Optimal. Leaf size=17 \[ \frac {2 (x+1)}{\sqrt {x^2+3 x+2}} \]
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Rubi [A] time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {2 (x+1)}{\sqrt {x^2+3 x+2}} \end {gather*}
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*}
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Mathematica [A] time = 0.04, size = 19, normalized size = 1.12 \begin {gather*} \frac {2 \sqrt {x^2+3 x+2}}{x+2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 19, normalized size = 1.12 \begin {gather*} \frac {2 \sqrt {x^2+3 x+2}}{x+2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 20, normalized size = 1.18 \begin {gather*} \frac {2 \, {\left (x + \sqrt {x^{2} + 3 \, x + 2} + 2\right )}}{x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 19, normalized size = 1.12 \begin {gather*} \frac {2}{x - \sqrt {x^{2} + 3 \, x + 2} + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 21, normalized size = 1.24 \begin {gather*} \frac {2 \left (x +1\right )^{2} \left (x +2\right )}{\left (x^{2}+3 x +2\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 26, normalized size = 1.53 \begin {gather*} \frac {2 \, x}{\sqrt {x^{2} + 3 \, x + 2}} + \frac {2}{\sqrt {x^{2} + 3 \, x + 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 17, normalized size = 1.00 \begin {gather*} \frac {2\,\sqrt {x^2+3\,x+2}}{x+2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 1}{\left (\left (x + 1\right ) \left (x + 2\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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