Optimal. Leaf size=125 \[ \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b}{6 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )}{6 c^3 d^2 \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 0.13, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4681, 266, 43} \[ \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b}{6 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )}{6 c^3 d^2 \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 4681
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \int \frac {x^3}{\left (1-c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1-c^2 x\right )^2} \, dx,x,x^2\right )}{6 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{c^2 \left (-1+c^2 x\right )^2}+\frac {1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{6 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b}{6 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )}{6 c^3 d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 103, normalized size = 0.82 \[ \frac {\sqrt {d-c^2 d x^2} \left (2 a c^3 x^3+2 b c^3 x^3 \sin ^{-1}(c x)-b \sqrt {1-c^2 x^2}-b \left (1-c^2 x^2\right )^{3/2} \log \left (c^2 x^2-1\right )\right )}{6 c^3 d^3 \left (c^2 x^2-1\right )^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} d x^{2} + d} {\left (b x^{2} \arcsin \left (c x\right ) + a x^{2}\right )}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.33, size = 1219, normalized size = 9.75 \[ \frac {a x}{3 c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {a x}{3 c^{2} d^{2} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} x^{5}}{3 d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right )}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-c^{2} x^{2}+1\right ) x^{3}}{6 d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right )}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} \left (-c^{2} x^{2}+1\right ) x^{5}}{6 d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{4} \arcsin \left (c x \right ) x^{7}}{d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right )}-\frac {2 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{4}}{d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c \sqrt {-c^{2} x^{2}+1}\, x^{4}}{2 d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right )}+\frac {4 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2}}{3 d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right ) c}-\frac {2 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )}{3 c^{3} d^{3} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} \arcsin \left (c x \right ) x^{5}}{d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right )}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3}}{6 d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, x^{2}}{2 d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right ) c}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{3 d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right ) c^{3}}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{3} \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{6}}{d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x^{3}}{3 d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}}{6 d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right ) c^{3}}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{4} x^{7}}{6 d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{3 c^{3} d^{3} \left (c^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.71, size = 153, normalized size = 1.22 \[ \frac {1}{6} \, b c {\left (\frac {1}{c^{6} d^{\frac {5}{2}} x^{2} - c^{4} d^{\frac {5}{2}}} - \frac {\log \left (c x + 1\right )}{c^{4} d^{\frac {5}{2}}} - \frac {\log \left (c x - 1\right )}{c^{4} d^{\frac {5}{2}}}\right )} - \frac {1}{3} \, b {\left (\frac {x}{\sqrt {-c^{2} d x^{2} + d} c^{2} d^{2}} - \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d}\right )} \arcsin \left (c x\right ) - \frac {1}{3} \, a {\left (\frac {x}{\sqrt {-c^{2} d x^{2} + d} c^{2} d^{2}} - \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d}\right )} \]
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 \[ \int \frac {x^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]
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 \[ \int \frac {x^{2} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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