Optimal. Leaf size=36 \[ -\frac{\tan ^{-1}\left (\frac{4-5 x}{2 \sqrt{2} \sqrt{3 x^2+5 x-2}}\right )}{\sqrt{2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0108348, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {724, 204} \[ -\frac{\tan ^{-1}\left (\frac{4-5 x}{2 \sqrt{2} \sqrt{3 x^2+5 x-2}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 724
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{-2+5 x+3 x^2}} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{-8-x^2} \, dx,x,\frac{-4+5 x}{\sqrt{-2+5 x+3 x^2}}\right )\right )\\ &=-\frac{\tan ^{-1}\left (\frac{4-5 x}{2 \sqrt{2} \sqrt{-2+5 x+3 x^2}}\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0092225, size = 30, normalized size = 0.83 \[ \frac{\tan ^{-1}\left (\frac{5 x-4}{2 \sqrt{6 x^2+10 x-4}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.048, size = 29, normalized size = 0.8 \begin{align*}{\frac{\sqrt{2}}{2}\arctan \left ({\frac{ \left ( 5\,x-4 \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x-2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.51143, size = 27, normalized size = 0.75 \begin{align*} \frac{1}{2} \, \sqrt{2} \arcsin \left (\frac{5 \, x}{7 \,{\left | x \right |}} - \frac{4}{7 \,{\left | x \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.05587, size = 88, normalized size = 2.44 \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (5 \, x - 4\right )}}{4 \, \sqrt{3 \, x^{2} + 5 \, x - 2}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{\left (x + 2\right ) \left (3 x - 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.11749, size = 41, normalized size = 1.14 \begin{align*} \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x - 2}\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]