Optimal. Leaf size=270 \[ \frac{f^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c \sqrt{d-c^2 d x^2}}-\frac{2 f g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^2 \sqrt{d-c^2 d x^2}}+\frac{g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c^3 \sqrt{d-c^2 d x^2}}-\frac{g^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 c^2 \sqrt{d-c^2 d x^2}}+\frac{2 b f g x \sqrt{1-c^2 x^2}}{c \sqrt{d-c^2 d x^2}}+\frac{b g^2 x^2 \sqrt{1-c^2 x^2}}{4 c \sqrt{d-c^2 d x^2}} \]
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Rubi [A] time = 0.433264, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {4777, 4763, 4641, 4677, 8, 4707, 30} \[ \frac{f^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c \sqrt{d-c^2 d x^2}}-\frac{2 f g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^2 \sqrt{d-c^2 d x^2}}+\frac{g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c^3 \sqrt{d-c^2 d x^2}}-\frac{g^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 c^2 \sqrt{d-c^2 d x^2}}+\frac{2 b f g x \sqrt{1-c^2 x^2}}{c \sqrt{d-c^2 d x^2}}+\frac{b g^2 x^2 \sqrt{1-c^2 x^2}}{4 c \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 4777
Rule 4763
Rule 4641
Rule 4677
Rule 8
Rule 4707
Rule 30
Rubi steps
\begin{align*} \int \frac{(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{d-c^2 d x^2}} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{d-c^2 d x^2}}\\ &=\frac{\sqrt{1-c^2 x^2} \int \left (\frac{f^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{2 f g x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{g^2 x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}\right ) \, dx}{\sqrt{d-c^2 d x^2}}\\ &=\frac{\left (f^2 \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{d-c^2 d x^2}}+\frac{\left (2 f g \sqrt{1-c^2 x^2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{d-c^2 d x^2}}+\frac{\left (g^2 \sqrt{1-c^2 x^2}\right ) \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{d-c^2 d x^2}}\\ &=-\frac{2 f g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^2 \sqrt{d-c^2 d x^2}}-\frac{g^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 c^2 \sqrt{d-c^2 d x^2}}+\frac{f^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c \sqrt{d-c^2 d x^2}}+\frac{\left (2 b f g \sqrt{1-c^2 x^2}\right ) \int 1 \, dx}{c \sqrt{d-c^2 d x^2}}+\frac{\left (g^2 \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{2 c^2 \sqrt{d-c^2 d x^2}}+\frac{\left (b g^2 \sqrt{1-c^2 x^2}\right ) \int x \, dx}{2 c \sqrt{d-c^2 d x^2}}\\ &=\frac{2 b f g x \sqrt{1-c^2 x^2}}{c \sqrt{d-c^2 d x^2}}+\frac{b g^2 x^2 \sqrt{1-c^2 x^2}}{4 c \sqrt{d-c^2 d x^2}}-\frac{2 f g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^2 \sqrt{d-c^2 d x^2}}-\frac{g^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 c^2 \sqrt{d-c^2 d x^2}}+\frac{f^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c \sqrt{d-c^2 d x^2}}+\frac{g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c^3 \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.692357, size = 266, normalized size = 0.99 \[ \frac{\sqrt{d} g \left (c^2 x^2-1\right ) \left (4 c \left (a \sqrt{1-c^2 x^2} (4 f+g x)-4 b c f x\right )+b g \cos \left (2 \sin ^{-1}(c x)\right )\right )-4 a \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (2 c^2 f^2+g^2\right ) \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-2 b \sqrt{d} \left (c^2 x^2-1\right ) \left (2 c^2 f^2+g^2\right ) \sin ^{-1}(c x)^2+2 b \sqrt{d} g \left (c^2 x^2-1\right ) \sin ^{-1}(c x) \left (8 c f \sqrt{1-c^2 x^2}+g \sin \left (2 \sin ^{-1}(c x)\right )\right )}{8 c^3 \sqrt{d} \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.37, size = 549, normalized size = 2. \begin{align*} -{\frac{a{g}^{2}x}{2\,{c}^{2}d}\sqrt{-{c}^{2}d{x}^{2}+d}}+{\frac{a{g}^{2}}{2\,{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}}}}-2\,{\frac{afg\sqrt{-{c}^{2}d{x}^{2}+d}}{{c}^{2}d}}+{a{f}^{2}\arctan \left ({x\sqrt{{c}^{2}d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}} \right ){\frac{1}{\sqrt{{c}^{2}d}}}}+2\,{\frac{b\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }fg\arcsin \left ( cx \right ) }{{c}^{2}d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{b{g}^{2}{x}^{2}}{4\,dc \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-2\,{\frac{b\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }fg\arcsin \left ( cx \right ){x}^{2}}{d \left ({c}^{2}{x}^{2}-1 \right ) }}-2\,{\frac{b\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }fg\sqrt{-{c}^{2}{x}^{2}+1}x}{dc \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{b \left ( \arcsin \left ( cx \right ) \right ) ^{2}{f}^{2}}{2\,dc \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}{g}^{2}}{4\,d{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{g}^{2}\arcsin \left ( cx \right ){x}^{3}}{2\,d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{b{g}^{2}\arcsin \left ( cx \right ) x}{2\,{c}^{2}d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{b{g}^{2}}{8\,d{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 (-\frac{\sqrt{-c^{2} d x^{2} + d}{\left (a g^{2} x^{2} + 2 \, a f g x + a f^{2} +{\left (b g^{2} x^{2} + 2 \, b f g x + b f^{2}\right )} \arcsin \left (c x\right )\right )}}{c^{2} d x^{2} - d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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 \frac{{\left (g x + f\right )}^{2}{\left (b \arcsin \left (c x\right ) + a\right )}}{\sqrt{-c^{2} d x^{2} + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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