Optimal. Leaf size=48 \[ -\frac {1}{8} (1+2 x) \sqrt {x+x^2}+\frac {1}{3} \left (x+x^2\right )^{3/2}+\frac {1}{8} \tanh ^{-1}\left (\frac {x}{\sqrt {x+x^2}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {654, 626, 634,
212} \begin {gather*} \frac {1}{3} \left (x^2+x\right )^{3/2}-\frac {1}{8} (2 x+1) \sqrt {x^2+x}+\frac {1}{8} \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+x}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 626
Rule 634
Rule 654
Rubi steps
\begin {align*} \int x \sqrt {x+x^2} \, dx &=\frac {1}{3} \left (x+x^2\right )^{3/2}-\frac {1}{2} \int \sqrt {x+x^2} \, dx\\ &=-\frac {1}{8} (1+2 x) \sqrt {x+x^2}+\frac {1}{3} \left (x+x^2\right )^{3/2}+\frac {1}{16} \int \frac {1}{\sqrt {x+x^2}} \, dx\\ &=-\frac {1}{8} (1+2 x) \sqrt {x+x^2}+\frac {1}{3} \left (x+x^2\right )^{3/2}+\frac {1}{8} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {x+x^2}}\right )\\ &=-\frac {1}{8} (1+2 x) \sqrt {x+x^2}+\frac {1}{3} \left (x+x^2\right )^{3/2}+\frac {1}{8} \tanh ^{-1}\left (\frac {x}{\sqrt {x+x^2}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.07, size = 49, normalized size = 1.02 \begin {gather*} \frac {1}{24} \sqrt {x (1+x)} \left (-3+2 x+8 x^2+\frac {3 \tanh ^{-1}\left (\sqrt {\frac {x}{1+x}}\right )}{\sqrt {x} \sqrt {1+x}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.40, size = 38, normalized size = 0.79
method | result | size |
trager | \(\left (\frac {1}{3} x^{2}+\frac {1}{12} x -\frac {1}{8}\right ) \sqrt {x^{2}+x}+\frac {\ln \left (2 x +1+2 \sqrt {x^{2}+x}\right )}{16}\) | \(37\) |
default | \(\frac {\left (x^{2}+x \right )^{\frac {3}{2}}}{3}-\frac {\left (2 x +1\right ) \sqrt {x^{2}+x}}{8}+\frac {\ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{16}\) | \(38\) |
risch | \(\frac {\left (8 x^{2}+2 x -3\right ) x \left (x +1\right )}{24 \sqrt {x \left (x +1\right )}}+\frac {\ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{16}\) | \(38\) |
meijerg | \(-\frac {\frac {\sqrt {\pi }\, \sqrt {x}\, \left (-40 x^{2}-10 x +15\right ) \sqrt {x +1}}{60}-\frac {\sqrt {\pi }\, \arcsinh \left (\sqrt {x}\right )}{4}}{2 \sqrt {\pi }}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 46, normalized size = 0.96 \begin {gather*} \frac {1}{3} \, {\left (x^{2} + x\right )}^{\frac {3}{2}} - \frac {1}{4} \, \sqrt {x^{2} + x} x - \frac {1}{8} \, \sqrt {x^{2} + x} + \frac {1}{16} \, \log \left (2 \, x + 2 \, \sqrt {x^{2} + x} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.56, size = 37, normalized size = 0.77 \begin {gather*} \frac {1}{24} \, {\left (8 \, x^{2} + 2 \, x - 3\right )} \sqrt {x^{2} + x} - \frac {1}{16} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \sqrt {x \left (x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.82, size = 38, normalized size = 0.79 \begin {gather*} \frac {1}{24} \, {\left (2 \, {\left (4 \, x + 1\right )} x - 3\right )} \sqrt {x^{2} + x} - \frac {1}{16} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} + x} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.08, size = 33, normalized size = 0.69 \begin {gather*} \frac {\ln \left (x+\sqrt {x\,\left (x+1\right )}+\frac {1}{2}\right )}{16}+\frac {\sqrt {x^2+x}\,\left (8\,x^2+2\,x-3\right )}{24} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________