Optimal. Leaf size=98 \[ \frac{(d+e x)^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e}-\frac{b \left (\frac{e^2}{c^2}+2 d^2\right ) \sin ^{-1}(c x)}{4 e}+\frac{b \sqrt{1-c^2 x^2} (d+e x)}{4 c}+\frac{3 b d \sqrt{1-c^2 x^2}}{4 c} \]
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Rubi [A] time = 0.0510849, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4743, 743, 641, 216} \[ \frac{(d+e x)^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e}-\frac{b \left (\frac{e^2}{c^2}+2 d^2\right ) \sin ^{-1}(c x)}{4 e}+\frac{b \sqrt{1-c^2 x^2} (d+e x)}{4 c}+\frac{3 b d \sqrt{1-c^2 x^2}}{4 c} \]
Antiderivative was successfully verified.
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Rule 4743
Rule 743
Rule 641
Rule 216
Rubi steps
\begin{align*} \int (d+e x) \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac{(d+e x)^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e}-\frac{(b c) \int \frac{(d+e x)^2}{\sqrt{1-c^2 x^2}} \, dx}{2 e}\\ &=\frac{b (d+e x) \sqrt{1-c^2 x^2}}{4 c}+\frac{(d+e x)^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e}+\frac{b \int \frac{-2 c^2 d^2-e^2-3 c^2 d e x}{\sqrt{1-c^2 x^2}} \, dx}{4 c e}\\ &=\frac{3 b d \sqrt{1-c^2 x^2}}{4 c}+\frac{b (d+e x) \sqrt{1-c^2 x^2}}{4 c}+\frac{(d+e x)^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e}-\frac{\left (b \left (2 c^2 d^2+e^2\right )\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{4 c e}\\ &=\frac{3 b d \sqrt{1-c^2 x^2}}{4 c}+\frac{b (d+e x) \sqrt{1-c^2 x^2}}{4 c}-\frac{b \left (2 d^2+\frac{e^2}{c^2}\right ) \sin ^{-1}(c x)}{4 e}+\frac{(d+e x)^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e}\\ \end{align*}
Mathematica [A] time = 0.0465409, size = 92, normalized size = 0.94 \[ a d x+\frac{1}{2} a e x^2+\frac{b d \sqrt{1-c^2 x^2}}{c}+\frac{b e x \sqrt{1-c^2 x^2}}{4 c}-\frac{b e \sin ^{-1}(c x)}{4 c^2}+b d x \sin ^{-1}(c x)+\frac{1}{2} b e x^2 \sin ^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 97, normalized size = 1. \begin{align*}{\frac{1}{c} \left ({\frac{a}{c} \left ({\frac{{x}^{2}{c}^{2}e}{2}}+d{c}^{2}x \right ) }+{\frac{b}{c} \left ({\frac{\arcsin \left ( cx \right ){c}^{2}{x}^{2}e}{2}}+\arcsin \left ( cx \right ) d{c}^{2}x-{\frac{e}{2} \left ( -{\frac{cx}{2}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{\arcsin \left ( cx \right ) }{2}} \right ) }+dc\sqrt{-{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 = 1.44807, size = 126, normalized size = 1.29 \begin{align*} \frac{1}{2} \, a e x^{2} + \frac{1}{4} \,{\left (2 \, x^{2} \arcsin \left (c x\right ) + c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x}{c^{2}} - \frac{\arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{2}}\right )}\right )} b e + a d x + \frac{{\left (c x \arcsin \left (c x\right ) + \sqrt{-c^{2} x^{2} + 1}\right )} b d}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.40443, size = 176, normalized size = 1.8 \begin{align*} \frac{2 \, a c^{2} e x^{2} + 4 \, a c^{2} d x +{\left (2 \, b c^{2} e x^{2} + 4 \, b c^{2} d x - b e\right )} \arcsin \left (c x\right ) +{\left (b c e x + 4 \, b c d\right )} \sqrt{-c^{2} x^{2} + 1}}{4 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.362445, size = 99, normalized size = 1.01 \begin{align*} \begin{cases} a d x + \frac{a e x^{2}}{2} + b d x \operatorname{asin}{\left (c x \right )} + \frac{b e x^{2} \operatorname{asin}{\left (c x \right )}}{2} + \frac{b d \sqrt{- c^{2} x^{2} + 1}}{c} + \frac{b e x \sqrt{- c^{2} x^{2} + 1}}{4 c} - \frac{b e \operatorname{asin}{\left (c x \right )}}{4 c^{2}} & \text{for}\: c \neq 0 \\a \left (d x + \frac{e x^{2}}{2}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29413, size = 138, normalized size = 1.41 \begin{align*} b d x \arcsin \left (c x\right ) + a d x + \frac{\sqrt{-c^{2} x^{2} + 1} b x e}{4 \, c} + \frac{{\left (c^{2} x^{2} - 1\right )} b \arcsin \left (c x\right ) e}{2 \, c^{2}} + \frac{\sqrt{-c^{2} x^{2} + 1} b d}{c} + \frac{{\left (c^{2} x^{2} - 1\right )} a e}{2 \, c^{2}} + \frac{b \arcsin \left (c x\right ) e}{4 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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