Optimal. Leaf size=106 \[ \frac {2}{15} \sqrt {3 x^2+2} (2 x+1)^4+\frac {13}{60} \sqrt {3 x^2+2} (2 x+1)^3-\frac {19}{540} \sqrt {3 x^2+2} (2 x+1)^2-\frac {1}{810} (2073 x+3937) \sqrt {3 x^2+2}+\frac {5 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.11, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1654, 833, 780, 215} \[ \frac {2}{15} \sqrt {3 x^2+2} (2 x+1)^4+\frac {13}{60} \sqrt {3 x^2+2} (2 x+1)^3-\frac {19}{540} \sqrt {3 x^2+2} (2 x+1)^2-\frac {1}{810} (2073 x+3937) \sqrt {3 x^2+2}+\frac {5 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 215
Rule 780
Rule 833
Rule 1654
Rubi steps
\begin {align*} \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx &=\frac {2}{15} (1+2 x)^4 \sqrt {2+3 x^2}+\frac {1}{60} \int \frac {(1+2 x)^3 (-68+156 x)}{\sqrt {2+3 x^2}} \, dx\\ &=\frac {13}{60} (1+2 x)^3 \sqrt {2+3 x^2}+\frac {2}{15} (1+2 x)^4 \sqrt {2+3 x^2}+\frac {1}{720} \int \frac {(-2688-228 x) (1+2 x)^2}{\sqrt {2+3 x^2}} \, dx\\ &=-\frac {19}{540} (1+2 x)^2 \sqrt {2+3 x^2}+\frac {13}{60} (1+2 x)^3 \sqrt {2+3 x^2}+\frac {2}{15} (1+2 x)^4 \sqrt {2+3 x^2}+\frac {\int \frac {(-22368-49752 x) (1+2 x)}{\sqrt {2+3 x^2}} \, dx}{6480}\\ &=-\frac {19}{540} (1+2 x)^2 \sqrt {2+3 x^2}+\frac {13}{60} (1+2 x)^3 \sqrt {2+3 x^2}+\frac {2}{15} (1+2 x)^4 \sqrt {2+3 x^2}-\frac {1}{810} (3937+2073 x) \sqrt {2+3 x^2}+\frac {5}{3} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=-\frac {19}{540} (1+2 x)^2 \sqrt {2+3 x^2}+\frac {13}{60} (1+2 x)^3 \sqrt {2+3 x^2}+\frac {2}{15} (1+2 x)^4 \sqrt {2+3 x^2}-\frac {1}{810} (3937+2073 x) \sqrt {2+3 x^2}+\frac {5 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 54, normalized size = 0.51 \[ \frac {1}{405} \left (\sqrt {3 x^2+2} \left (864 x^4+2430 x^3+2292 x^2-135 x-1841\right )+225 \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 60, normalized size = 0.57 \[ \frac {1}{405} \, {\left (864 \, x^{4} + 2430 \, x^{3} + 2292 \, x^{2} - 135 \, x - 1841\right )} \sqrt {3 \, x^{2} + 2} + \frac {5}{18} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 54, normalized size = 0.51 \[ \frac {1}{405} \, {\left (3 \, {\left (2 \, {\left (9 \, {\left (16 \, x + 45\right )} x + 382\right )} x - 45\right )} x - 1841\right )} \sqrt {3 \, x^{2} + 2} - \frac {5}{9} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 79, normalized size = 0.75 \[ \frac {32 \sqrt {3 x^{2}+2}\, x^{4}}{15}+6 \sqrt {3 x^{2}+2}\, x^{3}+\frac {764 \sqrt {3 x^{2}+2}\, x^{2}}{135}-\frac {\sqrt {3 x^{2}+2}\, x}{3}+\frac {5 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{9}-\frac {1841 \sqrt {3 x^{2}+2}}{405} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 78, normalized size = 0.74 \[ \frac {32}{15} \, \sqrt {3 \, x^{2} + 2} x^{4} + 6 \, \sqrt {3 \, x^{2} + 2} x^{3} + \frac {764}{135} \, \sqrt {3 \, x^{2} + 2} x^{2} - \frac {1}{3} \, \sqrt {3 \, x^{2} + 2} x + \frac {5}{9} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) - \frac {1841}{405} \, \sqrt {3 \, x^{2} + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 45, normalized size = 0.42 \[ \frac {5\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{9}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {32\,x^4}{5}+18\,x^3+\frac {764\,x^2}{45}-x-\frac {1841}{135}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.20, size = 94, normalized size = 0.89 \[ \frac {32 x^{4} \sqrt {3 x^{2} + 2}}{15} + 6 x^{3} \sqrt {3 x^{2} + 2} + \frac {764 x^{2} \sqrt {3 x^{2} + 2}}{135} - \frac {x \sqrt {3 x^{2} + 2}}{3} - \frac {1841 \sqrt {3 x^{2} + 2}}{405} + \frac {5 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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