Optimal. Leaf size=60 \[ \frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )+\frac{b \left (1-c^2 x^2\right )^{3/2}}{9 c^3}-\frac{b \sqrt{1-c^2 x^2}}{3 c^3} \]
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Rubi [A] time = 0.0418091, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4628, 266, 43} \[ \frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )+\frac{b \left (1-c^2 x^2\right )^{3/2}}{9 c^3}-\frac{b \sqrt{1-c^2 x^2}}{3 c^3} \]
Antiderivative was successfully verified.
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Rule 4628
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^2 \left (a+b \cos ^{-1}(c x)\right ) \, dx &=\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{3} (b c) \int \frac{x^3}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{6} (b c) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )\\ &=\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{6} (b c) \operatorname{Subst}\left (\int \left (\frac{1}{c^2 \sqrt{1-c^2 x}}-\frac{\sqrt{1-c^2 x}}{c^2}\right ) \, dx,x,x^2\right )\\ &=-\frac{b \sqrt{1-c^2 x^2}}{3 c^3}+\frac{b \left (1-c^2 x^2\right )^{3/2}}{9 c^3}+\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )\\ \end{align*}
Mathematica [A] time = 0.0417632, size = 55, normalized size = 0.92 \[ \frac{a x^3}{3}+b \left (-\frac{2}{9 c^3}-\frac{x^2}{9 c}\right ) \sqrt{1-c^2 x^2}+\frac{1}{3} b x^3 \cos ^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 64, normalized size = 1.1 \begin{align*}{\frac{1}{{c}^{3}} \left ({\frac{{c}^{3}{x}^{3}a}{3}}+b \left ({\frac{{c}^{3}{x}^{3}\arccos \left ( cx \right ) }{3}}-{\frac{{c}^{2}{x}^{2}}{9}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{2}{9}\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 = 1.4429, size = 81, normalized size = 1.35 \begin{align*} \frac{1}{3} \, a 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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.59659, size = 119, normalized size = 1.98 \begin{align*} \frac{3 \, b c^{3} x^{3} \arccos \left (c x\right ) + 3 \, a c^{3} x^{3} -{\left (b c^{2} x^{2} + 2 \, b\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 = 0.642933, size = 70, normalized size = 1.17 \begin{align*} \begin{cases} \frac{a x^{3}}{3} + \frac{b x^{3} \operatorname{acos}{\left (c x \right )}}{3} - \frac{b x^{2} \sqrt{- c^{2} x^{2} + 1}}{9 c} - \frac{2 b \sqrt{- c^{2} x^{2} + 1}}{9 c^{3}} & \text{for}\: c \neq 0 \\\frac{x^{3} \left (a + \frac{\pi b}{2}\right )}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1333, size = 76, normalized size = 1.27 \begin{align*} \frac{1}{3} \, b x^{3} \arccos \left (c x\right ) + \frac{1}{3} \, a x^{3} - \frac{\sqrt{-c^{2} x^{2} + 1} b x^{2}}{9 \, c} - \frac{2 \, \sqrt{-c^{2} x^{2} + 1} b}{9 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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