Optimal. Leaf size=60 \[ -\frac {7 (2-7 x) (2 x+3)}{6 \sqrt {3 x^2+2}}-\frac {53}{9} \sqrt {3 x^2+2}+\frac {8 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.02, antiderivative size = 60, 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 = {819, 641, 215} \begin {gather*} -\frac {7 (2-7 x) (2 x+3)}{6 \sqrt {3 x^2+2}}-\frac {53}{9} \sqrt {3 x^2+2}+\frac {8 \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 641
Rule 819
Rubi steps
\begin {align*} \int \frac {(5-x) (3+2 x)^2}{\left (2+3 x^2\right )^{3/2}} \, dx &=-\frac {7 (2-7 x) (3+2 x)}{6 \sqrt {2+3 x^2}}+\frac {1}{6} \int \frac {16-106 x}{\sqrt {2+3 x^2}} \, dx\\ &=-\frac {7 (2-7 x) (3+2 x)}{6 \sqrt {2+3 x^2}}-\frac {53}{9} \sqrt {2+3 x^2}+\frac {8}{3} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=-\frac {7 (2-7 x) (3+2 x)}{6 \sqrt {2+3 x^2}}-\frac {53}{9} \sqrt {2+3 x^2}+\frac {8 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 48, normalized size = 0.80 \begin {gather*} -\frac {24 x^2-16 \sqrt {9 x^2+6} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-357 x+338}{18 \sqrt {3 x^2+2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.33, size = 56, normalized size = 0.93 \begin {gather*} \frac {-24 x^2+357 x-338}{18 \sqrt {3 x^2+2}}-\frac {8 \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 = 0.42, size = 68, normalized size = 1.13 \begin {gather*} \frac {8 \, \sqrt {3} {\left (3 \, x^{2} + 2\right )} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) - {\left (24 \, x^{2} - 357 \, x + 338\right )} \sqrt {3 \, x^{2} + 2}}{18 \, {\left (3 \, x^{2} + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 44, normalized size = 0.73 \begin {gather*} -\frac {8}{9} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) - \frac {3 \, {\left (8 \, x - 119\right )} x + 338}{18 \, \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 = 51, normalized size = 0.85 \begin {gather*} -\frac {4 x^{2}}{3 \sqrt {3 x^{2}+2}}+\frac {119 x}{6 \sqrt {3 x^{2}+2}}+\frac {8 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{9}-\frac {169}{9 \sqrt {3 x^{2}+2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 50, normalized size = 0.83 \begin {gather*} -\frac {4 \, x^{2}}{3 \, \sqrt {3 \, x^{2} + 2}} + \frac {8}{9} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {119 \, x}{6 \, \sqrt {3 \, x^{2} + 2}} - \frac {169}{9 \, \sqrt {3 \, x^{2} + 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.75, size = 100, normalized size = 1.67 \begin {gather*} \frac {8\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{9}-\frac {4\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-966+\sqrt {6}\,357{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{648\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (966+\sqrt {6}\,357{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{648\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {51 x}{3 x^{2} \sqrt {3 x^{2} + 2} + 2 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \left (- \frac {8 x^{2}}{3 x^{2} \sqrt {3 x^{2} + 2} + 2 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \frac {4 x^{3}}{3 x^{2} \sqrt {3 x^{2} + 2} + 2 \sqrt {3 x^{2} + 2}}\, dx - \int \left (- \frac {45}{3 x^{2} \sqrt {3 x^{2} + 2} + 2 \sqrt {3 x^{2} + 2}}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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