Integrand size = 28, antiderivative size = 125 \[ \int \frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{x^{11/2}} \, dx=\frac {2 \left (-1+\sqrt {x}\right )^{3/2} \left (1+\sqrt {x}\right )^{3/2}}{9 x^{9/2}}+\frac {4 \left (-1+\sqrt {x}\right )^{3/2} \left (1+\sqrt {x}\right )^{3/2}}{21 x^{7/2}}+\frac {16 \left (-1+\sqrt {x}\right )^{3/2} \left (1+\sqrt {x}\right )^{3/2}}{105 x^{5/2}}+\frac {32 \left (-1+\sqrt {x}\right )^{3/2} \left (1+\sqrt {x}\right )^{3/2}}{315 x^{3/2}} \] Output:
2/9*(-1+x^(1/2))^(3/2)*(1+x^(1/2))^(3/2)/x^(9/2)+4/21*(-1+x^(1/2))^(3/2)*( 1+x^(1/2))^(3/2)/x^(7/2)+16/105*(-1+x^(1/2))^(3/2)*(1+x^(1/2))^(3/2)/x^(5/ 2)+32/315*(-1+x^(1/2))^(3/2)*(1+x^(1/2))^(3/2)/x^(3/2)
Time = 10.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.37 \[ \int \frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{x^{11/2}} \, dx=\frac {2 \left (-1+\sqrt {x}\right )^{3/2} \left (1+\sqrt {x}\right )^{3/2} \left (35+30 x+24 x^2+16 x^3\right )}{315 x^{9/2}} \] Input:
Integrate[(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/x^(11/2),x]
Output:
(2*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2)*(35 + 30*x + 24*x^2 + 16*x^3)) /(315*x^(9/2))
Time = 0.38 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {804, 804, 804, 797}
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.
\(\displaystyle \int \frac {\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}{x^{11/2}} \, dx\) |
\(\Big \downarrow \) 804 |
\(\displaystyle \frac {2}{3} \int \frac {\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}{x^{9/2}}dx+\frac {2 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{9 x^{9/2}}\) |
\(\Big \downarrow \) 804 |
\(\displaystyle \frac {2}{3} \left (\frac {4}{7} \int \frac {\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}{x^{7/2}}dx+\frac {2 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{7 x^{7/2}}\right )+\frac {2 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{9 x^{9/2}}\) |
\(\Big \downarrow \) 804 |
\(\displaystyle \frac {2}{3} \left (\frac {4}{7} \left (\frac {2}{5} \int \frac {\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}{x^{5/2}}dx+\frac {2 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{5 x^{5/2}}\right )+\frac {2 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{7 x^{7/2}}\right )+\frac {2 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{9 x^{9/2}}\) |
\(\Big \downarrow \) 797 |
\(\displaystyle \frac {2 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{9 x^{9/2}}+\frac {2}{3} \left (\frac {2 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{7 x^{7/2}}+\frac {4}{7} \left (\frac {4 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{15 x^{3/2}}+\frac {2 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{5 x^{5/2}}\right )\right )\) |
Input:
Int[(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/x^(11/2),x]
Output:
(2*((4*((2*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2))/(5*x^(5/2)) + (4*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2))/(15*x^(3/2))))/7 + (2*(-1 + Sqrt[x]) ^(3/2)*(1 + Sqrt[x])^(3/2))/(7*x^(7/2))))/3 + (2*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2))/(9*x^(9/2))
Int[((c_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_) ^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*(a1 + b1*x^n)^(p + 1)*((a2 + b 2*x^n)^(p + 1)/(a1*a2*c*(m + 1))), x] /; FreeQ[{a1, b1, a2, b2, c, m, n, p} , x] && EqQ[a2*b1 + a1*b2, 0] && EqQ[(m + 1)/(2*n) + p + 1, 0] && NeQ[m, -1 ]
Int[(x_)^(m_)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p _), x_Symbol] :> Simp[x^(m + 1)*(a1 + b1*x^n)^(p + 1)*((a2 + b2*x^n)^(p + 1 )/(a1*a2*(m + 1))), x] - Simp[b1*b2*((m + 2*n*(p + 1) + 1)/(a1*a2*(m + 1))) Int[x^(m + 2*n)*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x], x] /; FreeQ[{a1, b1 , a2, b2, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && ILtQ[Simplify[(m + 1)/(2 *n) + p + 1], 0] && NeQ[m, -1]
Time = 0.10 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.30
method | result | size |
derivativedivides | \(\frac {2 \sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\, \left (x -1\right ) \left (16 x^{3}+24 x^{2}+30 x +35\right )}{315 x^{\frac {9}{2}}}\) | \(38\) |
default | \(\frac {2 \sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\, \left (x -1\right ) \left (16 x^{3}+24 x^{2}+30 x +35\right )}{315 x^{\frac {9}{2}}}\) | \(38\) |
