Optimal. Leaf size=15 \[ \frac {2}{3} (x+1)^{3/2}-x \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {371} \[ \frac {2}{3} (x+1)^{3/2}-x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 371
Rubi steps
\begin {align*} \int \frac {x}{1+\sqrt {1+x}} \, dx &=\operatorname {Subst}\left (\int \left (-1+\sqrt {x}\right ) \, dx,x,1+x\right )\\ &=-x+\frac {2}{3} (1+x)^{3/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 19, normalized size = 1.27 \[ 2 \left (\frac {1}{3} (x+1)^{3/2}-\frac {x}{2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.01, size = 15, normalized size = 1.00 \[ \frac {2}{3} (x+1)^{3/2}-x \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.04, size = 11, normalized size = 0.73 \[ \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.92, size = 12, normalized size = 0.80 \[ \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - x - 1 \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 13, normalized size = 0.87
method | result | size |
derivativedivides | \(\frac {2 \left (1+x \right )^{\frac {3}{2}}}{3}-1-x\) | \(13\) |
default | \(\frac {2 \left (1+x \right )^{\frac {3}{2}}}{3}-1-x\) | \(13\) |
trager | \(-x +\left (\frac {2}{3}+\frac {2 x}{3}\right ) \sqrt {1+x}\) | \(16\) |
meijerg | \(\frac {-\frac {\sqrt {\pi }\, \left (12 x +8\right )}{6}+\frac {\sqrt {\pi }\, \left (8+8 x \right ) \sqrt {1+x}}{6}}{2 \sqrt {\pi }}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.52, size = 12, normalized size = 0.80 \[ \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - x - 1 \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 11, normalized size = 0.73 \[ \frac {2\,{\left (x+1\right )}^{3/2}}{3}-x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.97, size = 22, normalized size = 1.47 \[ \frac {2 x \sqrt {x + 1}}{3} - x + \frac {2 \sqrt {x + 1}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________