Optimal. Leaf size=131 \[ \frac {1}{5} c^4 d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{3} c^2 d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {b d^2 \left (1-c^2 x^2\right )^{5/2}}{25 c}+\frac {4 b d^2 \left (1-c^2 x^2\right )^{3/2}}{45 c}+\frac {8 b d^2 \sqrt {1-c^2 x^2}}{15 c} \]
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Rubi [A] time = 0.10, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {194, 4645, 12, 1247, 698} \[ \frac {1}{5} c^4 d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{3} c^2 d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {b d^2 \left (1-c^2 x^2\right )^{5/2}}{25 c}+\frac {4 b d^2 \left (1-c^2 x^2\right )^{3/2}}{45 c}+\frac {8 b d^2 \sqrt {1-c^2 x^2}}{15 c} \]
Antiderivative was successfully verified.
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Rule 12
Rule 194
Rule 698
Rule 1247
Rule 4645
Rubi steps
\begin {align*} \int \left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=d^2 x \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{3} c^2 d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} c^4 d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {d^2 x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{15 \sqrt {1-c^2 x^2}} \, dx\\ &=d^2 x \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{3} c^2 d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} c^4 d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{15} \left (b c d^2\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=d^2 x \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{3} c^2 d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} c^4 d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{30} \left (b c d^2\right ) \operatorname {Subst}\left (\int \frac {15-10 c^2 x+3 c^4 x^2}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=d^2 x \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{3} c^2 d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} c^4 d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{30} \left (b c d^2\right ) \operatorname {Subst}\left (\int \left (\frac {8}{\sqrt {1-c^2 x}}+4 \sqrt {1-c^2 x}+3 \left (1-c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )\\ &=\frac {8 b d^2 \sqrt {1-c^2 x^2}}{15 c}+\frac {4 b d^2 \left (1-c^2 x^2\right )^{3/2}}{45 c}+\frac {b d^2 \left (1-c^2 x^2\right )^{5/2}}{25 c}+d^2 x \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{3} c^2 d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} c^4 d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.11, size = 95, normalized size = 0.73 \[ \frac {d^2 \left (15 a c x \left (3 c^4 x^4-10 c^2 x^2+15\right )+b \sqrt {1-c^2 x^2} \left (9 c^4 x^4-38 c^2 x^2+149\right )+15 b c x \left (3 c^4 x^4-10 c^2 x^2+15\right ) \sin ^{-1}(c x)\right )}{225 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 121, normalized size = 0.92 \[ \frac {45 \, a c^{5} d^{2} x^{5} - 150 \, a c^{3} d^{2} x^{3} + 225 \, a c d^{2} x + 15 \, {\left (3 \, b c^{5} d^{2} x^{5} - 10 \, b c^{3} d^{2} x^{3} + 15 \, b c d^{2} x\right )} \arcsin \left (c x\right ) + {\left (9 \, b c^{4} d^{2} x^{4} - 38 \, b c^{2} d^{2} x^{2} + 149 \, b d^{2}\right )} \sqrt {-c^{2} x^{2} + 1}}{225 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.56, size = 158, normalized size = 1.21 \[ \frac {1}{5} \, a c^{4} d^{2} x^{5} - \frac {2}{3} \, a c^{2} d^{2} x^{3} + \frac {1}{5} \, {\left (c^{2} x^{2} - 1\right )}^{2} b d^{2} x \arcsin \left (c x\right ) - \frac {4}{15} \, {\left (c^{2} x^{2} - 1\right )} b d^{2} x \arcsin \left (c x\right ) + \frac {8}{15} \, b d^{2} x \arcsin \left (c x\right ) + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d^{2}}{25 \, c} + a d^{2} x + \frac {4 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{2}}{45 \, c} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} b d^{2}}{15 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 122, normalized size = 0.93 \[ \frac {d^{2} a \left (\frac {1}{5} c^{5} x^{5}-\frac {2}{3} c^{3} x^{3}+c x \right )+d^{2} b \left (\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}-\frac {2 c^{3} x^{3} \arcsin \left (c x \right )}{3}+c x \arcsin \left (c x \right )+\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {38 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{225}+\frac {149 \sqrt {-c^{2} x^{2}+1}}{225}\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 196, normalized size = 1.50 \[ \frac {1}{5} \, a c^{4} d^{2} x^{5} + \frac {1}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b c^{4} d^{2} - \frac {2}{3} \, a c^{2} d^{2} x^{3} - \frac {2}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b c^{2} d^{2} + a d^{2} x + \frac {{\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b d^{2}}{c} \]
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 \[ \int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.14, size = 165, normalized size = 1.26 \[ \begin {cases} \frac {a c^{4} d^{2} x^{5}}{5} - \frac {2 a c^{2} d^{2} x^{3}}{3} + a d^{2} x + \frac {b c^{4} d^{2} x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {b c^{3} d^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{25} - \frac {2 b c^{2} d^{2} x^{3} \operatorname {asin}{\left (c x \right )}}{3} - \frac {38 b c d^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{225} + b d^{2} x \operatorname {asin}{\left (c x \right )} + \frac {149 b d^{2} \sqrt {- c^{2} x^{2} + 1}}{225 c} & \text {for}\: c \neq 0 \\a d^{2} x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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