Integrand size = 28, antiderivative size = 63 \[ \int \frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{x^{7/2}} \, dx=\frac {2 \left (-1+\sqrt {x}\right )^{3/2} \left (1+\sqrt {x}\right )^{3/2}}{5 x^{5/2}}+\frac {4 \left (-1+\sqrt {x}\right )^{3/2} \left (1+\sqrt {x}\right )^{3/2}}{15 x^{3/2}} \] Output:
2/5*(-1+x^(1/2))^(3/2)*(1+x^(1/2))^(3/2)/x^(5/2)+4/15*(-1+x^(1/2))^(3/2)*( 1+x^(1/2))^(3/2)/x^(3/2)
Leaf count is larger than twice the leaf count of optimal. \(693\) vs. \(2(63)=126\).
Time = 8.07 (sec) , antiderivative size = 693, normalized size of antiderivative = 11.00 \[ \int \frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{x^{7/2}} \, dx=\frac {\left (-1+\sqrt {-1+\sqrt {x}}\right ) \left (\sqrt {3}-\sqrt {1+\sqrt {x}}\right ) \left (-2+\sqrt {-1+\sqrt {x}}+\sqrt {3} \sqrt {1+\sqrt {x}}-\sqrt {x}\right ) \left (-384 \left (97-168 \sqrt {-1+\sqrt {x}}-56 \sqrt {3} \sqrt {1+\sqrt {x}}+97 \sqrt {3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}\right )-192 \left (-499-1112 \sqrt {-1+\sqrt {x}}+344 \sqrt {3} \sqrt {1+\sqrt {x}}+545 \sqrt {3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}\right ) \sqrt {x}+32 \left (9925+1656 \sqrt {-1+\sqrt {x}}-4616 \sqrt {3} \sqrt {1+\sqrt {x}}+535 \sqrt {3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}\right ) x+16 \left (2243-20096 \sqrt {-1+\sqrt {x}}+2720 \sqrt {3} \sqrt {1+\sqrt {x}}+10385 \sqrt {3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}\right ) x^{3/2}+8 \left (-33645-39152 \sqrt {-1+\sqrt {x}}+15056 \sqrt {3} \sqrt {1+\sqrt {x}}+13135 \sqrt {3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}\right ) x^2+8 \left (-21349-18772 \sqrt {-1+\sqrt {x}}+6180 \sqrt {3} \sqrt {1+\sqrt {x}}+6379 \sqrt {3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}\right ) x^{5/2}+4 \left (-22053-18788 \sqrt {-1+\sqrt {x}}+8788 \sqrt {3} \sqrt {1+\sqrt {x}}+5745 \sqrt {3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}\right ) x^3+2 \left (-19920-7252 \sqrt {-1+\sqrt {x}}+4188 \sqrt {3} \sqrt {1+\sqrt {x}}+715 \sqrt {3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}\right ) x^{7/2}-2477 x^4\right )}{240 \left (-3-2 \sqrt {-1+\sqrt {x}}+2 \sqrt {3} \sqrt {1+\sqrt {x}}+\sqrt {3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}-2 \sqrt {x}\right )^5 x^{5/2}} \] Input:
Integrate[(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/x^(7/2),x]
Output:
((-1 + Sqrt[-1 + Sqrt[x]])*(Sqrt[3] - Sqrt[1 + Sqrt[x]])*(-2 + Sqrt[-1 + S qrt[x]] + Sqrt[3]*Sqrt[1 + Sqrt[x]] - Sqrt[x])*(-384*(97 - 168*Sqrt[-1 + S qrt[x]] - 56*Sqrt[3]*Sqrt[1 + Sqrt[x]] + 97*Sqrt[3]*Sqrt[-1 + Sqrt[x]]*Sqr t[1 + Sqrt[x]]) - 192*(-499 - 1112*Sqrt[-1 + Sqrt[x]] + 344*Sqrt[3]*Sqrt[1 + Sqrt[x]] + 545*Sqrt[3]*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])*Sqrt[x] + 32*(9925 + 1656*Sqrt[-1 + Sqrt[x]] - 4616*Sqrt[3]*Sqrt[1 + Sqrt[x]] + 535* Sqrt[3]*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])*x + 16*(2243 - 20096*Sqrt[-1 + Sqrt[x]] + 2720*Sqrt[3]*Sqrt[1 + Sqrt[x]] + 10385*Sqrt[3]*Sqrt[-1 + Sqr t[x]]*Sqrt[1 + Sqrt[x]])*x^(3/2) + 8*(-33645 - 39152*Sqrt[-1 + Sqrt[x]] + 15056*Sqrt[3]*Sqrt[1 + Sqrt[x]] + 13135*Sqrt[3]*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])*x^2 + 8*(-21349 - 18772*Sqrt[-1 + Sqrt[x]] + 6180*Sqrt[3]*Sqrt [1 + Sqrt[x]] + 6379*Sqrt[3]*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])*x^(5/2) + 4*(-22053 - 18788*Sqrt[-1 + Sqrt[x]] + 8788*Sqrt[3]*Sqrt[1 + Sqrt[x]] + 5745*Sqrt[3]*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])*x^3 + 2*(-19920 - 7252 *Sqrt[-1 + Sqrt[x]] + 4188*Sqrt[3]*Sqrt[1 + Sqrt[x]] + 715*Sqrt[3]*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])*x^(7/2) - 2477*x^4))/(240*(-3 - 2*Sqrt[-1 + Sqrt[x]] + 2*Sqrt[3]*Sqrt[1 + Sqrt[x]] + Sqrt[3]*Sqrt[-1 + Sqrt[x]]*Sqrt[ 1 + Sqrt[x]] - 2*Sqrt[x])^5*x^(5/2))
Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {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.