orering | \(\frac {2 \sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\, \left (x -1\right ) \left (16 x^{3}+24 x^{2}+30 x +35\right )}{315 x^{\frac {9}{2}}}\) | \(38\) |
Input:
int((-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)/x^(11/2),x,method=_RETURNVERBOSE)
Output:
2/315*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)*(x-1)*(16*x^3+24*x^2+30*x+35)/x ^(9/2)
Time = 0.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.39 \[ \int \frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{x^{11/2}} \, dx=\frac {2 \, {\left (16 \, x^{5} + {\left (16 \, x^{4} + 8 \, x^{3} + 6 \, x^{2} + 5 \, x - 35\right )} \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1}\right )}}{315 \, x^{5}} \] Input:
integrate((-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)/x^(11/2),x, algorithm="fric as")
Output:
2/315*(16*x^5 + (16*x^4 + 8*x^3 + 6*x^2 + 5*x - 35)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1))/x^5
Timed out. \[ \int \frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{x^{11/2}} \, dx=\text {Timed out} \] Input:
integrate((-1+x**(1/2))**(1/2)*(1+x**(1/2))**(1/2)/x**(11/2),x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.33 \[ \int \frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{x^{11/2}} \, dx=\frac {32 \, {\left (x - 1\right )}^{\frac {3}{2}}}{315 \, x^{\frac {3}{2}}} + \frac {16 \, {\left (x - 1\right )}^{\frac {3}{2}}}{105 \, x^{\frac {5}{2}}} + \frac {4 \, {\left (x - 1\right )}^{\frac {3}{2}}}{21 \, x^{\frac {7}{2}}} + \frac {2 \, {\left (x - 1\right )}^{\frac {3}{2}}}{9 \, x^{\frac {9}{2}}} \] Input:
integrate((-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)/x^(11/2),x, algorithm="maxi ma")
Output:
32/315*(x - 1)^(3/2)/x^(3/2) + 16/105*(x - 1)^(3/2)/x^(5/2) + 4/21*(x - 1) ^(3/2)/x^(7/2) + 2/9*(x - 1)^(3/2)/x^(9/2)
Time = 0.14 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{x^{11/2}} \, dx=\frac {16384 \, {\left (315 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{20} - 756 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{16} + 1344 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{12} + 2304 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{8} + 2304 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{4} + 1024\right )}}{315 \, {\left ({\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{4} + 4\right )}^{9}} \] Input:
integrate((-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)/x^(11/2),x, algorithm="giac ")
Output:
16384/315*(315*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^20 - 756*(sqrt(sqrt (x) + 1) - sqrt(sqrt(x) - 1))^16 + 1344*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^12 + 2304*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^8 + 2304*(sqrt(sqr t(x) + 1) - sqrt(sqrt(x) - 1))^4 + 1024)/((sqrt(sqrt(x) + 1) - sqrt(sqrt(x ) - 1))^4 + 4)^9
Time = 3.91 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.54 \[ \int \frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{x^{11/2}} \, dx=\frac {\sqrt {\sqrt {x}-1}\,\left (\frac {2\,x\,\sqrt {\sqrt {x}+1}}{63}-\frac {2\,\sqrt {\sqrt {x}+1}}{9}+\frac {4\,x^2\,\sqrt {\sqrt {x}+1}}{105}+\frac {16\,x^3\,\sqrt {\sqrt {x}+1}}{315}+\frac {32\,x^4\,\sqrt {\sqrt {x}+1}}{315}\right )}{x^{9/2}} \] Input:
int(((x^(1/2) - 1)^(1/2)*(x^(1/2) + 1)^(1/2))/x^(11/2),x)
Output:
((x^(1/2) - 1)^(1/2)*((2*x*(x^(1/2) + 1)^(1/2))/63 - (2*(x^(1/2) + 1)^(1/2 ))/9 + (4*x^2*(x^(1/2) + 1)^(1/2))/105 + (16*x^3*(x^(1/2) + 1)^(1/2))/315 + (32*x^4*(x^(1/2) + 1)^(1/2))/315))/x^(9/2)
Time = 0.21 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{x^{11/2}} \, dx=\frac {\frac {32 \sqrt {\sqrt {x}+1}\, \sqrt {\sqrt {x}-1}\, x^{4}}{315}+\frac {16 \sqrt {\sqrt {x}+1}\, \sqrt {\sqrt {x}-1}\, x^{3}}{315}+\frac {4 \sqrt {\sqrt {x}+1}\, \sqrt {\sqrt {x}-1}\, x^{2}}{105}+\frac {2 \sqrt {\sqrt {x}+1}\, \sqrt {\sqrt {x}-1}\, x}{63}-\frac {2 \sqrt {\sqrt {x}+1}\, \sqrt {\sqrt {x}-1}}{9}-\frac {32 \sqrt {x}\, x^{4}}{315}}{\sqrt {x}\, x^{4}} \] Input:
int((-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)/x^(11/2),x)
Output:
(2*(16*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)*x**4 + 8*sqrt(sqrt(x) + 1)*sqrt (sqrt(x) - 1)*x**3 + 6*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)*x**2 + 5*sqrt(s qrt(x) + 1)*sqrt(sqrt(x) - 1)*x - 35*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - 16*sqrt(x)*x**4))/(315*sqrt(x)*x**4)