Optimal. Leaf size=22 \[ \sqrt {x} \sqrt {1+x}-\sinh ^{-1}\left (\sqrt {x}\right ) \]
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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 = {1978, 52, 56,
221} \begin {gather*} \sqrt {x} \sqrt {x+1}-\sinh ^{-1}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 221
Rule 1978
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}-\text {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 [B] Leaf count is larger than twice the leaf count of optimal. \(48\) vs. \(2(22)=44\).
time = 0.02, size = 48, normalized size = 2.18 \begin {gather*} \frac {\sqrt {\frac {x}{1+x}} \left (\sqrt {x} (1+x)-\sqrt {1+x} \tanh ^{-1}\left (\sqrt {\frac {x}{1+x}}\right )\right )}{\sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(44\) vs.
\(2(16)=32\).
time = 0.05, size = 45, normalized size = 2.05
method | result | size |
default | \(\frac {\sqrt {\frac {x}{1+x}}\, \left (1+x \right ) \left (2 \sqrt {x^{2}+x}-\ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )\right )}{2 \sqrt {x \left (1+x \right )}}\) | \(45\) |
risch | \(\left (1+x \right ) \sqrt {\frac {x}{1+x}}-\frac {\ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right ) \sqrt {\frac {x}{1+x}}\, \sqrt {x \left (1+x \right )}}{2 x}\) | \(47\) |
trager | \(2 \left (\frac {1}{2}+\frac {x}{2}\right ) \sqrt {\frac {x}{1+x}}-\frac {\ln \left (2 \sqrt {\frac {x}{1+x}}\, x +2 \sqrt {\frac {x}{1+x}}+2 x +1\right )}{2}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 51 vs.
\(2 (16) = 32\).
time = 0.27, size = 51, normalized size = 2.32 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 42 vs.
\(2 (16) = 32\).
time = 0.35, size = 42, normalized size = 1.91 \begin {gather*} {\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 ) \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 \sqrt {\frac {x}{x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs.
\(2 (16) = 32\).
time = 5.00, size = 35, normalized size = 1.59 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 35, normalized size = 1.59 \begin {gather*} -\mathrm {atanh}\left (\sqrt {\frac {x}{x+1}}\right )-\frac {\sqrt {\frac {x}{x+1}}}{\frac {x}{x+1}-1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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