Optimal. Leaf size=81 \[ d x \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \cos ^{-1}(c x)\right )-\frac{b \sqrt{1-c^2 x^2} \left (3 c^2 d+e\right )}{3 c^3}+\frac{b e \left (1-c^2 x^2\right )^{3/2}}{9 c^3} \]
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Rubi [A] time = 0.0674421, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {4666, 444, 43} \[ d x \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \cos ^{-1}(c x)\right )-\frac{b \sqrt{1-c^2 x^2} \left (3 c^2 d+e\right )}{3 c^3}+\frac{b e \left (1-c^2 x^2\right )^{3/2}}{9 c^3} \]
Antiderivative was successfully verified.
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Rule 4666
Rule 444
Rule 43
Rubi steps
\begin{align*} \int \left (d+e x^2\right ) \left (a+b \cos ^{-1}(c x)\right ) \, dx &=d x \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \cos ^{-1}(c x)\right )+(b c) \int \frac{x \left (d+\frac{e x^2}{3}\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=d x \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{2} (b c) \operatorname{Subst}\left (\int \frac{d+\frac{e x}{3}}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )\\ &=d x \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{2} (b c) \operatorname{Subst}\left (\int \left (\frac{3 c^2 d+e}{3 c^2 \sqrt{1-c^2 x}}-\frac{e \sqrt{1-c^2 x}}{3 c^2}\right ) \, dx,x,x^2\right )\\ &=-\frac{b \left (3 c^2 d+e\right ) \sqrt{1-c^2 x^2}}{3 c^3}+\frac{b e \left (1-c^2 x^2\right )^{3/2}}{9 c^3}+d x \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \cos ^{-1}(c x)\right )\\ \end{align*}
Mathematica [A] time = 0.07248, size = 91, normalized size = 1.12 \[ a d x+\frac{1}{3} a e x^3-\frac{b d \sqrt{1-c^2 x^2}}{c}+b e \left (-\frac{2}{9 c^3}-\frac{x^2}{9 c}\right ) \sqrt{1-c^2 x^2}+b d x \cos ^{-1}(c x)+\frac{1}{3} b e x^3 \cos ^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 112, normalized size = 1.4 \begin{align*}{\frac{1}{c} \left ({\frac{a}{{c}^{2}} \left ({\frac{{c}^{3}{x}^{3}e}{3}}+d{c}^{3}x \right ) }+{\frac{b}{{c}^{2}} \left ({\frac{\arccos \left ( cx \right ){c}^{3}{x}^{3}e}{3}}+\arccos \left ( cx \right ) d{c}^{3}x+{\frac{e}{3} \left ( -{\frac{{c}^{2}{x}^{2}}{3}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{2}{3}\sqrt{-{c}^{2}{x}^{2}+1}} \right ) }-{c}^{2}d\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 = 1.65647, size = 127, normalized size = 1.57 \begin{align*} \frac{1}{3} \, a e x^{3} + \frac{1}{9} \,{\left (3 \, x^{3} \arccos \left (c x\right ) - c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b e + a d x + \frac{{\left (c x \arccos \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 = 2.1665, size = 186, normalized size = 2.3 \begin{align*} \frac{3 \, a c^{3} e x^{3} + 9 \, a c^{3} d x + 3 \,{\left (b c^{3} e x^{3} + 3 \, b c^{3} d x\right )} \arccos \left (c x\right ) -{\left (b c^{2} e x^{2} + 9 \, b c^{2} d + 2 \, b e\right )} \sqrt{-c^{2} x^{2} + 1}}{9 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.678229, size = 114, normalized size = 1.41 \begin{align*} \begin{cases} a d x + \frac{a e x^{3}}{3} + b d x \operatorname{acos}{\left (c x \right )} + \frac{b e x^{3} \operatorname{acos}{\left (c x \right )}}{3} - \frac{b d \sqrt{- c^{2} x^{2} + 1}}{c} - \frac{b e x^{2} \sqrt{- c^{2} x^{2} + 1}}{9 c} - \frac{2 b e \sqrt{- c^{2} x^{2} + 1}}{9 c^{3}} & \text{for}\: c \neq 0 \\\left (a + \frac{\pi b}{2}\right ) \left (d x + \frac{e x^{3}}{3}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16234, size = 128, normalized size = 1.58 \begin{align*} \frac{1}{3} \, b x^{3} \arccos \left (c x\right ) e + \frac{1}{3} \, a x^{3} e + b d x \arccos \left (c x\right ) - \frac{\sqrt{-c^{2} x^{2} + 1} b x^{2} e}{9 \, c} + a d x - \frac{\sqrt{-c^{2} x^{2} + 1} b d}{c} - \frac{2 \, \sqrt{-c^{2} x^{2} + 1} b e}{9 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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