Optimal. Leaf size=84 \[ -\frac {1}{12} \sqrt {3 x^2+2} (2 x+3)^3+\frac {31}{36} \sqrt {3 x^2+2} (2 x+3)^2+\frac {5}{54} (171 x+809) \sqrt {3 x^2+2}+\frac {275 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.04, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, 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}{12} \sqrt {3 x^2+2} (2 x+3)^3+\frac {31}{36} \sqrt {3 x^2+2} (2 x+3)^2+\frac {5}{54} (171 x+809) \sqrt {3 x^2+2}+\frac {275 \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)^3}{\sqrt {2+3 x^2}} \, dx &=-\frac {1}{12} (3+2 x)^3 \sqrt {2+3 x^2}+\frac {1}{12} \int \frac {(3+2 x)^2 (192+93 x)}{\sqrt {2+3 x^2}} \, dx\\ &=\frac {31}{36} (3+2 x)^2 \sqrt {2+3 x^2}-\frac {1}{12} (3+2 x)^3 \sqrt {2+3 x^2}+\frac {1}{108} \int \frac {(3+2 x) (4440+5130 x)}{\sqrt {2+3 x^2}} \, dx\\ &=\frac {31}{36} (3+2 x)^2 \sqrt {2+3 x^2}-\frac {1}{12} (3+2 x)^3 \sqrt {2+3 x^2}+\frac {5}{54} (809+171 x) \sqrt {2+3 x^2}+\frac {275}{3} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=\frac {31}{36} (3+2 x)^2 \sqrt {2+3 x^2}-\frac {1}{12} (3+2 x)^3 \sqrt {2+3 x^2}+\frac {5}{54} (809+171 x) \sqrt {2+3 x^2}+\frac {275 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 50, normalized size = 0.60 \begin {gather*} \frac {1}{27} \left (825 \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-\sqrt {3 x^2+2} \left (18 x^3-12 x^2-585 x-2171\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.32, size = 61, normalized size = 0.73 \begin {gather*} \frac {1}{27} \sqrt {3 x^2+2} \left (-18 x^3+12 x^2+585 x+2171\right )-\frac {275 \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 = 0.43, size = 55, normalized size = 0.65 \begin {gather*} -\frac {1}{27} \, {\left (18 \, x^{3} - 12 \, x^{2} - 585 \, x - 2171\right )} \sqrt {3 \, x^{2} + 2} + \frac {275}{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 = 0.25, size = 49, normalized size = 0.58 \begin {gather*} -\frac {1}{27} \, {\left (3 \, {\left (2 \, {\left (3 \, x - 2\right )} x - 195\right )} x - 2171\right )} \sqrt {3 \, x^{2} + 2} - \frac {275}{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 = 0.06, size = 65, normalized size = 0.77 \begin {gather*} -\frac {2 \sqrt {3 x^{2}+2}\, x^{3}}{3}+\frac {4 \sqrt {3 x^{2}+2}\, x^{2}}{9}+\frac {65 \sqrt {3 x^{2}+2}\, x}{3}+\frac {275 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{9}+\frac {2171 \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 = 1.46, size = 64, normalized size = 0.76 \begin {gather*} -\frac {2}{3} \, \sqrt {3 \, x^{2} + 2} x^{3} + \frac {4}{9} \, \sqrt {3 \, x^{2} + 2} x^{2} + \frac {65}{3} \, \sqrt {3 \, x^{2} + 2} x + \frac {275}{9} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {2171}{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 = 0.04, size = 40, normalized size = 0.48 \begin {gather*} \frac {275\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{9}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (-2\,x^3+\frac {4\,x^2}{3}+65\,x+\frac {2171}{9}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.08, size = 80, normalized size = 0.95 \begin {gather*} - \frac {2 x^{3} \sqrt {3 x^{2} + 2}}{3} + \frac {4 x^{2} \sqrt {3 x^{2} + 2}}{9} + \frac {65 x \sqrt {3 x^{2} + 2}}{3} + \frac {2171 \sqrt {3 x^{2} + 2}}{27} + \frac {275 \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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