Optimal. Leaf size=72 \[ \frac {1}{45} (21-5 x) \left (3 x^2+2\right )^{5/2}+\frac {137}{36} x \left (3 x^2+2\right )^{3/2}+\frac {137}{12} x \sqrt {3 x^2+2}+\frac {137 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{6 \sqrt {3}} \]
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Rubi [A] time = 0.02, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {780, 195, 215} \begin {gather*} \frac {1}{45} (21-5 x) \left (3 x^2+2\right )^{5/2}+\frac {137}{36} x \left (3 x^2+2\right )^{3/2}+\frac {137}{12} x \sqrt {3 x^2+2}+\frac {137 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{6 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 195
Rule 215
Rule 780
Rubi steps
\begin {align*} \int (5-x) (3+2 x) \left (2+3 x^2\right )^{3/2} \, dx &=\frac {1}{45} (21-5 x) \left (2+3 x^2\right )^{5/2}+\frac {137}{9} \int \left (2+3 x^2\right )^{3/2} \, dx\\ &=\frac {137}{36} x \left (2+3 x^2\right )^{3/2}+\frac {1}{45} (21-5 x) \left (2+3 x^2\right )^{5/2}+\frac {137}{6} \int \sqrt {2+3 x^2} \, dx\\ &=\frac {137}{12} x \sqrt {2+3 x^2}+\frac {137}{36} x \left (2+3 x^2\right )^{3/2}+\frac {1}{45} (21-5 x) \left (2+3 x^2\right )^{5/2}+\frac {137}{6} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=\frac {137}{12} x \sqrt {2+3 x^2}+\frac {137}{36} x \left (2+3 x^2\right )^{3/2}+\frac {1}{45} (21-5 x) \left (2+3 x^2\right )^{5/2}+\frac {137 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{6 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 60, normalized size = 0.83 \begin {gather*} \frac {1}{60} \sqrt {3 x^2+2} \left (-60 x^5+252 x^4+605 x^3+336 x^2+1115 x+112\right )+\frac {137 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{6 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.26, size = 71, normalized size = 0.99 \begin {gather*} \frac {1}{60} \sqrt {3 x^2+2} \left (-60 x^5+252 x^4+605 x^3+336 x^2+1115 x+112\right )-\frac {137 \log \left (\sqrt {3 x^2+2}-\sqrt {3} x\right )}{6 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 65, normalized size = 0.90 \begin {gather*} -\frac {1}{60} \, {\left (60 \, x^{5} - 252 \, x^{4} - 605 \, x^{3} - 336 \, x^{2} - 1115 \, x - 112\right )} \sqrt {3 \, x^{2} + 2} + \frac {137}{36} \, \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 = 0.19, size = 56, normalized size = 0.78 \begin {gather*} -\frac {1}{60} \, {\left ({\left ({\left ({\left (12 \, {\left (5 \, x - 21\right )} x - 605\right )} x - 336\right )} x - 1115\right )} x - 112\right )} \sqrt {3 \, x^{2} + 2} - \frac {137}{18} \, \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 = 61, normalized size = 0.85 \begin {gather*} -\frac {\left (3 x^{2}+2\right )^{\frac {5}{2}} x}{9}+\frac {137 \left (3 x^{2}+2\right )^{\frac {3}{2}} x}{36}+\frac {137 \sqrt {3 x^{2}+2}\, x}{12}+\frac {137 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{18}+\frac {7 \left (3 x^{2}+2\right )^{\frac {5}{2}}}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 60, normalized size = 0.83 \begin {gather*} -\frac {1}{9} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x + \frac {7}{15} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} + \frac {137}{36} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {137}{12} \, \sqrt {3 \, x^{2} + 2} x + \frac {137}{18} \, \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 = 1.74, size = 50, normalized size = 0.69 \begin {gather*} \frac {137\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{18}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (-3\,x^5+\frac {63\,x^4}{5}+\frac {121\,x^3}{4}+\frac {84\,x^2}{5}+\frac {223\,x}{4}+\frac {28}{5}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.00, size = 110, normalized size = 1.53 \begin {gather*} - x^{5} \sqrt {3 x^{2} + 2} + \frac {21 x^{4} \sqrt {3 x^{2} + 2}}{5} + \frac {121 x^{3} \sqrt {3 x^{2} + 2}}{12} + \frac {28 x^{2} \sqrt {3 x^{2} + 2}}{5} + \frac {223 x \sqrt {3 x^{2} + 2}}{12} + \frac {28 \sqrt {3 x^{2} + 2}}{15} + \frac {137 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{18} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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