Optimal. Leaf size=277 \[ -\frac {x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d}+\frac {2 b x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 d}+\frac {4 a b x \sqrt {1-c^2 x^2}}{3 c^3 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \left (1-c^2 x^2\right )^2}{27 c^4 \sqrt {d-c^2 d x^2}}+\frac {14 b^2 \left (1-c^2 x^2\right )}{9 c^4 \sqrt {d-c^2 d x^2}}+\frac {4 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{3 c^3 \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 0.33, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {4707, 4677, 4619, 261, 4627, 266, 43} \[ \frac {4 a b x \sqrt {1-c^2 x^2}}{3 c^3 \sqrt {d-c^2 d x^2}}+\frac {2 b x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c \sqrt {d-c^2 d x^2}}-\frac {x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d}-\frac {2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 d}-\frac {2 b^2 \left (1-c^2 x^2\right )^2}{27 c^4 \sqrt {d-c^2 d x^2}}+\frac {14 b^2 \left (1-c^2 x^2\right )}{9 c^4 \sqrt {d-c^2 d x^2}}+\frac {4 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{3 c^3 \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 261
Rule 266
Rule 4619
Rule 4627
Rule 4677
Rule 4707
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {d-c^2 d x^2}} \, dx &=-\frac {x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d}+\frac {2 \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {d-c^2 d x^2}} \, dx}{3 c^2}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int x^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 c \sqrt {d-c^2 d x^2}}\\ &=\frac {2 b x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 d}-\frac {x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d}-\frac {\left (2 b^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x^3}{\sqrt {1-c^2 x^2}} \, dx}{9 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b \sqrt {1-c^2 x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 c^3 \sqrt {d-c^2 d x^2}}\\ &=\frac {4 a b x \sqrt {1-c^2 x^2}}{3 c^3 \sqrt {d-c^2 d x^2}}+\frac {2 b x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 d}-\frac {x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d}-\frac {\left (b^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{9 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b^2 \sqrt {1-c^2 x^2}\right ) \int \sin ^{-1}(c x) \, dx}{3 c^3 \sqrt {d-c^2 d x^2}}\\ &=\frac {4 a b x \sqrt {1-c^2 x^2}}{3 c^3 \sqrt {d-c^2 d x^2}}+\frac {4 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{3 c^3 \sqrt {d-c^2 d x^2}}+\frac {2 b x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 d}-\frac {x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d}-\frac {\left (b^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{c^2 \sqrt {1-c^2 x}}-\frac {\sqrt {1-c^2 x}}{c^2}\right ) \, dx,x,x^2\right )}{9 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{3 c^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {4 a b x \sqrt {1-c^2 x^2}}{3 c^3 \sqrt {d-c^2 d x^2}}+\frac {14 b^2 \left (1-c^2 x^2\right )}{9 c^4 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \left (1-c^2 x^2\right )^2}{27 c^4 \sqrt {d-c^2 d x^2}}+\frac {4 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{3 c^3 \sqrt {d-c^2 d x^2}}+\frac {2 b x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 d}-\frac {x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 176, normalized size = 0.64 \[ \frac {9 a^2 \left (c^4 x^4+c^2 x^2-2\right )+6 a b c x \sqrt {1-c^2 x^2} \left (c^2 x^2+6\right )+6 b \sin ^{-1}(c x) \left (3 a \left (c^4 x^4+c^2 x^2-2\right )+b c x \sqrt {1-c^2 x^2} \left (c^2 x^2+6\right )\right )-2 b^2 \left (c^4 x^4+19 c^2 x^2-20\right )+9 b^2 \left (c^4 x^4+c^2 x^2-2\right ) \sin ^{-1}(c x)^2}{27 c^4 \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 210, normalized size = 0.76 \[ -\frac {6 \, {\left (a b c^{3} x^{3} + 6 \, a b c x + {\left (b^{2} c^{3} x^{3} + 6 \, b^{2} c x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + {\left ({\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{4} x^{4} + {\left (9 \, a^{2} - 38 \, b^{2}\right )} c^{2} x^{2} + 9 \, {\left (b^{2} c^{4} x^{4} + b^{2} c^{2} x^{2} - 2 \, b^{2}\right )} \arcsin \left (c x\right )^{2} - 18 \, a^{2} + 40 \, b^{2} + 18 \, {\left (a b c^{4} x^{4} + a b c^{2} x^{2} - 2 \, a b\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{27 \, {\left (c^{6} d x^{2} - c^{4} d\right )}} \]
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 = 0.59, size = 812, normalized size = 2.93 \[ a^{2} \left (-\frac {x^{2} \sqrt {-c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {-c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{2} x^{2}-2 i \sqrt {-c^{2} x^{2}+1}\, x c -1\right ) \left (6 i \arcsin \left (c x \right )+9 \arcsin \left (c x \right )^{2}-2\right )}{432 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, x c -1\right ) \left (\arcsin \left (c x \right )^{2}-2+2 i \arcsin \left (c x \right )\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )^{2}-2-2 i \arcsin \left (c x \right )\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, x c +2 c^{2} x^{2}-1\right ) \left (-6 i \arcsin \left (c x \right )+9 \arcsin \left (c x \right )^{2}-2\right )}{432 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (9 \arcsin \left (c x \right )^{2}-2\right ) \cos \left (4 \arcsin \left (c x \right )\right )}{216 c^{4} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \sin \left (4 \arcsin \left (c x \right )\right )}{36 c^{4} d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{2} x^{2}-2 i \sqrt {-c^{2} x^{2}+1}\, x c -1\right ) \left (i+3 \arcsin \left (c x \right )\right )}{144 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, x c -1\right ) \left (i+\arcsin \left (c x \right )\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, x c +2 c^{2} x^{2}-1\right ) \left (-i+3 \arcsin \left (c x \right )\right )}{144 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{24 c^{4} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sin \left (4 \arcsin \left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}-1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 251, normalized size = 0.91 \[ -\frac {1}{3} \, b^{2} {\left (\frac {\sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{2} d} + \frac {2 \, \sqrt {-c^{2} d x^{2} + d}}{c^{4} d}\right )} \arcsin \left (c x\right )^{2} - \frac {2}{3} \, a b {\left (\frac {\sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{2} d} + \frac {2 \, \sqrt {-c^{2} d x^{2} + d}}{c^{4} d}\right )} \arcsin \left (c x\right ) - \frac {1}{3} \, a^{2} {\left (\frac {\sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{2} d} + \frac {2 \, \sqrt {-c^{2} d x^{2} + d}}{c^{4} d}\right )} + \frac {2}{27} \, b^{2} {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2} + \frac {20 \, \sqrt {-c^{2} x^{2} + 1}}{c^{2}}}{c^{2} \sqrt {d}} + \frac {3 \, {\left (c^{2} x^{3} + 6 \, x\right )} \arcsin \left (c x\right )}{c^{3} \sqrt {d}}\right )} + \frac {2 \, {\left (c^{2} x^{3} + 6 \, x\right )} a b}{9 \, c^{3} \sqrt {d}} \]
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 \[ \int \frac {x^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{\sqrt {d-c^2\,d\,x^2}} \,d x \]
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 \[ \int \frac {x^{3} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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