Integrand size = 65, antiderivative size = 20 \[ \int \frac {\sqrt {x} \sqrt {1+x}+\sqrt {x} \sqrt {2+x}+\sqrt {1+x} \sqrt {2+x}}{2 \sqrt {x} \sqrt {1+x} \sqrt {2+x}} \, dx=\sqrt {x}+\sqrt {1+x}+\sqrt {2+x} \] Output:
x^(1/2)+(1+x)^(1/2)+(2+x)^(1/2)
Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.50 \[ \int \frac {\sqrt {x} \sqrt {1+x}+\sqrt {x} \sqrt {2+x}+\sqrt {1+x} \sqrt {2+x}}{2 \sqrt {x} \sqrt {1+x} \sqrt {2+x}} \, dx=\frac {1}{2} \left (2 \sqrt {x}+2 \sqrt {1+x}+2 \sqrt {2+x}\right ) \] Input:
Integrate[(Sqrt[x]*Sqrt[1 + x] + Sqrt[x]*Sqrt[2 + x] + Sqrt[1 + x]*Sqrt[2 + x])/(2*Sqrt[x]*Sqrt[1 + x]*Sqrt[2 + x]),x]
Output:
(2*Sqrt[x] + 2*Sqrt[1 + x] + 2*Sqrt[2 + x])/2
Time = 0.49 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {27, 2035, 7239, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sqrt {x} \sqrt {x+1}+\sqrt {x+2} \sqrt {x+1}+\sqrt {x} \sqrt {x+2}}{2 \sqrt {x} \sqrt {x+1} \sqrt {x+2}} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int \frac {\sqrt {x} \sqrt {x+1}+\sqrt {x+2} \sqrt {x+1}+\sqrt {x} \sqrt {x+2}}{\sqrt {x} \sqrt {x+1} \sqrt {x+2}}dx\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle \int \frac {\sqrt {x} \sqrt {x+1}+\sqrt {x+2} \sqrt {x+1}+\sqrt {x} \sqrt {x+2}}{\sqrt {x+1} \sqrt {x+2}}d\sqrt {x}\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \left (\sqrt {x} \left (\frac {1}{\sqrt {x+2}}+\frac {1}{\sqrt {x+1}}\right )+1\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \sqrt {x}+\sqrt {x+1}+\sqrt {x+2}\) |
Input:
Int[(Sqrt[x]*Sqrt[1 + x] + Sqrt[x]*Sqrt[2 + x] + Sqrt[1 + x]*Sqrt[2 + x])/ (2*Sqrt[x]*Sqrt[1 + x]*Sqrt[2 + x]),x]
Output:
Sqrt[x] + Sqrt[1 + x] + Sqrt[2 + x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75
method | result | size |
default | \(\sqrt {x}+\sqrt {1+x}+\sqrt {2+x}\) | \(15\) |
orering | \(\frac {\left (6 x +6\right ) \left (\sqrt {x}\, \sqrt {1+x}+\sqrt {x}\, \sqrt {2+x}+\sqrt {1+x}\, \sqrt {2+x}\right )}{2 \sqrt {x}\, \sqrt {1+x}\, \sqrt {2+x}}+\left (12 x^{2}+24 x +8\right ) \left (\frac {\frac {\sqrt {1+x}}{2 \sqrt {x}}+\frac {\sqrt {x}}{2 \sqrt {1+x}}+\frac {\sqrt {2+x}}{2 \sqrt {x}}+\frac {\sqrt {x}}{2 \sqrt {2+x}}+\frac {\sqrt {2+x}}{2 \sqrt {1+x}}+\frac {\sqrt {1+x}}{2 \sqrt {2+x}}}{2 \sqrt {x}\, \sqrt {1+x}\, \sqrt {2+x}}-\frac {\sqrt {x}\, \sqrt {1+x}+\sqrt {x}\, \sqrt {2+x}+\sqrt {1+x}\, \sqrt {2+x}}{4 x^{\frac {3}{2}} \sqrt {1+x}\, \sqrt {2+x}}-\frac {\sqrt {x}\, \sqrt {1+x}+\sqrt {x}\, \sqrt {2+x}+\sqrt {1+x}\, \sqrt {2+x}}{4 \sqrt {x}\, \left (1+x \right )^{\frac {3}{2}} \sqrt {2+x}}-\frac {\sqrt {x}\, \sqrt {1+x}+\sqrt {x}\, \sqrt {2+x}+\sqrt {1+x}\, \sqrt {2+x}}{4 \sqrt {x}\, \sqrt {1+x}\, \left (2+x \right )^{\frac {3}{2}}}\right )+\frac {8 x \left (2+x \right ) \left (1+x \right ) \left (\frac {-\frac {\sqrt {1+x}}{4 x^{\frac {3}{2}}}+\frac {1}{2 \sqrt {x}\, \sqrt {1+x}}-\frac {\sqrt {x}}{4 \left (1+x \right )^{\frac {3}{2}}}-\frac {\sqrt {2+x}}{4 x^{\frac {3}{2}}}+\frac {1}{2 \sqrt {x}\, \sqrt {2+x}}-\frac {\sqrt {x}}{4 \left (2+x \right )^{\frac {3}{2}}}-\frac {\sqrt {2+x}}{4 \left (1+x \right )^{\frac {3}{2}}}+\frac {1}{2 \sqrt {1+x}\, \sqrt {2+x}}-\frac {\sqrt {1+x}}{4 \left (2+x \right )^{\frac {3}{2}}}}{2 \sqrt {x}\, \sqrt {1+x}\, \sqrt {2+x}}-\frac {\frac {\sqrt {1+x}}{2 \sqrt {x}}+\frac {\sqrt {x}}{2 \sqrt {1+x}}+\frac {\sqrt {2+x}}{2 \sqrt {x}}+\frac {\sqrt {x}}{2 \sqrt {2+x}}+\frac {\sqrt {2+x}}{2 \sqrt {1+x}}+\frac {\sqrt {1+x}}{2 \sqrt {2+x}}}{2 x^{\frac {3}{2}} \sqrt {1+x}\, \sqrt {2+x}}-\frac {\frac {\sqrt {1+x}}{2 \sqrt {x}}+\frac {\sqrt {x}}{2 \sqrt {1+x}}+\frac {\sqrt {2+x}}{2 \sqrt {x}}+\frac {\sqrt {x}}{2 \sqrt {2+x}}+\frac {\sqrt {2+x}}{2 \sqrt {1+x}}+\frac {\sqrt {1+x}}{2 \sqrt {2+x}}}{2 \sqrt {x}\, \left (1+x \right )^{\frac {3}{2}} \sqrt {2+x}}-\frac {\frac {\sqrt {1+x}}{2 \sqrt {x}}+\frac {\sqrt {x}}{2 \sqrt {1+x}}+\frac {\sqrt {2+x}}{2 \sqrt {x}}+\frac {\sqrt {x}}{2 \sqrt {2+x}}+\frac {\sqrt {2+x}}{2 \sqrt {1+x}}+\frac {\sqrt {1+x}}{2 \sqrt {2+x}}}{2 \sqrt {x}\, \sqrt {1+x}\, \left (2+x \right )^{\frac {3}{2}}}+\frac {\frac {3 \sqrt {x}\, \sqrt {1+x}}{8}+\frac {3 \sqrt {x}\, \sqrt {2+x}}{8}+\frac {3 \sqrt {1+x}\, \sqrt {2+x}}{8}}{x^{\frac {5}{2}} \sqrt {1+x}\, \sqrt {2+x}}+\frac {\sqrt {x}\, \sqrt {1+x}+\sqrt {x}\, \sqrt {2+x}+\sqrt {1+x}\, \sqrt {2+x}}{4 x^{\frac {3}{2}} \left (1+x \right )^{\frac {3}{2}} \sqrt {2+x}}+\frac {\sqrt {x}\, \sqrt {1+x}+\sqrt {x}\, \sqrt {2+x}+\sqrt {1+x}\, \sqrt {2+x}}{4 x^{\frac {3}{2}} \sqrt {1+x}\, \left (2+x \right )^{\frac {3}{2}}}+\frac {\frac {3 \sqrt {x}\, \sqrt {1+x}}{8}+\frac {3 \sqrt {x}\, \sqrt {2+x}}{8}+\frac {3 \sqrt {1+x}\, \sqrt {2+x}}{8}}{\sqrt {x}\, \left (1+x \right )^{\frac {5}{2}} \sqrt {2+x}}+\frac {\sqrt {x}\, \sqrt {1+x}+\sqrt {x}\, \sqrt {2+x}+\sqrt {1+x}\, \sqrt {2+x}}{4 \sqrt {x}\, \left (1+x \right )^{\frac {3}{2}} \left (2+x \right )^{\frac {3}{2}}}+\frac {\frac {3 \sqrt {x}\, \sqrt {1+x}}{8}+\frac {3 \sqrt {x}\, \sqrt {2+x}}{8}+\frac {3 \sqrt {1+x}\, \sqrt {2+x}}{8}}{\sqrt {x}\, \sqrt {1+x}\, \left (2+x \right )^{\frac {5}{2}}}\right )}{3}\) | \(911\) |
Input:
int(1/2*(x^(1/2)*(1+x)^(1/2)+x^(1/2)*(2+x)^(1/2)+(1+x)^(1/2)*(2+x)^(1/2))/ x^(1/2)/(1+x)^(1/2)/(2+x)^(1/2),x,method=_RETURNVERBOSE)
