Optimal. Leaf size=26 \[ \sqrt {x^2+2 x}-\tan ^{-1}\left (\sqrt {x^2+2 x}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {685, 688, 203} \[ \sqrt {x^2+2 x}-\tan ^{-1}\left (\sqrt {x^2+2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 685
Rule 688
Rubi steps
\begin {align*} \int \frac {\sqrt {2 x+x^2}}{1+x} \, dx &=\sqrt {2 x+x^2}-\int \frac {1}{(1+x) \sqrt {2 x+x^2}} \, dx\\ &=\sqrt {2 x+x^2}-4 \operatorname {Subst}\left (\int \frac {1}{4+4 x^2} \, dx,x,\sqrt {2 x+x^2}\right )\\ &=\sqrt {2 x+x^2}-\tan ^{-1}\left (\sqrt {2 x+x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 38, normalized size = 1.46 \[ \sqrt {x (x+2)} \left (1-\frac {2 \tan ^{-1}\left (\sqrt {\frac {x}{x+2}}\right )}{\sqrt {x} \sqrt {x+2}}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.86, size = 27, normalized size = 1.04 \[ \sqrt {x^{2} + 2 \, x} - 2 \, \arctan \left (-x + \sqrt {x^{2} + 2 \, x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 27, normalized size = 1.04 \[ \sqrt {x^{2} + 2 \, x} - 2 \, \arctan \left (-x + \sqrt {x^{2} + 2 \, x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 21, normalized size = 0.81 \[ \arctan \left (\frac {1}{\sqrt {\left (x +1\right )^{2}-1}}\right )+\sqrt {\left (x +1\right )^{2}-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.02, size = 17, normalized size = 0.65 \[ \sqrt {x^{2} + 2 \, x} + \arcsin \left (\frac {1}{{\left | x + 1 \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\sqrt {x^2+2\,x}}{x+1} \,d x \]
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 \[ \int \frac {\sqrt {x \left (x + 2\right )}}{x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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