Optimal. Leaf size=43 \[ \frac {1}{4} \sqrt {x} \sqrt {1+x}+\frac {1}{2} x^{3/2} \sqrt {1+x}-\frac {1}{4} \sinh ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {52, 56, 221}
\begin {gather*} \frac {1}{2} \sqrt {x+1} x^{3/2}+\frac {1}{4} \sqrt {x+1} \sqrt {x}-\frac {1}{4} \sinh ^{-1}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 221
Rubi steps
\begin {align*} \int \sqrt {x} \sqrt {1+x} \, dx &=\frac {1}{2} x^{3/2} \sqrt {1+x}+\frac {1}{4} \int \frac {\sqrt {x}}{\sqrt {1+x}} \, dx\\ &=\frac {1}{4} \sqrt {x} \sqrt {1+x}+\frac {1}{2} x^{3/2} \sqrt {1+x}-\frac {1}{8} \int \frac {1}{\sqrt {x} \sqrt {1+x}} \, dx\\ &=\frac {1}{4} \sqrt {x} \sqrt {1+x}+\frac {1}{2} x^{3/2} \sqrt {1+x}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {1}{4} \sqrt {x} \sqrt {1+x}+\frac {1}{2} x^{3/2} \sqrt {1+x}-\frac {1}{4} \sinh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 37, normalized size = 0.86 \begin {gather*} \frac {1}{4} \left (\sqrt {x} \sqrt {1+x} (1+2 x)-\tanh ^{-1}\left (\sqrt {\frac {x}{1+x}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 50, normalized size = 1.16
method | result | size |
meijerg | \(-\frac {-\frac {\sqrt {\pi }\, \sqrt {x}\, \left (3+6 x \right ) \sqrt {1+x}}{6}+\frac {\sqrt {\pi }\, \arcsinh \left (\sqrt {x}\right )}{2}}{2 \sqrt {\pi }}\) | \(34\) |
risch | \(\frac {\left (1+2 x \right ) \sqrt {x}\, \sqrt {1+x}}{4}-\frac {\sqrt {x \left (1+x \right )}\, \ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{8 \sqrt {1+x}\, \sqrt {x}}\) | \(45\) |
default | \(\frac {\sqrt {x}\, \left (1+x \right )^{\frac {3}{2}}}{2}-\frac {\sqrt {x}\, \sqrt {1+x}}{4}-\frac {\sqrt {x \left (1+x \right )}\, \ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{8 \sqrt {1+x}\, \sqrt {x}}\) | \(50\) |
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. 71 vs.
\(2 (27) = 54\).
time = 2.34, size = 71, normalized size = 1.65 \begin {gather*} \frac {\frac {{\left (x + 1\right )}^{\frac {3}{2}}}{x^{\frac {3}{2}}} + \frac {\sqrt {x + 1}}{\sqrt {x}}}{4 \, {\left (\frac {{\left (x + 1\right )}^{2}}{x^{2}} - \frac {2 \, {\left (x + 1\right )}}{x} + 1\right )}} - \frac {1}{8} \, \log \left (\frac {\sqrt {x + 1}}{\sqrt {x}} + 1\right ) + \frac {1}{8} \, \log \left (\frac {\sqrt {x + 1}}{\sqrt {x}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.74, size = 34, normalized size = 0.79 \begin {gather*} \frac {1}{4} \, {\left (2 \, x + 1\right )} \sqrt {x + 1} \sqrt {x} + \frac {1}{8} \, \log \left (2 \, \sqrt {x + 1} \sqrt {x} - 2 \, x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.57, size = 119, normalized size = 2.77 \begin {gather*} \begin {cases} - \frac {\operatorname {acosh}{\left (\sqrt {x + 1} \right )}}{4} + \frac {\left (x + 1\right )^{\frac {5}{2}}}{2 \sqrt {x}} - \frac {3 \left (x + 1\right )^{\frac {3}{2}}}{4 \sqrt {x}} + \frac {\sqrt {x + 1}}{4 \sqrt {x}} & \text {for}\: \left |{x + 1}\right | > 1 \\\frac {i \operatorname {asin}{\left (\sqrt {x + 1} \right )}}{4} - \frac {i \left (x + 1\right )^{\frac {5}{2}}}{2 \sqrt {- x}} + \frac {3 i \left (x + 1\right )^{\frac {3}{2}}}{4 \sqrt {- x}} - \frac {i \sqrt {x + 1}}{4 \sqrt {- x}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.50, size = 39, normalized size = 0.91 \begin {gather*} \frac {1}{4} \, {\left (2 \, x - 3\right )} \sqrt {x + 1} \sqrt {x} + \sqrt {x + 1} \sqrt {x} + \frac {1}{4} \, \log \left (\sqrt {x + 1} - \sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.22, size = 30, normalized size = 0.70 \begin {gather*} \sqrt {x}\,\left (\frac {x}{2}+\frac {1}{4}\right )\,\sqrt {x+1}-\frac {\ln \left (x+\sqrt {x}\,\sqrt {x+1}+\frac {1}{2}\right )}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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