Optimal. Leaf size=188 \[ -\frac{2 b c x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}+\frac{2 b x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d}+\frac{2 b^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{27 c^2}+\frac{4 b^2 \sqrt{d-c^2 d x^2}}{9 c^2} \]
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Rubi [A] time = 0.159379, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {4677, 4645, 444, 43} \[ -\frac{2 b c x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}+\frac{2 b x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d}+\frac{2 b^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{27 c^2}+\frac{4 b^2 \sqrt{d-c^2 d x^2}}{9 c^2} \]
Antiderivative was successfully verified.
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Rule 4677
Rule 4645
Rule 444
Rule 43
Rubi steps
\begin{align*} \int x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d}+\frac{\left (2 b \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 c \sqrt{1-c^2 x^2}}\\ &=\frac{2 b x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt{1-c^2 x^2}}-\frac{2 b c x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d}-\frac{\left (2 b^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x \left (1-\frac{c^2 x^2}{3}\right )}{\sqrt{1-c^2 x^2}} \, dx}{3 \sqrt{1-c^2 x^2}}\\ &=\frac{2 b x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt{1-c^2 x^2}}-\frac{2 b c x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d}-\frac{\left (b^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{c^2 x}{3}}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )}{3 \sqrt{1-c^2 x^2}}\\ &=\frac{2 b x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt{1-c^2 x^2}}-\frac{2 b c x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d}-\frac{\left (b^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{2}{3 \sqrt{1-c^2 x}}+\frac{1}{3} \sqrt{1-c^2 x}\right ) \, dx,x,x^2\right )}{3 \sqrt{1-c^2 x^2}}\\ &=\frac{4 b^2 \sqrt{d-c^2 d x^2}}{9 c^2}+\frac{2 b^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{27 c^2}+\frac{2 b x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt{1-c^2 x^2}}-\frac{2 b c x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d}\\ \end{align*}
Mathematica [A] time = 0.259402, size = 120, normalized size = 0.64 \[ \frac{\sqrt{d-c^2 d x^2} \left (\left (c^2 x^2-1\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2 b \left (3 a c x \left (c^2 x^2-3\right )+b \sqrt{1-c^2 x^2} \left (c^2 x^2-7\right )+3 b c x \left (c^2 x^2-3\right ) \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}\right )}{3 c^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.226, size = 700, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.68768, size = 254, normalized size = 1.35 \begin{align*} -\frac{2}{27} \, b^{2}{\left (\frac{\sqrt{-c^{2} x^{2} + 1} d^{\frac{3}{2}} x^{2} - \frac{7 \, \sqrt{-c^{2} x^{2} + 1} d^{\frac{3}{2}}}{c^{2}}}{d} + \frac{3 \,{\left (c^{2} d^{\frac{3}{2}} x^{3} - 3 \, d^{\frac{3}{2}} x\right )} \arcsin \left (c x\right )}{c d}\right )} - \frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} b^{2} \arcsin \left (c x\right )^{2}}{3 \, c^{2} d} - \frac{2 \,{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} a b \arcsin \left (c x\right )}{3 \, c^{2} d} - \frac{2 \,{\left (c^{2} d^{\frac{3}{2}} x^{3} - 3 \, d^{\frac{3}{2}} x\right )} a b}{9 \, c d} - \frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} a^{2}}{3 \, c^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.49072, size = 450, normalized size = 2.39 \begin{align*} \frac{6 \,{\left (a b c^{3} x^{3} - 3 \, a b c x +{\left (b^{2} c^{3} x^{3} - 3 \, 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 ({\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{4} x^{4} - 2 \,{\left (9 \, a^{2} - 8 \, b^{2}\right )} c^{2} x^{2} + 9 \,{\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \arcsin \left (c x\right )^{2} + 9 \, a^{2} - 14 \, b^{2} + 18 \,{\left (a b c^{4} x^{4} - 2 \, a b c^{2} x^{2} + a b\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{27 \,{\left (c^{4} x^{2} - c^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-c^{2} d x^{2} + d}{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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