Optimal. Leaf size=62 \[ -\frac {1}{9} \sqrt {3 x^2+2} (2 x+3)^2+\frac {2}{27} (36 x+251) \sqrt {3 x^2+2}+\frac {127 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.02, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {833, 780, 215} \begin {gather*} -\frac {1}{9} \sqrt {3 x^2+2} (2 x+3)^2+\frac {2}{27} (36 x+251) \sqrt {3 x^2+2}+\frac {127 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 215
Rule 780
Rule 833
Rubi steps
\begin {align*} \int \frac {(5-x) (3+2 x)^2}{\sqrt {2+3 x^2}} \, dx &=-\frac {1}{9} (3+2 x)^2 \sqrt {2+3 x^2}+\frac {1}{9} \int \frac {(3+2 x) (143+72 x)}{\sqrt {2+3 x^2}} \, dx\\ &=-\frac {1}{9} (3+2 x)^2 \sqrt {2+3 x^2}+\frac {2}{27} (251+36 x) \sqrt {2+3 x^2}+\frac {127}{3} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=-\frac {1}{9} (3+2 x)^2 \sqrt {2+3 x^2}+\frac {2}{27} (251+36 x) \sqrt {2+3 x^2}+\frac {127 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 45, normalized size = 0.73 \begin {gather*} \frac {1}{27} \left (381 \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-\sqrt {3 x^2+2} \left (12 x^2-36 x-475\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.24, size = 56, normalized size = 0.90 \begin {gather*} \frac {1}{27} \left (-12 x^2+36 x+475\right ) \sqrt {3 x^2+2}-\frac {127 \log \left (\sqrt {3 x^2+2}-\sqrt {3} x\right )}{3 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 50, normalized size = 0.81 \begin {gather*} -\frac {1}{27} \, {\left (12 \, x^{2} - 36 \, x - 475\right )} \sqrt {3 \, x^{2} + 2} + \frac {127}{18} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 42, normalized size = 0.68 \begin {gather*} -\frac {1}{27} \, {\left (12 \, {\left (x - 3\right )} x - 475\right )} \sqrt {3 \, x^{2} + 2} - \frac {127}{9} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 51, normalized size = 0.82 \begin {gather*} -\frac {4 \sqrt {3 x^{2}+2}\, x^{2}}{9}+\frac {4 \sqrt {3 x^{2}+2}\, x}{3}+\frac {127 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{9}+\frac {475 \sqrt {3 x^{2}+2}}{27} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 50, normalized size = 0.81 \begin {gather*} -\frac {4}{9} \, \sqrt {3 \, x^{2} + 2} x^{2} + \frac {4}{3} \, \sqrt {3 \, x^{2} + 2} x + \frac {127}{9} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {475}{27} \, \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 = 35, normalized size = 0.56 \begin {gather*} \frac {127\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{9}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (-\frac {4\,x^2}{3}+4\,x+\frac {475}{9}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.54, size = 63, normalized size = 1.02 \begin {gather*} - \frac {4 x^{2} \sqrt {3 x^{2} + 2}}{9} + \frac {4 x \sqrt {3 x^{2} + 2}}{3} + \frac {475 \sqrt {3 x^{2} + 2}}{27} + \frac {127 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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