Optimal. Leaf size=37 \[ \frac{5 x}{6 \sqrt{3 x^2+2}}+\frac{15 x+2}{18 \left (3 x^2+2\right )^{3/2}} \]
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Rubi [A] time = 0.0065831, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {639, 191} \[ \frac{5 x}{6 \sqrt{3 x^2+2}}+\frac{15 x+2}{18 \left (3 x^2+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 639
Rule 191
Rubi steps
\begin{align*} \int \frac{5-x}{\left (2+3 x^2\right )^{5/2}} \, dx &=\frac{2+15 x}{18 \left (2+3 x^2\right )^{3/2}}+\frac{5}{3} \int \frac{1}{\left (2+3 x^2\right )^{3/2}} \, dx\\ &=\frac{2+15 x}{18 \left (2+3 x^2\right )^{3/2}}+\frac{5 x}{6 \sqrt{2+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0103427, size = 25, normalized size = 0.68 \[ \frac{45 x^3+45 x+2}{18 \left (3 x^2+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 22, normalized size = 0.6 \begin{align*}{\frac{45\,{x}^{3}+45\,x+2}{18} \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 = 1.00243, size = 49, normalized size = 1.32 \begin{align*} \frac{5 \, x}{6 \, \sqrt{3 \, x^{2} + 2}} + \frac{5 \, x}{6 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} + \frac{1}{9 \,{\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 = 1.50362, size = 85, normalized size = 2.3 \begin{align*} \frac{{\left (45 \, x^{3} + 45 \, x + 2\right )} \sqrt{3 \, x^{2} + 2}}{18 \,{\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 = 51.7392, size = 90, normalized size = 2.43 \begin{align*} \frac{5 x^{3}}{6 x^{2} \sqrt{3 x^{2} + 2} + 4 \sqrt{3 x^{2} + 2}} + \frac{5 x}{6 x^{2} \sqrt{3 x^{2} + 2} + 4 \sqrt{3 x^{2} + 2}} + \frac{1}{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 = 1.2109, size = 28, normalized size = 0.76 \begin{align*} \frac{45 \,{\left (x^{2} + 1\right )} x + 2}{18 \,{\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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