Optimal. Leaf size=19 \[ -\frac{2 (2 x+3)}{\sqrt{x^2+3 x+2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0023999, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {613} \[ -\frac{2 (2 x+3)}{\sqrt{x^2+3 x+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 613
Rubi steps
\begin{align*} \int \frac{1}{\left (2+3 x+x^2\right )^{3/2}} \, dx &=-\frac{2 (3+2 x)}{\sqrt{2+3 x+x^2}}\\ \end{align*}
Mathematica [A] time = 0.0055414, size = 19, normalized size = 1. \[ -\frac{2 (2 x+3)}{\sqrt{x^2+3 x+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.049, size = 24, normalized size = 1.3 \begin{align*} -2\,{\frac{ \left ( 2+x \right ) \left ( 1+x \right ) \left ( 3+2\,x \right ) }{ \left ({x}^{2}+3\,x+2 \right ) ^{3/2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.17317, size = 35, normalized size = 1.84 \begin{align*} -\frac{4 \, x}{\sqrt{x^{2} + 3 \, x + 2}} - \frac{6}{\sqrt{x^{2} + 3 \, x + 2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.87356, size = 95, normalized size = 5. \begin{align*} -\frac{2 \,{\left (2 \, x^{2} + \sqrt{x^{2} + 3 \, x + 2}{\left (2 \, x + 3\right )} + 6 \, x + 4\right )}}{x^{2} + 3 \, x + 2} \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}{\left (x^{2} + 3 x + 2\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.19274, size = 23, normalized size = 1.21 \begin{align*} -\frac{2 \,{\left (2 \, x + 3\right )}}{\sqrt{x^{2} + 3 \, x + 2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]