\(\displaystyle \int \frac {\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}{x^{7/2}} \, dx\) |
\(\Big \downarrow \) 804 |
\(\displaystyle \frac {2}{5} \int \frac {\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}{x^{5/2}}dx+\frac {2 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{5 x^{5/2}}\) |
\(\Big \downarrow \) 797 |
\(\displaystyle \frac {4 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{15 x^{3/2}}+\frac {2 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{5 x^{5/2}}\) |
Input:
Int[(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/x^(7/2),x]
Output:
(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))
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 = 0.09 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.44
method | result | size |
derivativedivides | \(\frac {2 \sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\, \left (x -1\right ) \left (2 x +3\right )}{15 x^{\frac {5}{2}}}\) | \(28\) |
default | \(\frac {2 \sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\, \left (x -1\right ) \left (2 x +3\right )}{15 x^{\frac {5}{2}}}\) | \(28\) |
orering | \(\frac {2 \sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\, \left (x -1\right ) \left (2 x +3\right )}{15 x^{\frac {5}{2}}}\) | \(28\) |
Input:
int((-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)/x^(7/2),x,method=_RETURNVERBOSE)
Output:
2/15*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)*(x-1)*(2*x+3)/x^(5/2)
Time = 0.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{x^{7/2}} \, dx=\frac {2 \, {\left (2 \, x^{3} + {\left (2 \, x^{2} + x - 3\right )} \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1}\right )}}{15 \, x^{3}} \] Input:
integrate((-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)/x^(7/2),x, algorithm="frica s")
Output:
2/15*(2*x^3 + (2*x^2 + x - 3)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)) /x^3
\[ \int \frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{x^{7/2}} \, dx=\int \frac {\sqrt {\sqrt {x} - 1} \sqrt {\sqrt {x} + 1}}{x^{\frac {7}{2}}}\, dx \] Input:
integrate((-1+x**(1/2))**(1/2)*(1+x**(1/2))**(1/2)/x**(7/2),x)
Output:
Integral(sqrt(sqrt(x) - 1)*sqrt(sqrt(x) + 1)/x**(7/2), x)
Time = 0.10 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.33 \[ \int \frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{x^{7/2}} \, dx=\frac {4 \, {\left (x - 1\right )}^{\frac {3}{2}}}{15 \, x^{\frac {3}{2}}} + \frac {2 \, {\left (x - 1\right )}^{\frac {3}{2}}}{5 \, x^{\frac {5}{2}}} \] Input:
integrate((-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)/x^(7/2),x, algorithm="maxim a")
Output:
4/15*(x - 1)^(3/2)/x^(3/2) + 2/5*(x - 1)^(3/2)/x^(5/2)
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (39) = 78\).
Time = 0.13 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.43 \[ \int \frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{x^{7/2}} \, dx=\frac {128 \, {\left (15 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{12} - 20 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{8} + 80 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{4} + 64\right )}}{15 \, {\left ({\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{4} + 4\right )}^{5}} \] Input:
integrate((-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)/x^(7/2),x, algorithm="giac" )
Output:
128/15*(15*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^12 - 20*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^8 + 80*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^4 + 64)/((sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^4 + 4)^5
Time = 3.77 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{x^{7/2}} \, dx=\frac {\sqrt {\sqrt {x}-1}\,\left (\frac {2\,x\,\sqrt {\sqrt {x}+1}}{15}-\frac {2\,\sqrt {\sqrt {x}+1}}{5}+\frac {4\,x^2\,\sqrt {\sqrt {x}+1}}{15}\right )}{x^{5/2}} \] Input:
int(((x^(1/2) - 1)^(1/2)*(x^(1/2) + 1)^(1/2))/x^(7/2),x)
Output:
((x^(1/2) - 1)^(1/2)*((2*x*(x^(1/2) + 1)^(1/2))/15 - (2*(x^(1/2) + 1)^(1/2 ))/5 + (4*x^2*(x^(1/2) + 1)^(1/2))/15))/x^(5/2)
Time = 0.21 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{x^{7/2}} \, dx=\frac {\frac {4 \sqrt {\sqrt {x}+1}\, \sqrt {\sqrt {x}-1}\, x^{2}}{15}+\frac {2 \sqrt {\sqrt {x}+1}\, \sqrt {\sqrt {x}-1}\, x}{15}-\frac {2 \sqrt {\sqrt {x}+1}\, \sqrt {\sqrt {x}-1}}{5}-\frac {4 \sqrt {x}\, x^{2}}{15}}{\sqrt {x}\, x^{2}} \] Input:
int((-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)/x^(7/2),x)
Output:
(2*(2*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)*x**2 + sqrt(sqrt(x) + 1)*sqrt(sq rt(x) - 1)*x - 3*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - 2*sqrt(x)*x**2))/(1 5*sqrt(x)*x**2)