Optimal. Leaf size=33 \[ \frac {5 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {3}}-\frac {1}{3} \sqrt {3 x^2+2} \]
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Rubi [A] time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {641, 215} \begin {gather*} \frac {5 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {3}}-\frac {1}{3} \sqrt {3 x^2+2} \end {gather*}
Antiderivative was successfully verified.
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Rule 215
Rule 641
Rubi steps
\begin {align*} \int \frac {5-x}{\sqrt {2+3 x^2}} \, dx &=-\frac {1}{3} \sqrt {2+3 x^2}+5 \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=-\frac {1}{3} \sqrt {2+3 x^2}+\frac {5 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 1.00 \begin {gather*} \frac {5 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {3}}-\frac {1}{3} \sqrt {3 x^2+2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.13, size = 44, normalized size = 1.33 \begin {gather*} -\frac {1}{3} \sqrt {3 x^2+2}-\frac {5 \log \left (\sqrt {3 x^2+2}-\sqrt {3} x\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 40, normalized size = 1.21 \begin {gather*} \frac {5}{6} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) - \frac {1}{3} \, \sqrt {3 \, x^{2} + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 34, normalized size = 1.03 \begin {gather*} -\frac {5}{3} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) - \frac {1}{3} \, \sqrt {3 \, x^{2} + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 25, normalized size = 0.76 \begin {gather*} \frac {5 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{3}-\frac {\sqrt {3 x^{2}+2}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 24, normalized size = 0.73 \begin {gather*} \frac {5}{3} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) - \frac {1}{3} \, \sqrt {3 \, x^{2} + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 25, normalized size = 0.76 \begin {gather*} \frac {5\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{3}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 29, normalized size = 0.88 \begin {gather*} - \frac {\sqrt {3 x^{2} + 2}}{3} + \frac {5 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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