Optimal. Leaf size=57 \[ \frac{1}{3} \sqrt{3 x^2+5 x-2}-\frac{5 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x-2}}\right )}{6 \sqrt{3}} \]
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Rubi [A] time = 0.0145413, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {640, 621, 206} \[ \frac{1}{3} \sqrt{3 x^2+5 x-2}-\frac{5 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x-2}}\right )}{6 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 640
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{x}{\sqrt{-2+5 x+3 x^2}} \, dx &=\frac{1}{3} \sqrt{-2+5 x+3 x^2}-\frac{5}{6} \int \frac{1}{\sqrt{-2+5 x+3 x^2}} \, dx\\ &=\frac{1}{3} \sqrt{-2+5 x+3 x^2}-\frac{5}{3} \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{-2+5 x+3 x^2}}\right )\\ &=\frac{1}{3} \sqrt{-2+5 x+3 x^2}-\frac{5 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{-2+5 x+3 x^2}}\right )}{6 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.018374, size = 52, normalized size = 0.91 \[ \frac{1}{18} \left (6 \sqrt{3 x^2+5 x-2}-5 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x-6}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 45, normalized size = 0.8 \begin{align*}{\frac{1}{3}\sqrt{3\,{x}^{2}+5\,x-2}}-{\frac{5\,\sqrt{3}}{18}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x-2} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45138, size = 58, normalized size = 1.02 \begin{align*} -\frac{5}{18} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x - 2} + 6 \, x + 5\right ) + \frac{1}{3} \, \sqrt{3 \, x^{2} + 5 \, x - 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01611, size = 150, normalized size = 2.63 \begin{align*} \frac{5}{36} \, \sqrt{3} \log \left (-4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x - 2}{\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 1\right ) + \frac{1}{3} \, \sqrt{3 \, x^{2} + 5 \, x - 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{\left (x + 2\right ) \left (3 x - 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11637, size = 66, normalized size = 1.16 \begin{align*} \frac{5}{18} \, \sqrt{3} \log \left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x - 2}\right )} - 5 \right |}\right ) + \frac{1}{3} \, \sqrt{3 \, x^{2} + 5 \, x - 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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