Optimal. Leaf size=119 \[ \frac {a+b \sin ^{-1}(c x)}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b x}{6 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \tanh ^{-1}(c x)}{6 c^2 d^2 \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 0.08, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {4677, 199, 206} \[ \frac {a+b \sin ^{-1}(c x)}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b x}{6 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \tanh ^{-1}(c x)}{6 c^2 d^2 \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 199
Rule 206
Rule 4677
Rubi steps
\begin {align*} \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac {a+b \sin ^{-1}(c x)}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\left (1-c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b x}{6 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {a+b \sin ^{-1}(c x)}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int \frac {1}{1-c^2 x^2} \, dx}{6 c d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b x}{6 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {a+b \sin ^{-1}(c x)}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b \sqrt {1-c^2 x^2} \tanh ^{-1}(c x)}{6 c^2 d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 85, normalized size = 0.71 \[ \frac {-2 a+b c x \sqrt {1-c^2 x^2}+b \left (1-c^2 x^2\right )^{3/2} \tanh ^{-1}(c x)-2 b \sin ^{-1}(c x)}{6 c^2 d^2 \left (c^2 x^2-1\right ) \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.22, size = 374, normalized size = 3.14 \[ \left [-\frac {4 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} b c x - {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt {d} \log \left (-\frac {c^{6} d x^{6} + 5 \, c^{4} d x^{4} - 5 \, c^{2} d x^{2} + 4 \, {\left (c^{3} x^{3} + c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} \sqrt {d} - d}{c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1}\right ) - 8 \, \sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}}{24 \, {\left (c^{6} d^{3} x^{4} - 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}\right )}}, -\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} b c x + {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt {-d} \arctan \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} c \sqrt {-d} x}{c^{4} d x^{4} - d}\right ) - 4 \, \sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}}{12 \, {\left (c^{6} d^{3} x^{4} - 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}\right )}}\right ] \]
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 \[ \int \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.18, size = 259, normalized size = 2.18 \[ \frac {a}{3 c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, x}{6 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{3 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{2}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{6 c^{2} d^{3} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{6 c^{2} d^{3} \left (c^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {x\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]
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 \[ \int \frac {x \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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