Optimal. Leaf size=17 \[ \frac {2 (1+x)}{\sqrt {2+3 x+x^2}} \]
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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 = {650}
\begin {gather*} \frac {2 (x+1)}{\sqrt {x^2+3 x+2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 650
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 = 0.08, size = 19, normalized size = 1.12 \begin {gather*} \frac {2 \sqrt {2+3 x+x^2}}{2+x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.66, size = 30, normalized size = 1.76
method | result | size |
risch | \(\frac {2+2 x}{\sqrt {x^{2}+3 x +2}}\) | \(16\) |
trager | \(\frac {2 \sqrt {x^{2}+3 x +2}}{2+x}\) | \(18\) |
gosper | \(\frac {2 \left (x +1\right )^{2} \left (2+x \right )}{\left (x^{2}+3 x +2\right )^{\frac {3}{2}}}\) | \(21\) |
default | \(-\frac {1}{\sqrt {x^{2}+3 x +2}}+\frac {2 x +3}{\sqrt {x^{2}+3 x +2}}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, 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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Fricas [A]
time = 2.42, 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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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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Giac [A]
time = 1.93, 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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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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