{128 a \sqrt {1-a^2 x^2}}-\frac {3 a^3 c x^4 \sqrt {c-a^2 c x^2}}{128 \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.32, antiderivative size = 365, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4649, 4647, 4641, 4627, 4707, 30, 4677, 14} \[ -\frac {3 a^3 c x^4 \sqrt {c-a^2 c x^2}}{128 \sqrt {1-a^2 x^2}}+\frac {51 a c x^2 \sqrt {c-a^2 c x^2}}{128 \sqrt {1-a^2 x^2}}-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{16 \sqrt {1-a^2 x^2}}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^3+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^3-\frac {45}{64} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)-\frac {3}{32} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^4}{32 a \sqrt {1-a^2 x^2}}+\frac {3 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{16 a}+\frac {27 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{128 a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 4627
Rule 4641
Rule 4647
Rule 4649
Rule 4677
Rule 4707
Rubi steps
\begin {align*} \int \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^3 \, dx &=\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^3+\frac {1}{4} (3 c) \int \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^3 \, dx-\frac {\left (3 a c \sqrt {c-a^2 c x^2}\right ) \int x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^2 \, dx}{4 \sqrt {1-a^2 x^2}}\\ &=\frac {3 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{16 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^3+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^3-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \int \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x) \, dx}{8 \sqrt {1-a^2 x^2}}+\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\sin ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{8 \sqrt {1-a^2 x^2}}-\frac {\left (9 a c \sqrt {c-a^2 c x^2}\right ) \int x \sin ^{-1}(a x)^2 \, dx}{8 \sqrt {1-a^2 x^2}}\\ &=-\frac {3}{32} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{16 \sqrt {1-a^2 x^2}}+\frac {3 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{16 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^3+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^3+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^4}{32 a \sqrt {1-a^2 x^2}}-\frac {\left (9 c \sqrt {c-a^2 c x^2}\right ) \int \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \, dx}{32 \sqrt {1-a^2 x^2}}+\frac {\left (3 a c \sqrt {c-a^2 c x^2}\right ) \int x \left (1-a^2 x^2\right ) \, dx}{32 \sqrt {1-a^2 x^2}}+\frac {\left (9 a^2 c \sqrt {c-a^2 c x^2}\right ) \int \frac {x^2 \sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{8 \sqrt {1-a^2 x^2}}\\ &=-\frac {45}{64} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)-\frac {3}{32} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{16 \sqrt {1-a^2 x^2}}+\frac {3 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{16 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^3+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^3+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^4}{32 a \sqrt {1-a^2 x^2}}-\frac {\left (9 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{64 \sqrt {1-a^2 x^2}}+\frac {\left (9 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{16 \sqrt {1-a^2 x^2}}+\frac {\left (3 a c \sqrt {c-a^2 c x^2}\right ) \int \left (x-a^2 x^3\right ) \, dx}{32 \sqrt {1-a^2 x^2}}+\frac {\left (9 a c \sqrt {c-a^2 c x^2}\right ) \int x \, dx}{64 \sqrt {1-a^2 x^2}}+\frac {\left (9 a c \sqrt {c-a^2 c x^2}\right ) \int x \, dx}{16 \sqrt {1-a^2 x^2}}\\ &=\frac {51 a c x^2 \sqrt {c-a^2 c x^2}}{128 \sqrt {1-a^2 x^2}}-\frac {3 a^3 c x^4 \sqrt {c-a^2 c x^2}}{128 \sqrt {1-a^2 x^2}}-\frac {45}{64} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)-\frac {3}{32} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)+\frac {27 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{128 a \sqrt {1-a^2 x^2}}-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{16 \sqrt {1-a^2 x^2}}+\frac {3 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{16 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^3+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^3+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^4}{32 a \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 138, normalized size = 0.38 \[ \frac {c \sqrt {c-a^2 c x^2} \left (96 \sin ^{-1}(a x)^4+32 \left (8 \sin \left (2 \sin ^{-1}(a x)\right )+\sin \left (4 \sin ^{-1}(a x)\right )\right ) \sin ^{-1}(a x)^3-12 \left (32 \sin \left (2 \sin ^{-1}(a x)\right )+\sin \left (4 \sin ^{-1}(a x)\right )\right ) \sin ^{-1}(a x)+24 \sin ^{-1}(a x)^2 \left (16 \cos \left (2 \sin ^{-1}(a x)\right )+\cos \left (4 \sin ^{-1}(a x)\right )\right )-3 \left (64 \cos \left (2 \sin ^{-1}(a x)\right )+\cos \left (4 \sin ^{-1}(a x)\right )\right )\right )}{1024 a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{2} c x^{2} - c\right )} \sqrt {-a^{2} c x^{2} + c} \arcsin \left (a x\right )^{3}, x\right ) \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.23, size = 474, normalized size = 1.30 \[ -\frac {3 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \arcsin \left (a x \right )^{4} c}{32 a \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (-8 i \sqrt {-a^{2} x^{2}+1}\, x^{4} a^{4}+8 a^{5} x^{5}+8 i \sqrt {-a^{2} x^{2}+1}\, x^{2} a^{2}-12 a^{3} x^{3}-i \sqrt {-a^{2} x^{2}+1}+4 a x \right ) \left (24 i \arcsin \left (a x \right )^{2}+32 \arcsin \left (a x \right )^{3}-3 i-12 \arcsin \left (a x \right )\right ) c}{2048 a \left (a^{2} x^{2}-1\right )}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-a^{2} x^{2}+1}\, x^{2} a^{2}+2 a^{3} x^{3}-i \sqrt {-a^{2} x^{2}+1}-2 a x \right ) \left (-6 i \arcsin \left (a x \right )^{2}+4 \arcsin \left (a x \right )^{3}+3 i-6 \arcsin \left (a x \right )\right ) c}{32 a \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (i x^{2} a^{2}-a x \sqrt {-a^{2} x^{2}+1}-i\right ) \left (408 i \arcsin \left (a x \right )^{2}+224 \arcsin \left (a x \right )^{3}-195 i-372 \arcsin \left (a x \right )\right ) \cos \left (3 \arcsin \left (a x \right )\right ) c}{2048 a \left (a^{2} x^{2}-1\right )}+\frac {9 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (i \sqrt {-a^{2} x^{2}+1}\, x a +a^{2} x^{2}-1\right ) \left (40 i \arcsin \left (a x \right )^{2}+32 \arcsin \left (a x \right )^{3}-21 i-44 \arcsin \left (a x \right )\right ) \sin \left (3 \arcsin \left (a x \right )\right ) c}{2048 a \left (a^{2} x^{2}-1\right )} \]
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 \[ \int {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arcsin \left (a x\right )^{3}\,{d x} \]
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 \[ \int {\mathrm {asin}\left (a\,x\right )}^3\,{\left (c-a^2\,c\,x^2\right )}^{3/2} \,d x \]
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 \[ \int \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {asin}^{3}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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