Output:
x^(1/2)+(1+x)^(1/2)+(2+x)^(1/2)
Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {x} \sqrt {1+x}+\sqrt {x} \sqrt {2+x}+\sqrt {1+x} \sqrt {2+x}}{2 \sqrt {x} \sqrt {1+x} \sqrt {2+x}} \, dx=\sqrt {x + 2} + \sqrt {x + 1} + \sqrt {x} \] Input:
integrate(1/2*(x^(1/2)*(1+x)^(1/2)+x^(1/2)*(2+x)^(1/2)+(1+x)^(1/2)*(2+x)^( 1/2))/x^(1/2)/(1+x)^(1/2)/(2+x)^(1/2),x, algorithm="fricas")
Output:
sqrt(x + 2) + sqrt(x + 1) + sqrt(x)
Time = 0.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {x} \sqrt {1+x}+\sqrt {x} \sqrt {2+x}+\sqrt {1+x} \sqrt {2+x}}{2 \sqrt {x} \sqrt {1+x} \sqrt {2+x}} \, dx=\sqrt {x} + \sqrt {x + 1} + \sqrt {x + 2} \] Input:
integrate(1/2*(x**(1/2)*(1+x)**(1/2)+x**(1/2)*(2+x)**(1/2)+(1+x)**(1/2)*(2 +x)**(1/2))/x**(1/2)/(1+x)**(1/2)/(2+x)**(1/2),x)
Output:
sqrt(x) + sqrt(x + 1) + sqrt(x + 2)
Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {x} \sqrt {1+x}+\sqrt {x} \sqrt {2+x}+\sqrt {1+x} \sqrt {2+x}}{2 \sqrt {x} \sqrt {1+x} \sqrt {2+x}} \, dx=\sqrt {x + 2} + \sqrt {x + 1} + \sqrt {x} \] Input:
integrate(1/2*(x^(1/2)*(1+x)^(1/2)+x^(1/2)*(2+x)^(1/2)+(1+x)^(1/2)*(2+x)^( 1/2))/x^(1/2)/(1+x)^(1/2)/(2+x)^(1/2),x, algorithm="maxima")
Output:
sqrt(x + 2) + sqrt(x + 1) + sqrt(x)
\[ \int \frac {\sqrt {x} \sqrt {1+x}+\sqrt {x} \sqrt {2+x}+\sqrt {1+x} \sqrt {2+x}}{2 \sqrt {x} \sqrt {1+x} \sqrt {2+x}} \, dx=\int { \frac {\sqrt {x + 2} \sqrt {x + 1} + \sqrt {x + 2} \sqrt {x} + \sqrt {x + 1} \sqrt {x}}{2 \, \sqrt {x + 2} \sqrt {x + 1} \sqrt {x}} \,d x } \] Input:
integrate(1/2*(x^(1/2)*(1+x)^(1/2)+x^(1/2)*(2+x)^(1/2)+(1+x)^(1/2)*(2+x)^( 1/2))/x^(1/2)/(1+x)^(1/2)/(2+x)^(1/2),x, algorithm="giac")
Output:
integrate(1/2*(sqrt(x + 2)*sqrt(x + 1) + sqrt(x + 2)*sqrt(x) + sqrt(x + 1) *sqrt(x))/(sqrt(x + 2)*sqrt(x + 1)*sqrt(x)), x)
Time = 0.16 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {x} \sqrt {1+x}+\sqrt {x} \sqrt {2+x}+\sqrt {1+x} \sqrt {2+x}}{2 \sqrt {x} \sqrt {1+x} \sqrt {2+x}} \, dx=\sqrt {x+1}+\sqrt {x+2}+\sqrt {x} \] Input:
int(((x^(1/2)*(x + 1)^(1/2))/2 + (x^(1/2)*(x + 2)^(1/2))/2 + ((x + 1)^(1/2 )*(x + 2)^(1/2))/2)/(x^(1/2)*(x + 1)^(1/2)*(x + 2)^(1/2)),x)
Output:
(x + 1)^(1/2) + (x + 2)^(1/2) + x^(1/2)
Time = 0.15 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.55 \[ \int \frac {\sqrt {x} \sqrt {1+x}+\sqrt {x} \sqrt {2+x}+\sqrt {1+x} \sqrt {2+x}}{2 \sqrt {x} \sqrt {1+x} \sqrt {2+x}} \, dx=\sqrt {x +2}+\sqrt {x +1}+\sqrt {x} \] Input:
int(1/2*(x^(1/2)*(1+x)^(1/2)+x^(1/2)*(2+x)^(1/2)+(1+x)^(1/2)*(2+x)^(1/2))/ x^(1/2)/(1+x)^(1/2)/(2+x)^(1/2),x)
Output:
sqrt(x + 2) + sqrt(x + 1) + sqrt(x)