Optimal. Leaf size=41 \[ \frac{2-51 x}{54 \left (3 x^2+2\right )^{3/2}}-\frac{16-13 x}{18 \sqrt{3 x^2+2}} \]
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Rubi [A] time = 0.0533089, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {1814, 637} \[ \frac{2-51 x}{54 \left (3 x^2+2\right )^{3/2}}-\frac{16-13 x}{18 \sqrt{3 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 1814
Rule 637
Rubi steps
\begin{align*} \int \frac{(1+2 x) \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{5/2}} \, dx &=\frac{2-51 x}{54 \left (2+3 x^2\right )^{3/2}}-\frac{1}{6} \int \frac{-\frac{26}{3}-16 x}{\left (2+3 x^2\right )^{3/2}} \, dx\\ &=\frac{2-51 x}{54 \left (2+3 x^2\right )^{3/2}}-\frac{16-13 x}{18 \sqrt{2+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.018033, size = 30, normalized size = 0.73 \[ \frac{117 x^3-144 x^2+27 x-94}{54 \left (3 x^2+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 27, normalized size = 0.7 \begin{align*}{\frac{117\,{x}^{3}-144\,{x}^{2}+27\,x-94}{54} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.970827, size = 68, normalized size = 1.66 \begin{align*} \frac{13 \, x}{18 \, \sqrt{3 \, x^{2} + 2}} - \frac{8 \, x^{2}}{3 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{17 \, x}{18 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{47}{27 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5486, size = 101, normalized size = 2.46 \begin{align*} \frac{{\left (117 \, x^{3} - 144 \, x^{2} + 27 \, x - 94\right )} \sqrt{3 \, x^{2} + 2}}{54 \,{\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 111.998, size = 180, normalized size = 4.39 \begin{align*} \frac{10 x^{3}}{18 x^{2} \sqrt{3 x^{2} + 2} + 12 \sqrt{3 x^{2} + 2}} + \frac{x^{3}}{6 x^{2} \sqrt{3 x^{2} + 2} + 4 \sqrt{3 x^{2} + 2}} - \frac{72 x^{2}}{81 x^{2} \sqrt{3 x^{2} + 2} + 54 \sqrt{3 x^{2} + 2}} + \frac{x}{6 x^{2} \sqrt{3 x^{2} + 2} + 4 \sqrt{3 x^{2} + 2}} - \frac{32}{81 x^{2} \sqrt{3 x^{2} + 2} + 54 \sqrt{3 x^{2} + 2}} - \frac{5}{27 x^{2} \sqrt{3 x^{2} + 2} + 18 \sqrt{3 x^{2} + 2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25106, size = 34, normalized size = 0.83 \begin{align*} \frac{9 \,{\left ({\left (13 \, x - 16\right )} x + 3\right )} x - 94}{54 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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