Optimal. Leaf size=23 \[ \frac{1}{3} \left (x^2-2\right )^{3/2}+\sqrt{x^2-2} \]
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Rubi [A] time = 0.0211766, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {1593, 444, 43} \[ \frac{1}{3} \left (x^2-2\right )^{3/2}+\sqrt{x^2-2} \]
Antiderivative was successfully verified.
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Rule 1593
Rule 444
Rule 43
Rubi steps
\begin{align*} \int \frac{-x+x^3}{\sqrt{-2+x^2}} \, dx &=\int \frac{x \left (-1+x^2\right )}{\sqrt{-2+x^2}} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{-1+x}{\sqrt{-2+x}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{\sqrt{-2+x}}+\sqrt{-2+x}\right ) \, dx,x,x^2\right )\\ &=\sqrt{-2+x^2}+\frac{1}{3} \left (-2+x^2\right )^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0081222, size = 18, normalized size = 0.78 \[ \frac{1}{3} \sqrt{x^2-2} \left (x^2+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 15, normalized size = 0.7 \begin{align*}{\frac{{x}^{2}+1}{3}\sqrt{{x}^{2}-2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.970398, size = 30, normalized size = 1.3 \begin{align*} \frac{1}{3} \, \sqrt{x^{2} - 2} x^{2} + \frac{1}{3} \, \sqrt{x^{2} - 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42835, size = 39, normalized size = 1.7 \begin{align*} \frac{1}{3} \,{\left (x^{2} + 1\right )} \sqrt{x^{2} - 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.343196, size = 22, normalized size = 0.96 \begin{align*} \frac{x^{2} \sqrt{x^{2} - 2}}{3} + \frac{\sqrt{x^{2} - 2}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14376, size = 23, normalized size = 1. \begin{align*} \frac{1}{3} \,{\left (x^{2} - 2\right )}^{\frac{3}{2}} + \sqrt{x^{2} - 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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