Optimal. Leaf size=211 \[ \frac{1}{5} d x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{4 b d x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c}-\frac{2 b d \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{25 c^3}+\frac{2 b d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 c^3}+\frac{8 b d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c^3}+\frac{2}{15} d x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{125} b^2 c^2 d x^5-\frac{52 b^2 d x}{225 c^2}-\frac{26}{675} b^2 d x^3 \]
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Rubi [A] time = 0.33748, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4699, 4627, 4707, 4677, 8, 30, 266, 43, 4689, 12} \[ \frac{1}{5} d x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{4 b d x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c}-\frac{2 b d \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{25 c^3}+\frac{2 b d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 c^3}+\frac{8 b d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c^3}+\frac{2}{15} d x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{125} b^2 c^2 d x^5-\frac{52 b^2 d x}{225 c^2}-\frac{26}{675} b^2 d x^3 \]
Antiderivative was successfully verified.
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Rule 4699
Rule 4627
Rule 4707
Rule 4677
Rule 8
Rule 30
Rule 266
Rule 43
Rule 4689
Rule 12
Rubi steps
\begin{align*} \int x^2 \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac{1}{5} d x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} (2 d) \int x^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac{1}{5} (2 b c d) \int x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=\frac{2 b d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 c^3}-\frac{2 b d \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{25 c^3}+\frac{2}{15} d x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} d x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{1}{15} (4 b c d) \int \frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx+\frac{1}{5} \left (2 b^2 c^2 d\right ) \int \frac{-2-c^2 x^2+3 c^4 x^4}{15 c^4} \, dx\\ &=\frac{4 b d x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c}+\frac{2 b d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 c^3}-\frac{2 b d \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{25 c^3}+\frac{2}{15} d x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} d x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{1}{45} \left (4 b^2 d\right ) \int x^2 \, dx+\frac{\left (2 b^2 d\right ) \int \left (-2-c^2 x^2+3 c^4 x^4\right ) \, dx}{75 c^2}-\frac{(8 b d) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{45 c}\\ &=-\frac{4 b^2 d x}{75 c^2}-\frac{26}{675} b^2 d x^3+\frac{2}{125} b^2 c^2 d x^5+\frac{8 b d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c^3}+\frac{4 b d x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c}+\frac{2 b d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 c^3}-\frac{2 b d \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{25 c^3}+\frac{2}{15} d x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} d x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (8 b^2 d\right ) \int 1 \, dx}{45 c^2}\\ &=-\frac{52 b^2 d x}{225 c^2}-\frac{26}{675} b^2 d x^3+\frac{2}{125} b^2 c^2 d x^5+\frac{8 b d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c^3}+\frac{4 b d x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c}+\frac{2 b d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 c^3}-\frac{2 b d \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{25 c^3}+\frac{2}{15} d x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} d x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2\\ \end{align*}
Mathematica [A] time = 0.21888, size = 179, normalized size = 0.85 \[ -\frac{d \left (225 a^2 c^3 x^3 \left (3 c^2 x^2-5\right )+30 a b \sqrt{1-c^2 x^2} \left (9 c^4 x^4-13 c^2 x^2-26\right )+30 b \sin ^{-1}(c x) \left (15 a c^3 x^3 \left (3 c^2 x^2-5\right )+b \sqrt{1-c^2 x^2} \left (9 c^4 x^4-13 c^2 x^2-26\right )\right )+b^2 \left (-54 c^5 x^5+130 c^3 x^3+780 c x\right )+225 b^2 c^3 x^3 \left (3 c^2 x^2-5\right ) \sin ^{-1}(c x)^2\right )}{3375 c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.087, size = 280, normalized size = 1.3 \begin{align*}{\frac{1}{{c}^{3}} \left ( -d{a}^{2} \left ({\frac{{c}^{5}{x}^{5}}{5}}-{\frac{{c}^{3}{x}^{3}}{3}} \right ) -d{b}^{2} \left ({\frac{ \left ( \arcsin \left ( cx \right ) \right ) ^{2} \left ({c}^{2}{x}^{2}-3 \right ) cx}{3}}+{\frac{4\,cx}{15}}-{\frac{4\,\arcsin \left ( cx \right ) }{15}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{2\, \left ({c}^{2}{x}^{2}-1 \right ) \arcsin \left ( cx \right ) }{45}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{ \left ( 