Optimal. Leaf size=21 \[ \frac {2}{3} (x+1)^{3/2}-\frac {2 x^{3/2}}{3} \]
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Rubi [A] time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2106, 30, 32} \[ \frac {2}{3} (x+1)^{3/2}-\frac {2 x^{3/2}}{3} \]
Antiderivative was successfully verified.
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Rule 30
Rule 32
Rule 2106
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x}+\sqrt {1+x}} \, dx &=-\int \sqrt {x} \, dx+\int \sqrt {1+x} \, dx\\ &=-\frac {2 x^{3/2}}{3}+\frac {2}{3} (1+x)^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 21, normalized size = 1.00 \[ \frac {2}{3} (x+1)^{3/2}-\frac {2 x^{3/2}}{3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 13, normalized size = 0.62 \[ \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - \frac {2}{3} \, x^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 13, normalized size = 0.62 \[ \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - \frac {2}{3} \, x^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 14, normalized size = 0.67 \[ -\frac {2 x^{\frac {3}{2}}}{3}+\frac {2 \left (x +1\right )^{\frac {3}{2}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x + 1} + \sqrt {x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.97, size = 21, normalized size = 1.00 \[ \frac {2\,x\,\sqrt {x+1}}{3}+\frac {2\,\sqrt {x+1}}{3}-\frac {2\,x^{3/2}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.94, size = 63, normalized size = 3.00 \[ \frac {2 \sqrt {x} \sqrt {x + 1}}{3 \sqrt {x} + 3 \sqrt {x + 1}} + \frac {4 x}{3 \sqrt {x} + 3 \sqrt {x + 1}} + \frac {2}{3 \sqrt {x} + 3 \sqrt {x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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