Optimal. Leaf size=189 \[ \frac{1}{4} x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c^2}+\frac{\sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b c^3 \sqrt{1-c^2 x^2}}-\frac{b c x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}+\frac{b x^2 \sqrt{d-c^2 d x^2}}{16 c \sqrt{1-c^2 x^2}} \]
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Rubi [A] time = 0.191156, antiderivative size = 189, 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 = {4697, 4707, 4641, 30} \[ \frac{1}{4} x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c^2}+\frac{\sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b c^3 \sqrt{1-c^2 x^2}}-\frac{b c x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}+\frac{b x^2 \sqrt{d-c^2 d x^2}}{16 c \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 4697
Rule 4707
Rule 4641
Rule 30
Rubi steps
\begin{align*} \int x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac{1}{4} x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{\sqrt{d-c^2 d x^2} \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{4 \sqrt{1-c^2 x^2}}-\frac{\left (b c \sqrt{d-c^2 d x^2}\right ) \int x^3 \, dx}{4 \sqrt{1-c^2 x^2}}\\ &=-\frac{b c x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}-\frac{x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c^2}+\frac{1}{4} x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{\sqrt{d-c^2 d x^2} \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{8 c^2 \sqrt{1-c^2 x^2}}+\frac{\left (b \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{8 c \sqrt{1-c^2 x^2}}\\ &=\frac{b x^2 \sqrt{d-c^2 d x^2}}{16 c \sqrt{1-c^2 x^2}}-\frac{b c x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}-\frac{x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c^2}+\frac{1}{4} x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{\sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b c^3 \sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0956823, size = 140, normalized size = 0.74 \[ \frac{\sqrt{d-c^2 d x^2} \left (a^2+2 a b c x \sqrt{1-c^2 x^2} \left (2 c^2 x^2-1\right )+2 b \sin ^{-1}(c x) \left (a+b c x \sqrt{1-c^2 x^2} \left (2 c^2 x^2-1\right )\right )+b^2 c^2 x^2 \left (1-c^2 x^2\right )+b^2 \sin ^{-1}(c x)^2\right )}{16 b c^3 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.206, size = 373, normalized size = 2. \begin{align*} -{\frac{ax}{4\,{c}^{2}d} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{ax}{8\,{c}^{2}}\sqrt{-{c}^{2}d{x}^{2}+d}}+{\frac{ad}{8\,{c}^{2}}\arctan \left ({x\sqrt{{c}^{2}d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}} \right ){\frac{1}{\sqrt{{c}^{2}d}}}}+{\frac{bc{x}^{4}}{16\,{c}^{2}{x}^{2}-16}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{b{x}^{2}}{16\,c \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{b{c}^{2}\arcsin \left ( cx \right ){x}^{5}}{4\,{c}^{2}{x}^{2}-4}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{3\,b\arcsin \left ( cx \right ){x}^{3}}{8\,{c}^{2}{x}^{2}-8}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{b\arcsin \left ( cx \right ) x}{8\,{c}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{b}{128\,{c}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{b \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{16\,{c}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{-c^{2} d x^{2} + d}{\left (b x^{2} \arcsin \left (c x\right ) + a x^{2}\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sqrt{- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname{asin}{\left (c x \right )}\right )\, dx \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 )} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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