Optimal. Leaf size=67 \[ \frac{b x \sqrt{1-c^2 x^2}}{c \sqrt{d-c^2 d x^2}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^2 d} \]
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Rubi [A] time = 0.0605404, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {4677, 8} \[ \frac{b x \sqrt{1-c^2 x^2}}{c \sqrt{d-c^2 d x^2}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^2 d} \]
Antiderivative was successfully verified.
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Rule 4677
Rule 8
Rubi steps
\begin{align*} \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{d-c^2 d x^2}} \, dx &=-\frac{\sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^2 d}+\frac{\left (b \sqrt{1-c^2 x^2}\right ) \int 1 \, dx}{c \sqrt{d-c^2 d x^2}}\\ &=\frac{b x \sqrt{1-c^2 x^2}}{c \sqrt{d-c^2 d x^2}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^2 d}\\ \end{align*}
Mathematica [A] time = 0.0315414, size = 64, normalized size = 0.96 \[ \frac{a \left (c^2 x^2-1\right )+b c x \sqrt{1-c^2 x^2}+b \left (c^2 x^2-1\right ) \sin ^{-1}(c x)}{c^2 \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.093, size = 159, normalized size = 2.4 \begin{align*} -{\frac{a}{{c}^{2}d}\sqrt{-{c}^{2}d{x}^{2}+d}}+b \left ( -{\frac{\arcsin \left ( cx \right ) +i}{2\,{c}^{2}d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) } \left ({c}^{2}{x}^{2}-i\sqrt{-{c}^{2}{x}^{2}+1}xc-1 \right ) }-{\frac{\arcsin \left ( cx \right ) -i}{2\,{c}^{2}d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) } \left ( i\sqrt{-{c}^{2}{x}^{2}+1}xc+{c}^{2}{x}^{2}-1 \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.68097, size = 78, normalized size = 1.16 \begin{align*} \frac{b x}{c \sqrt{d}} - \frac{\sqrt{-c^{2} d x^{2} + d} b \arcsin \left (c x\right )}{c^{2} d} - \frac{\sqrt{-c^{2} d x^{2} + d} a}{c^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80182, size = 188, normalized size = 2.81 \begin{align*} -\frac{\sqrt{-c^{2} d x^{2} + d} \sqrt{-c^{2} x^{2} + 1} b c x +{\left (a c^{2} x^{2} +{\left (b c^{2} x^{2} - b\right )} \arcsin \left (c x\right ) - a\right )} \sqrt{-c^{2} d x^{2} + d}}{c^{4} d x^{2} - c^{2} d} \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 (b \arcsin \left (c x\right ) + a\right )} x}{\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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