Optimal. Leaf size=37 \[ \frac{43 x}{18 \sqrt{3 x^2+2}}-\frac{7 (2-7 x)}{18 \left (3 x^2+2\right )^{3/2}} \]
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Rubi [A] time = 0.0099111, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {778, 191} \[ \frac{43 x}{18 \sqrt{3 x^2+2}}-\frac{7 (2-7 x)}{18 \left (3 x^2+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 778
Rule 191
Rubi steps
\begin{align*} \int \frac{(5-x) (3+2 x)}{\left (2+3 x^2\right )^{5/2}} \, dx &=-\frac{7 (2-7 x)}{18 \left (2+3 x^2\right )^{3/2}}+\frac{43}{9} \int \frac{1}{\left (2+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{7 (2-7 x)}{18 \left (2+3 x^2\right )^{3/2}}+\frac{43 x}{18 \sqrt{2+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0154025, size = 25, normalized size = 0.68 \[ -\frac{-129 x^3-135 x+14}{18 \left (3 x^2+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 22, normalized size = 0.6 \begin{align*}{\frac{129\,{x}^{3}+135\,x-14}{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 = 0.982255, size = 49, normalized size = 1.32 \begin{align*} \frac{43 \, x}{18 \, \sqrt{3 \, x^{2} + 2}} + \frac{49 \, x}{18 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{7}{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 = 1.47716, size = 89, normalized size = 2.41 \begin{align*} \frac{{\left (129 \, x^{3} + 135 \, x - 14\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 = 91.9838, size = 122, normalized size = 3.3 \begin{align*} - \frac{2 x^{3}}{18 x^{2} \sqrt{3 x^{2} + 2} + 12 \sqrt{3 x^{2} + 2}} + \frac{15 x^{3}}{6 x^{2} \sqrt{3 x^{2} + 2} + 4 \sqrt{3 x^{2} + 2}} + \frac{15 x}{6 x^{2} \sqrt{3 x^{2} + 2} + 4 \sqrt{3 x^{2} + 2}} - \frac{7}{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 = 1.15975, size = 31, normalized size = 0.84 \begin{align*} \frac{3 \,{\left (43 \, x^{2} + 45\right )} x - 14}{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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