\(\int \frac {x}{1+\sqrt {1+x}} \, dx\) [211]

Optimal result

Integrand size = 13, antiderivative size = 15 \[ \int \frac {x}{1+\sqrt {1+x}} \, dx=-x+\frac {2}{3} (1+x)^{3/2} \]

[Out]

-x+2/3*(1+x)^(3/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {378} \[ \int \frac {x}{1+\sqrt {1+x}} \, dx=\frac {2}{3} (x+1)^{3/2}-x \]

[In]

Int[x/(1 + Sqrt[1 + x]),x]

[Out]

-x + (2*(1 + x)^(3/2))/3

Rule 378

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coefficient[v, x, 0], d = Coefficient[v,
x, 1]}, Dist[1/d^(m + 1), Subst[Int[SimplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]
] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \text {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] (verified)

Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.20 \[ \int \frac {x}{1+\sqrt {1+x}} \, dx=\frac {1}{3} (1+x) \left (-3+2 \sqrt {1+x}\right ) \]

[In]

Integrate[x/(1 + Sqrt[1 + x]),x]

[Out]

((1 + x)*(-3 + 2*Sqrt[1 + x]))/3

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 13, normalized size of antiderivative = 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	\(\frac {-\frac {\sqrt {\pi }\, \left (12 x +8\right )}{6}+\frac {\sqrt {\pi }\, \left (8+8 x \right ) \sqrt {1+x}}{6}}{2 \sqrt {\pi }}\)	\(32\)

[In]

int(x/(1+(1+x)^(1/2)),x,method=_RETURNVERBOSE)

[Out]

2/3*(1+x)^(3/2)-1-x

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \frac {x}{1+\sqrt {1+x}} \, dx=\frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - x \]

[In]

integrate(x/(1+(1+x)^(1/2)),x, algorithm="fricas")

[Out]

2/3*(x + 1)^(3/2) - x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (10) = 20\).

Time = 0.49 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.47 \[ \int \frac {x}{1+\sqrt {1+x}} \, dx=\frac {2 x \sqrt {x + 1}}{3} - x + \frac {2 \sqrt {x + 1}}{3} \]

[In]

integrate(x/(1+(1+x)**(1/2)),x)

[Out]

2*x*sqrt(x + 1)/3 - x + 2*sqrt(x + 1)/3

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {x}{1+\sqrt {1+x}} \, dx=\frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - x - 1 \]

[In]

integrate(x/(1+(1+x)^(1/2)),x, algorithm="maxima")

[Out]

2/3*(x + 1)^(3/2) - x - 1

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {x}{1+\sqrt {1+x}} \, dx=\frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - x - 1 \]

[In]

integrate(x/(1+(1+x)^(1/2)),x, algorithm="giac")

[Out]

2/3*(x + 1)^(3/2) - x - 1

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \frac {x}{1+\sqrt {1+x}} \, dx=\frac {2\,{\left (x+1\right )}^{3/2}}{3}-x \]

[In]

int(x/((x + 1)^(1/2) + 1),x)

[Out]

(2*(x + 1)^(3/2))/3 - x