Optimal. Leaf size=17 \[ \frac{2 (x+1)}{\sqrt{x^2+3 x+2}} \]
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Rubi [A] time = 0.0045294, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {636} \[ \frac{2 (x+1)}{\sqrt{x^2+3 x+2}} \]
Antiderivative was successfully verified.
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Rule 636
Rubi steps
\begin{align*} \int \frac{1+x}{\left (2+3 x+x^2\right )^{3/2}} \, dx &=\frac{2 (1+x)}{\sqrt{2+3 x+x^2}}\\ \end{align*}
Mathematica [A] time = 0.0395123, size = 19, normalized size = 1.12 \[ \frac{2 \sqrt{x^2+3 x+2}}{x+2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 21, normalized size = 1.2 \begin{align*} 2\,{\frac{ \left ( 1+x \right ) ^{2} \left ( 2+x \right ) }{ \left ({x}^{2}+3\,x+2 \right ) ^{3/2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.96237, size = 35, normalized size = 2.06 \begin{align*} \frac{2 \, x}{\sqrt{x^{2} + 3 \, x + 2}} + \frac{2}{\sqrt{x^{2} + 3 \, x + 2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01473, size = 55, normalized size = 3.24 \begin{align*} \frac{2 \,{\left (x + \sqrt{x^{2} + 3 \, x + 2} + 2\right )}}{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 + 1}{\left (\left (x + 1\right ) \left (x + 2\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27919, size = 26, normalized size = 1.53 \begin{align*} \frac{2}{x - \sqrt{x^{2} + 3 \, x + 2} + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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