Optimal. Leaf size=19 \[ x-2 \sqrt{x}+2 \log \left (\sqrt{x}+1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0107918, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ x-2 \sqrt{x}+2 \log \left (\sqrt{x}+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\sqrt{x}}{1+\sqrt{x}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^2}{1+x} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-1+x+\frac{1}{1+x}\right ) \, dx,x,\sqrt{x}\right )\\ &=-2 \sqrt{x}+x+2 \log \left (1+\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0075446, size = 19, normalized size = 1. \[ x-2 \sqrt{x}+2 \log \left (\sqrt{x}+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 16, normalized size = 0.8 \begin{align*} x+2\,\ln \left ( \sqrt{x}+1 \right ) -2\,\sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.961382, size = 30, normalized size = 1.58 \begin{align*}{\left (\sqrt{x} + 1\right )}^{2} - 4 \, \sqrt{x} + 2 \, \log \left (\sqrt{x} + 1\right ) - 4 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.36597, size = 49, normalized size = 2.58 \begin{align*} x - 2 \, \sqrt{x} + 2 \, \log \left (\sqrt{x} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.12822, size = 17, normalized size = 0.89 \begin{align*} - 2 \sqrt{x} + x + 2 \log{\left (\sqrt{x} + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.0978, size = 20, normalized size = 1.05 \begin{align*} x - 2 \, \sqrt{x} + 2 \, \log \left (\sqrt{x} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]