Optimal. Leaf size=22 \[ \sqrt {x^2+x}+\tanh ^{-1}\left (\frac {x}{\sqrt {x^2+x}}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {664, 620, 206} \[ \sqrt {x^2+x}+\tanh ^{-1}\left (\frac {x}{\sqrt {x^2+x}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 664
Rubi steps
\begin {align*} \int \frac {\sqrt {x+x^2}}{x} \, dx &=\sqrt {x+x^2}+\frac {1}{2} \int \frac {1}{\sqrt {x+x^2}} \, dx\\ &=\sqrt {x+x^2}+\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {x+x^2}}\right )\\ &=\sqrt {x+x^2}+\tanh ^{-1}\left (\frac {x}{\sqrt {x+x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 31, normalized size = 1.41 \[ \sqrt {x (x+1)} \left (\frac {\sinh ^{-1}\left (\sqrt {x}\right )}{\sqrt {x} \sqrt {x+1}}+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 25, normalized size = 1.14 \[ \sqrt {x^{2} + x} - \frac {1}{2} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.02, size = 26, normalized size = 1.18 \[ \sqrt {x^{2} + x} - \frac {1}{2} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} + x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 22, normalized size = 1.00 \[ \frac {\ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{2}+\sqrt {x^{2}+x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 25, normalized size = 1.14 \[ \sqrt {x^{2} + x} + \frac {1}{2} \, \log \left (2 \, x + 2 \, \sqrt {x^{2} + x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 21, normalized size = 0.95 \[ \frac {\ln \left (x+\sqrt {x\,\left (x+1\right )}+\frac {1}{2}\right )}{2}+\sqrt {x^2+x} \]
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 \[ \int \frac {\sqrt {x \left (x + 1\right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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