3.211 \(\int \frac{x}{1+\sqrt{1+x}} \, dx\)

Optimal. Leaf size=15 \[ \frac{2}{3} (x+1)^{3/2}-x \]

[Out]

-x + (2*(1 + x)^(3/2))/3

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Rubi [A]  time = 0.0063466, antiderivative size = 15, normalized size of antiderivative = 1., 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]

Int[x/(1 + Sqrt[1 + x]),x]

[Out]

-x + (2*(1 + x)^(3/2))/3

Rule 371

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*} \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 = 0.0150741, size = 19, normalized size = 1.27 \[ 2 \left (\frac{1}{3} (x+1)^{3/2}-\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x/(1 + Sqrt[1 + x]),x]

[Out]

2*(-x/2 + (1 + x)^(3/2)/3)

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Maple [A]  time = 0.002, size = 13, normalized size = 0.9 \begin{align*}{\frac{2}{3} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-1-x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(1+(1+x)^(1/2)),x)

[Out]

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

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Maxima [A]  time = 0.932082, size = 16, normalized size = 1.07 \begin{align*} \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} - x - 1 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x + 1)^(3/2) - x - 1

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Fricas [A]  time = 1.74389, size = 31, normalized size = 2.07 \begin{align*} \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} - x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x + 1)^(3/2) - x

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Sympy [B]  time = 0.751358, size = 22, normalized size = 1.47 \begin{align*} \frac{2 x \sqrt{x + 1}}{3} - x + \frac{2 \sqrt{x + 1}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.04315, size = 16, normalized size = 1.07 \begin{align*} \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} - x - 1 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x + 1)^(3/2) - x - 1