Optimal. Leaf size=374 \[ -\frac {x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^2}-\frac {2 b c x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {2 b x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c \sqrt {1-c^2 x^2}}-\frac {2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^4}+\frac {4 a b x \sqrt {d-c^2 d x^2}}{15 c^3 \sqrt {1-c^2 x^2}}-\frac {2 b^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 c^4}+\frac {26 b^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{675 c^4}+\frac {52 b^2 \sqrt {d-c^2 d x^2}}{225 c^4}+\frac {4 b^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{15 c^3 \sqrt {1-c^2 x^2}} \]
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Rubi [A] time = 0.47, antiderivative size = 374, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {4697, 4707, 4677, 4619, 261, 4627, 266, 43} \[ \frac {4 a b x \sqrt {d-c^2 d x^2}}{15 c^3 \sqrt {1-c^2 x^2}}-\frac {2 b c x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {2 b x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c \sqrt {1-c^2 x^2}}-\frac {x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^2}-\frac {2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^4}-\frac {2 b^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 c^4}+\frac {26 b^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{675 c^4}+\frac {52 b^2 \sqrt {d-c^2 d x^2}}{225 c^4}+\frac {4 b^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{15 c^3 \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 261
Rule 266
Rule 4619
Rule 4627
Rule 4677
Rule 4697
Rule 4707
Rubi steps
\begin {align*} \int x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{5 \sqrt {1-c^2 x^2}}-\frac {\left (2 b c \sqrt {d-c^2 d x^2}\right ) \int x^4 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{5 \sqrt {1-c^2 x^2}}\\ &=-\frac {2 b c x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}-\frac {x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^2}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\left (2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{15 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (2 b \sqrt {d-c^2 d x^2}\right ) \int x^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{15 c \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5}{\sqrt {1-c^2 x^2}} \, dx}{25 \sqrt {1-c^2 x^2}}\\ &=\frac {2 b x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c \sqrt {1-c^2 x^2}}-\frac {2 b c x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}-\frac {2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^4}-\frac {x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^2}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (2 b^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^3}{\sqrt {1-c^2 x^2}} \, dx}{45 \sqrt {1-c^2 x^2}}+\frac {\left (4 b \sqrt {d-c^2 d x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{15 c^3 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{25 \sqrt {1-c^2 x^2}}\\ &=\frac {4 a b x \sqrt {d-c^2 d x^2}}{15 c^3 \sqrt {1-c^2 x^2}}+\frac {2 b x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c \sqrt {1-c^2 x^2}}-\frac {2 b c x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}-\frac {2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^4}-\frac {x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^2}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (b^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{45 \sqrt {1-c^2 x^2}}+\frac {\left (4 b^2 \sqrt {d-c^2 d x^2}\right ) \int \sin ^{-1}(c x) \, dx}{15 c^3 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{c^4 \sqrt {1-c^2 x}}-\frac {2 \sqrt {1-c^2 x}}{c^4}+\frac {\left (1-c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )}{25 \sqrt {1-c^2 x^2}}\\ &=-\frac {2 b^2 \sqrt {d-c^2 d x^2}}{25 c^4}+\frac {4 a b x \sqrt {d-c^2 d x^2}}{15 c^3 \sqrt {1-c^2 x^2}}+\frac {4 b^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{75 c^4}-\frac {2 b^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 c^4}+\frac {4 b^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{15 c^3 \sqrt {1-c^2 x^2}}+\frac {2 b x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c \sqrt {1-c^2 x^2}}-\frac {2 b c x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}-\frac {2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^4}-\frac {x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^2}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (b^2 \sqrt {d-c^2 d 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 )}{45 \sqrt {1-c^2 x^2}}-\frac {\left (4 b^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{15 c^2 \sqrt {1-c^2 x^2}}\\ &=\frac {52 b^2 \sqrt {d-c^2 d x^2}}{225 c^4}+\frac {4 a b x \sqrt {d-c^2 d x^2}}{15 c^3 \sqrt {1-c^2 x^2}}+\frac {26 b^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{675 c^4}-\frac {2 b^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 c^4}+\frac {4 b^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{15 c^3 \sqrt {1-c^2 x^2}}+\frac {2 b x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c \sqrt {1-c^2 x^2}}-\frac {2 b c x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}-\frac {2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^4}-\frac {x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^2}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2\\ \end {align*}
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Mathematica [A] time = 0.30, size = 242, normalized size = 0.65 \[ \frac {\sqrt {d-c^2 d x^2} \left (225 a^2 \sqrt {1-c^2 x^2} \left (3 c^4 x^4-c^2 x^2-2\right )-30 a b c x \left (9 c^4 x^4-5 c^2 x^2-30\right )-30 b \sin ^{-1}(c x) \left (15 a \sqrt {1-c^2 x^2} \left (-3 c^4 x^4+c^2 x^2+2\right )+b c x \left (9 c^4 x^4-5 c^2 x^2-30\right )\right )-2 b^2 \sqrt {1-c^2 x^2} \left (27 c^4 x^4+11 c^2 x^2-428\right )+225 b^2 \sqrt {1-c^2 x^2} \left (3 c^4 x^4-c^2 x^2-2\right ) \sin ^{-1}(c x)^2\right )}{3375 c^4 \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 277, normalized size = 0.74 \[ \frac {30 \, {\left (9 \, a b c^{5} x^{5} - 5 \, a b c^{3} x^{3} - 30 \, a b c x + {\left (9 \, b^{2} c^{5} x^{5} - 5 \, b^{2} c^{3} x^{3} - 30 \, b^{2} c x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + {\left (27 \, {\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{6} x^{6} - 4 \, {\left (225 \, a^{2} - 8 \, b^{2}\right )} c^{4} x^{4} - {\left (225 \, a^{2} - 878 \, b^{2}\right )} c^{2} x^{2} + 225 \, {\left (3 \, b^{2} c^{6} x^{6} - 4 \, b^{2} c^{4} x^{4} - b^{2} c^{2} x^{2} + 2 \, b^{2}\right )} \arcsin \left (c x\right )^{2} + 450 \, a^{2} - 856 \, b^{2} + 450 \, {\left (3 \, a b c^{6} x^{6} - 4 \, a b c^{4} x^{4} - a b c^{2} x^{2} + 2 \, a b\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{3375 \, {\left (c^{6} x^{2} - c^{4}\right )}} \]
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 = 0.58, size = 1165, normalized size = 3.11 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 311, normalized size = 0.83 \[ -\frac {1}{15} \, b^{2} {\left (\frac {3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{4} d}\right )} \arcsin \left (c x\right )^{2} - \frac {2}{15} \, a b {\left (\frac {3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{4} d}\right )} \arcsin \left (c x\right ) - \frac {1}{15} \, a^{2} {\left (\frac {3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{4} d}\right )} - \frac {2}{3375} \, b^{2} {\left (\frac {27 \, \sqrt {-c^{2} x^{2} + 1} c^{2} \sqrt {d} x^{4} + 11 \, \sqrt {-c^{2} x^{2} + 1} \sqrt {d} x^{2} - \frac {428 \, \sqrt {-c^{2} x^{2} + 1} \sqrt {d}}{c^{2}}}{c^{2}} + \frac {15 \, {\left (9 \, c^{4} \sqrt {d} x^{5} - 5 \, c^{2} \sqrt {d} x^{3} - 30 \, \sqrt {d} x\right )} \arcsin \left (c x\right )}{c^{3}}\right )} - \frac {2 \, {\left (9 \, c^{4} \sqrt {d} x^{5} - 5 \, c^{2} \sqrt {d} x^{3} - 30 \, \sqrt {d} x\right )} a b}{225 \, c^{3}} \]
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 \[ \int x^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {d-c^2\,d\,x^2} \,d x \]
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 \[ \int x^{3} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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