Optimal. Leaf size=22 \[ \sqrt {x} \sqrt {x+1}-\sinh ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1958, 50, 54, 215} \[ \sqrt {x} \sqrt {x+1}-\sinh ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 215
Rule 1958
Rubi steps
\begin {align*} \int \sqrt {\frac {x}{1+x}} \, dx &=\int \frac {\sqrt {x}}{\sqrt {1+x}} \, dx\\ &=\sqrt {x} \sqrt {1+x}-\frac {1}{2} \int \frac {1}{\sqrt {x} \sqrt {1+x}} \, dx\\ &=\sqrt {x} \sqrt {1+x}-\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\sqrt {x}\right )\\ &=\sqrt {x} \sqrt {1+x}-\sinh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 42, normalized size = 1.91 \[ \frac {\sqrt {\frac {x}{x+1}} \left (\sqrt {x} (x+1)-\sqrt {x+1} \sinh ^{-1}\left (\sqrt {x}\right )\right )}{\sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 42, normalized size = 1.91 \[ {\left (x + 1\right )} \sqrt {\frac {x}{x + 1}} - \frac {1}{2} \, \log \left (\sqrt {\frac {x}{x + 1}} + 1\right ) + \frac {1}{2} \, \log \left (\sqrt {\frac {x}{x + 1}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.44, size = 35, normalized size = 1.59 \[ \frac {1}{2} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} + x} - 1 \right |}\right ) \mathrm {sgn}\left (x + 1\right ) + \sqrt {x^{2} + x} \mathrm {sgn}\left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 45, normalized size = 2.05 \[ \frac {\sqrt {\frac {x}{x +1}}\, \left (x +1\right ) \left (-\ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )+2 \sqrt {x^{2}+x}\right )}{2 \sqrt {\left (x +1\right ) x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.82, size = 51, normalized size = 2.32 \[ -\frac {\sqrt {\frac {x}{x + 1}}}{\frac {x}{x + 1} - 1} - \frac {1}{2} \, \log \left (\sqrt {\frac {x}{x + 1}} + 1\right ) + \frac {1}{2} \, \log \left (\sqrt {\frac {x}{x + 1}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.12, size = 35, normalized size = 1.59 \[ -\mathrm {atanh}\left (\sqrt {\frac {x}{x+1}}\right )-\frac {\sqrt {\frac {x}{x+1}}}{\frac {x}{x+1}-1} \]
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 \[ \int \sqrt {\frac {x}{x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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