2\,{c}^{2}{x}^{2}-6 \right ) cx}{135}}+{\frac{ \left ( \arcsin \left ( cx \right ) \right ) ^{2} \left ( 3\,{c}^{4}{x}^{4}-10\,{c}^{2}{x}^{2}+15 \right ) cx}{15}}+{\frac{2\,\arcsin \left ( cx \right ) \left ({c}^{2}{x}^{2}-1 \right ) ^{2}}{25}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{ \left ( 6\,{c}^{4}{x}^{4}-20\,{c}^{2}{x}^{2}+30 \right ) cx}{375}} \right ) -2\,dab \left ( 1/5\,\arcsin \left ( cx \right ){c}^{5}{x}^{5}-1/3\,{c}^{3}{x}^{3}\arcsin \left ( cx \right ) +1/25\,{c}^{4}{x}^{4}\sqrt{-{c}^{2}{x}^{2}+1}-{\frac{13\,{c}^{2}{x}^{2}\sqrt{-{c}^{2}{x}^{2}+1}}{225}}-{\frac{26\,\sqrt{-{c}^{2}{x}^{2}+1}}{225}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.65787, size = 478, normalized size = 2.27 \begin{align*} -\frac{1}{5} \, b^{2} c^{2} d x^{5} \arcsin \left (c x\right )^{2} - \frac{1}{5} \, a^{2} c^{2} d x^{5} + \frac{1}{3} \, b^{2} d x^{3} \arcsin \left (c x\right )^{2} - \frac{2}{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 )} a b c^{2} d - \frac{2}{1125} \,{\left (15 \,{\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 \arcsin \left (c x\right ) - \frac{9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} c^{2} d + \frac{1}{3} \, a^{2} d 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 )} a b d + \frac{2}{27} \,{\left (3 \, c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{c^{4}}\right )} \arcsin \left (c x\right ) - \frac{c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} d \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79953, size = 456, normalized size = 2.16 \begin{align*} -\frac{27 \,{\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{5} d x^{5} - 5 \,{\left (225 \, a^{2} - 26 \, b^{2}\right )} c^{3} d x^{3} + 780 \, b^{2} c d x + 225 \,{\left (3 \, b^{2} c^{5} d x^{5} - 5 \, b^{2} c^{3} d x^{3}\right )} \arcsin \left (c x\right )^{2} + 450 \,{\left (3 \, a b c^{5} d x^{5} - 5 \, a b c^{3} d x^{3}\right )} \arcsin \left (c x\right ) + 30 \,{\left (9 \, a b c^{4} d x^{4} - 13 \, a b c^{2} d x^{2} - 26 \, a b d +{\left (9 \, b^{2} c^{4} d x^{4} - 13 \, b^{2} c^{2} d x^{2} - 26 \, b^{2} d\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} x^{2} + 1}}{3375 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.33685, size = 313, normalized size = 1.48 \begin{align*} \begin{cases} - \frac{a^{2} c^{2} d x^{5}}{5} + \frac{a^{2} d x^{3}}{3} - \frac{2 a b c^{2} d x^{5} \operatorname{asin}{\left (c x \right )}}{5} - \frac{2 a b c d x^{4} \sqrt{- c^{2} x^{2} + 1}}{25} + \frac{2 a b d x^{3} \operatorname{asin}{\left (c x \right )}}{3} + \frac{26 a b d x^{2} \sqrt{- c^{2} x^{2} + 1}}{225 c} + \frac{52 a b d \sqrt{- c^{2} x^{2} + 1}}{225 c^{3}} - \frac{b^{2} c^{2} d x^{5} \operatorname{asin}^{2}{\left (c x \right )}}{5} + \frac{2 b^{2} c^{2} d x^{5}}{125} - \frac{2 b^{2} c d x^{4} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{25} + \frac{b^{2} d x^{3} \operatorname{asin}^{2}{\left (c x \right )}}{3} - \frac{26 b^{2} d x^{3}}{675} + \frac{26 b^{2} d x^{2} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{225 c} - \frac{52 b^{2} d x}{225 c^{2}} + \frac{52 b^{2} d \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{225 c^{3}} & \text{for}\: c \neq 0 \\\frac{a^{2} d x^{3}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.51352, size = 481, normalized size = 2.28 \begin{align*} -\frac{1}{5} \, a^{2} c^{2} d x^{5} + \frac{1}{3} \, a^{2} d x^{3} - \frac{{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d x \arcsin \left (c x\right )^{2}}{5 \, c^{2}} - \frac{2 \,{\left (c^{2} x^{2} - 1\right )}^{2} a b d x \arcsin \left (c x\right )}{5 \, c^{2}} - \frac{{\left (c^{2} x^{2} - 1\right )} b^{2} d x \arcsin \left (c x\right )^{2}}{15 \, c^{2}} + \frac{2 \,{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d x}{125 \, c^{2}} - \frac{2 \,{\left (c^{2} x^{2} - 1\right )} a b d x \arcsin \left (c x\right )}{15 \, c^{2}} + \frac{2 \, b^{2} d x \arcsin \left (c x\right )^{2}}{15 \, c^{2}} - \frac{2 \,{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} b^{2} d \arcsin \left (c x\right )}{25 \, c^{3}} - \frac{22 \,{\left (c^{2} x^{2} - 1\right )} b^{2} d x}{3375 \, c^{2}} + \frac{4 \, a b d x \arcsin \left (c x\right )}{15 \, c^{2}} - \frac{2 \,{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} a b d}{25 \, c^{3}} + \frac{2 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b^{2} d \arcsin \left (c x\right )}{45 \, c^{3}} - \frac{856 \, b^{2} d x}{3375 \, c^{2}} + \frac{2 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} a b d}{45 \, c^{3}} + \frac{4 \, \sqrt{-c^{2} x^{2} + 1} b^{2} d \arcsin \left (c x\right )}{15 \, c^{3}} + \frac{4 \, \sqrt{-c^{2} x^{2} + 1} a b d}{15 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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