Optimal. Leaf size=45 \[ \frac{a x^2}{2}+\frac{b \sqrt{1-c^2 x^4}}{2 c}+\frac{1}{2} b x^2 \sin ^{-1}\left (c x^2\right ) \]
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Rubi [A] time = 0.0388268, antiderivative size = 45, 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 = {6715, 4619, 261} \[ \frac{a x^2}{2}+\frac{b \sqrt{1-c^2 x^4}}{2 c}+\frac{1}{2} b x^2 \sin ^{-1}\left (c x^2\right ) \]
Antiderivative was successfully verified.
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Rule 6715
Rule 4619
Rule 261
Rubi steps
\begin{align*} \int x \left (a+b \sin ^{-1}\left (c x^2\right )\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \left (a+b \sin ^{-1}(c x)\right ) \, dx,x,x^2\right )\\ &=\frac{a x^2}{2}+\frac{1}{2} b \operatorname{Subst}\left (\int \sin ^{-1}(c x) \, dx,x,x^2\right )\\ &=\frac{a x^2}{2}+\frac{1}{2} b x^2 \sin ^{-1}\left (c x^2\right )-\frac{1}{2} (b c) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-c^2 x^2}} \, dx,x,x^2\right )\\ &=\frac{a x^2}{2}+\frac{b \sqrt{1-c^2 x^4}}{2 c}+\frac{1}{2} b x^2 \sin ^{-1}\left (c x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0084366, size = 43, normalized size = 0.96 \[ \frac{a x^2}{2}+\frac{1}{2} b \left (\frac{\sqrt{1-c^2 x^4}}{c}+x^2 \sin ^{-1}\left (c x^2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 39, normalized size = 0.9 \begin{align*}{\frac{1}{2\,c} \left ( a{x}^{2}c+b \left ({x}^{2}c\arcsin \left ( c{x}^{2} \right ) +\sqrt{-{c}^{2}{x}^{4}+1} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47575, size = 50, normalized size = 1.11 \begin{align*} \frac{1}{2} \, a x^{2} + \frac{{\left (c x^{2} \arcsin \left (c x^{2}\right ) + \sqrt{-c^{2} x^{4} + 1}\right )} b}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.51839, size = 86, normalized size = 1.91 \begin{align*} \frac{b c x^{2} \arcsin \left (c x^{2}\right ) + a c x^{2} + \sqrt{-c^{2} x^{4} + 1} b}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.248994, size = 42, normalized size = 0.93 \begin{align*} \begin{cases} \frac{a x^{2}}{2} + \frac{b x^{2} \operatorname{asin}{\left (c x^{2} \right )}}{2} + \frac{b \sqrt{- c^{2} x^{4} + 1}}{2 c} & \text{for}\: c \neq 0 \\\frac{a x^{2}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15716, size = 51, normalized size = 1.13 \begin{align*} \frac{a c x^{2} +{\left (c x^{2} \arcsin \left (c x^{2}\right ) + \sqrt{-c^{2} x^{4} + 1}\right )} b}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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