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 = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {1}{3} \left (x^2-2\right )^{3/2}+\sqrt {x^2-2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 444
Rule 1593
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*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 0.78 \begin {gather*} \frac {1}{3} \sqrt {x^2-2} \left (x^2+1\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 18, normalized size = 0.78 \begin {gather*} \frac {1}{3} \sqrt {x^2-2} \left (x^2+1\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 14, normalized size = 0.61 \begin {gather*} \frac {1}{3} \, {\left (x^{2} + 1\right )} \sqrt {x^{2} - 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 17, normalized size = 0.74 \begin {gather*} \frac {1}{3} \, {\left (x^{2} - 2\right )}^{\frac {3}{2}} + \sqrt {x^{2} - 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 15, normalized size = 0.65 \begin {gather*} \frac {\left (x^{2}+1\right ) \sqrt {x^{2}-2}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 22, normalized size = 0.96 \begin {gather*} \frac {1}{3} \, \sqrt {x^{2} - 2} x^{2} + \frac {1}{3} \, \sqrt {x^{2} - 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 14, normalized size = 0.61 \begin {gather*} \frac {\left (x^2+1\right )\,\sqrt {x^2-2}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.69, size = 22, normalized size = 0.96 \begin {gather*} \frac {x^{2} \sqrt {x^{2} - 2}}{3} + \frac {\sqrt {x^{2} - 2}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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