Optimal. Leaf size=21 \[ \frac{2}{3} (x+1)^{3/2}-\frac{2 x^{3/2}}{3} \]
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Rubi [A] time = 0.0055358, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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 2106
Rule 30
Rule 32
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*}
Mathematica [A] time = 0.0158995, size = 21, normalized size = 1. \[ \frac{2}{3} (x+1)^{3/2}-\frac{2 x^{3/2}}{3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 14, normalized size = 0.7 \begin{align*} -{\frac{2}{3}{x}^{{\frac{3}{2}}}}+{\frac{2}{3} \left ( 1+x \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x + 1} + \sqrt{x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84066, size = 45, normalized size = 2.14 \begin{align*} \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} - \frac{2}{3} \, x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.351887, size = 63, normalized size = 3. \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05009, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} - \frac{2}{3} \, x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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