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

Optimal. Leaf size=21 \[ \frac {2}{3} (x+1)^{3/2}-\frac {2 x^{3/2}}{3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2106, 30, 32} \[ \frac {2}{3} (x+1)^{3/2}-\frac {2 x^{3/2}}{3} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[x] + Sqrt[1 + x])^(-1),x]

[Out]

(-2*x^(3/2))/3 + (2*(1 + x)^(3/2))/3

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2106

Int[(u_.)/((d_.)*(x_)^(n_.) + (c_.)*Sqrt[(a_.) + (b_.)*(x_)^(p_.)]), x_Symbol] :> -Dist[b/(a*d), Int[u*x^n, x]
, x] + Dist[1/(a*c), Int[u*Sqrt[a + b*x^(2*n)], x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 2*n] && EqQ[b*c^
2 - d^2, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {x}+\sqrt {1+x}} \, dx &=-\int \sqrt {x} \, dx+\int \sqrt {1+x} \, dx\\ &=-\frac {2 x^{3/2}}{3}+\frac {2}{3} (1+x)^{3/2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 21, normalized size = 1.00 \[ \frac {2}{3} (x+1)^{3/2}-\frac {2 x^{3/2}}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[x] + Sqrt[1 + x])^(-1),x]

[Out]

(-2*x^(3/2))/3 + (2*(1 + x)^(3/2))/3

________________________________________________________________________________________

fricas [A]  time = 0.45, size = 13, normalized size = 0.62 \[ \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - \frac {2}{3} \, x^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x + 1)^(3/2) - 2/3*x^(3/2)

________________________________________________________________________________________

giac [A]  time = 0.23, size = 13, normalized size = 0.62 \[ \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - \frac {2}{3} \, x^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x + 1)^(3/2) - 2/3*x^(3/2)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 14, normalized size = 0.67 \[ -\frac {2 x^{\frac {3}{2}}}{3}+\frac {2 \left (x +1\right )^{\frac {3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x + 1} + \sqrt {x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(x + 1) + sqrt(x)), x)

________________________________________________________________________________________

mupad [B]  time = 2.97, size = 21, normalized size = 1.00 \[ \frac {2\,x\,\sqrt {x+1}}{3}+\frac {2\,\sqrt {x+1}}{3}-\frac {2\,x^{3/2}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((x + 1)^(1/2) + x^(1/2)),x)

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 0.94, size = 63, normalized size = 3.00 \[ \frac {2 \sqrt {x} \sqrt {x + 1}}{3 \sqrt {x} + 3 \sqrt {x + 1}} + \frac {4 x}{3 \sqrt {x} + 3 \sqrt {x + 1}} + \frac {2}{3 \sqrt {x} + 3 \sqrt {x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x)*sqrt(x + 1)/(3*sqrt(x) + 3*sqrt(x + 1)) + 4*x/(3*sqrt(x) + 3*sqrt(x + 1)) + 2/(3*sqrt(x) + 3*sqrt(x
+ 1))

________________________________________________________________________________________