Optimal. Leaf size=78 \[ -\frac {1}{15} \left (3 x^2+2\right )^{3/2} (2 x+3)^2+\frac {2}{135} (99 x+431) \left (3 x^2+2\right )^{3/2}+\frac {131}{6} x \sqrt {3 x^2+2}+\frac {131 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.03, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {833, 780, 195, 215} \begin {gather*} -\frac {1}{15} \left (3 x^2+2\right )^{3/2} (2 x+3)^2+\frac {2}{135} (99 x+431) \left (3 x^2+2\right )^{3/2}+\frac {131}{6} x \sqrt {3 x^2+2}+\frac {131 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 195
Rule 215
Rule 780
Rule 833
Rubi steps
\begin {align*} \int (5-x) (3+2 x)^2 \sqrt {2+3 x^2} \, dx &=-\frac {1}{15} (3+2 x)^2 \left (2+3 x^2\right )^{3/2}+\frac {1}{15} \int (3+2 x) (233+132 x) \sqrt {2+3 x^2} \, dx\\ &=-\frac {1}{15} (3+2 x)^2 \left (2+3 x^2\right )^{3/2}+\frac {2}{135} (431+99 x) \left (2+3 x^2\right )^{3/2}+\frac {131}{3} \int \sqrt {2+3 x^2} \, dx\\ &=\frac {131}{6} x \sqrt {2+3 x^2}-\frac {1}{15} (3+2 x)^2 \left (2+3 x^2\right )^{3/2}+\frac {2}{135} (431+99 x) \left (2+3 x^2\right )^{3/2}+\frac {131}{3} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=\frac {131}{6} x \sqrt {2+3 x^2}-\frac {1}{15} (3+2 x)^2 \left (2+3 x^2\right )^{3/2}+\frac {2}{135} (431+99 x) \left (2+3 x^2\right )^{3/2}+\frac {131 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 55, normalized size = 0.71 \begin {gather*} \frac {1}{270} \left (3930 \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-\sqrt {3 x^2+2} \left (216 x^4-540 x^3-4542 x^2-6255 x-3124\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.22, size = 66, normalized size = 0.85 \begin {gather*} \frac {1}{270} \sqrt {3 x^2+2} \left (-216 x^4+540 x^3+4542 x^2+6255 x+3124\right )-\frac {131 \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 = 60, normalized size = 0.77 \begin {gather*} -\frac {1}{270} \, {\left (216 \, x^{4} - 540 \, x^{3} - 4542 \, x^{2} - 6255 \, x - 3124\right )} \sqrt {3 \, x^{2} + 2} + \frac {131}{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 = 0.16, size = 54, normalized size = 0.69 \begin {gather*} -\frac {1}{270} \, {\left (3 \, {\left (2 \, {\left (18 \, {\left (2 \, x - 5\right )} x - 757\right )} x - 2085\right )} x - 3124\right )} \sqrt {3 \, x^{2} + 2} - \frac {131}{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 = 63, normalized size = 0.81 \begin {gather*} -\frac {4 \left (3 x^{2}+2\right )^{\frac {3}{2}} x^{2}}{15}+\frac {2 \left (3 x^{2}+2\right )^{\frac {3}{2}} x}{3}+\frac {131 \sqrt {3 x^{2}+2}\, x}{6}+\frac {131 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{9}+\frac {781 \left (3 x^{2}+2\right )^{\frac {3}{2}}}{135} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 62, normalized size = 0.79 \begin {gather*} -\frac {4}{15} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x^{2} + \frac {2}{3} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {781}{135} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} + \frac {131}{6} \, \sqrt {3 \, x^{2} + 2} x + \frac {131}{9} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.71, size = 45, normalized size = 0.58 \begin {gather*} \frac {131\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{9}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (-\frac {12\,x^4}{5}+6\,x^3+\frac {757\,x^2}{15}+\frac {139\,x}{2}+\frac {1562}{45}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.93, size = 95, normalized size = 1.22 \begin {gather*} - \frac {4 x^{4} \sqrt {3 x^{2} + 2}}{5} + 2 x^{3} \sqrt {3 x^{2} + 2} + \frac {757 x^{2} \sqrt {3 x^{2} + 2}}{45} + \frac {139 x \sqrt {3 x^{2} + 2}}{6} + \frac {1562 \sqrt {3 x^{2} + 2}}{135} + \frac {131 \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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