Optimal. Leaf size=29 \[ -x+\sqrt{x}+x \log \left (x+\sqrt{x}\right )-\log \left (\sqrt{x}+1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0180576, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2548, 376, 77} \[ -x+\sqrt{x}+x \log \left (x+\sqrt{x}\right )-\log \left (\sqrt{x}+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2548
Rule 376
Rule 77
Rubi steps
\begin{align*} \int \log \left (\sqrt{x}+x\right ) \, dx &=x \log \left (\sqrt{x}+x\right )-\int \frac{1+2 \sqrt{x}}{2+2 \sqrt{x}} \, dx\\ &=x \log \left (\sqrt{x}+x\right )-2 \operatorname{Subst}\left (\int \frac{x (1+2 x)}{2+2 x} \, dx,x,\sqrt{x}\right )\\ &=x \log \left (\sqrt{x}+x\right )-2 \operatorname{Subst}\left (\int \left (-\frac{1}{2}+x+\frac{1}{2 (1+x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\sqrt{x}-x-\log \left (1+\sqrt{x}\right )+x \log \left (\sqrt{x}+x\right )\\ \end{align*}
Mathematica [A] time = 0.0116458, size = 29, normalized size = 1. \[ -x+\sqrt{x}+x \log \left (x+\sqrt{x}\right )-\log \left (\sqrt{x}+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.002, size = 24, normalized size = 0.8 \begin{align*} -x-\ln \left ( 1+\sqrt{x} \right ) +x\ln \left ( x+\sqrt{x} \right ) +\sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.03459, size = 31, normalized size = 1.07 \begin{align*} x \log \left (x + \sqrt{x}\right ) - x + \sqrt{x} - \log \left (\sqrt{x} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.23564, size = 103, normalized size = 3.55 \begin{align*}{\left (x + 1\right )} \log \left (x + \sqrt{x}\right ) - x + \sqrt{x} - 2 \, \log \left (\sqrt{x} + 1\right ) - \log \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 4.50717, size = 24, normalized size = 0.83 \begin{align*} \sqrt{x} + x \log{\left (\sqrt{x} + x \right )} - x - \log{\left (\sqrt{x} + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.34515, size = 31, normalized size = 1.07 \begin{align*} x \log \left (x + \sqrt{x}\right ) - x + \sqrt{x} - \log \left (\sqrt{x} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]