Optimal. Leaf size=106 \[ -\frac {1}{15} \sqrt {3 x^2+2} (2 x+3)^4+\frac {19}{30} \sqrt {3 x^2+2} (2 x+3)^3+\frac {1477}{270} \sqrt {3 x^2+2} (2 x+3)^2+\frac {49}{81} (99 x+383) \sqrt {3 x^2+2}+\frac {343 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.06, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 5, 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}{15} \sqrt {3 x^2+2} (2 x+3)^4+\frac {19}{30} \sqrt {3 x^2+2} (2 x+3)^3+\frac {1477}{270} \sqrt {3 x^2+2} (2 x+3)^2+\frac {49}{81} (99 x+383) \sqrt {3 x^2+2}+\frac {343 \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)^4}{\sqrt {2+3 x^2}} \, dx &=-\frac {1}{15} (3+2 x)^4 \sqrt {2+3 x^2}+\frac {1}{15} \int \frac {(3+2 x)^3 (241+114 x)}{\sqrt {2+3 x^2}} \, dx\\ &=\frac {19}{30} (3+2 x)^3 \sqrt {2+3 x^2}-\frac {1}{15} (3+2 x)^4 \sqrt {2+3 x^2}+\frac {1}{180} \int \frac {(3+2 x)^2 (7308+8862 x)}{\sqrt {2+3 x^2}} \, dx\\ &=\frac {1477}{270} (3+2 x)^2 \sqrt {2+3 x^2}+\frac {19}{30} (3+2 x)^3 \sqrt {2+3 x^2}-\frac {1}{15} (3+2 x)^4 \sqrt {2+3 x^2}+\frac {\int \frac {(3+2 x) (126420+291060 x)}{\sqrt {2+3 x^2}} \, dx}{1620}\\ &=\frac {1477}{270} (3+2 x)^2 \sqrt {2+3 x^2}+\frac {19}{30} (3+2 x)^3 \sqrt {2+3 x^2}-\frac {1}{15} (3+2 x)^4 \sqrt {2+3 x^2}+\frac {49}{81} (383+99 x) \sqrt {2+3 x^2}+\frac {343}{3} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=\frac {1477}{270} (3+2 x)^2 \sqrt {2+3 x^2}+\frac {19}{30} (3+2 x)^3 \sqrt {2+3 x^2}-\frac {1}{15} (3+2 x)^4 \sqrt {2+3 x^2}+\frac {49}{81} (383+99 x) \sqrt {2+3 x^2}+\frac {343 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 55, normalized size = 0.52 \begin {gather*} \frac {1}{405} \left (15435 \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-\sqrt {3 x^2+2} \left (432 x^4+540 x^3-12264 x^2-58860 x-118513\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.34, size = 66, normalized size = 0.62 \begin {gather*} \frac {1}{405} \sqrt {3 x^2+2} \left (-432 x^4-540 x^3+12264 x^2+58860 x+118513\right )-\frac {343 \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 = 60, normalized size = 0.57 \begin {gather*} -\frac {1}{405} \, {\left (432 \, x^{4} + 540 \, x^{3} - 12264 \, x^{2} - 58860 \, x - 118513\right )} \sqrt {3 \, x^{2} + 2} + \frac {343}{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 = 0.19, size = 53, normalized size = 0.50 \begin {gather*} -\frac {1}{405} \, {\left (12 \, {\left ({\left (9 \, {\left (4 \, x + 5\right )} x - 1022\right )} x - 4905\right )} x - 118513\right )} \sqrt {3 \, x^{2} + 2} - \frac {343}{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 = 79, normalized size = 0.75 \begin {gather*} -\frac {16 \sqrt {3 x^{2}+2}\, x^{4}}{15}-\frac {4 \sqrt {3 x^{2}+2}\, x^{3}}{3}+\frac {4088 \sqrt {3 x^{2}+2}\, x^{2}}{135}+\frac {436 \sqrt {3 x^{2}+2}\, x}{3}+\frac {343 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{9}+\frac {118513 \sqrt {3 x^{2}+2}}{405} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 78, normalized size = 0.74 \begin {gather*} -\frac {16}{15} \, \sqrt {3 \, x^{2} + 2} x^{4} - \frac {4}{3} \, \sqrt {3 \, x^{2} + 2} x^{3} + \frac {4088}{135} \, \sqrt {3 \, x^{2} + 2} x^{2} + \frac {436}{3} \, \sqrt {3 \, x^{2} + 2} x + \frac {343}{9} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {118513}{405} \, \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 = 45, normalized size = 0.42 \begin {gather*} \frac {343\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{9}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (-\frac {16\,x^4}{5}-4\,x^3+\frac {4088\,x^2}{45}+436\,x+\frac {118513}{135}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.93, size = 97, normalized size = 0.92 \begin {gather*} - \frac {16 x^{4} \sqrt {3 x^{2} + 2}}{15} - \frac {4 x^{3} \sqrt {3 x^{2} + 2}}{3} + \frac {4088 x^{2} \sqrt {3 x^{2} + 2}}{135} + \frac {436 x \sqrt {3 x^{2} + 2}}{3} + \frac {118513 \sqrt {3 x^{2} + 2}}{405} + \frac {343 \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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