Integrand size = 28, antiderivative size = 104 \[ \int \frac {x^{5/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx=\frac {5}{8} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}+\frac {5}{12} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}+\frac {1}{3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {5 \text {arccosh}\left (\sqrt {x}\right )}{8} \]
5/8*arccosh(x^(1/2))+5/12*x^(3/2)*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)+1/3 *x^(5/2)*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)+5/8*x^(1/2)*(-1+x^(1/2))^(1/ 2)*(1+x^(1/2))^(1/2)
Time = 1.36 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.85 \[ \int \frac {x^{5/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx=\frac {1}{24} \sqrt {\frac {-1+\sqrt {x}}{1+\sqrt {x}}} \sqrt {x} \left (15+15 \sqrt {x}+10 x+10 x^{3/2}+8 x^2+8 x^{5/2}\right )+\frac {5}{4} \text {arctanh}\left (\sqrt {\frac {-1+\sqrt {x}}{1+\sqrt {x}}}\right ) \]
(Sqrt[(-1 + Sqrt[x])/(1 + Sqrt[x])]*Sqrt[x]*(15 + 15*Sqrt[x] + 10*x + 10*x ^(3/2) + 8*x^2 + 8*x^(5/2)))/24 + (5*ArcTanh[Sqrt[(-1 + Sqrt[x])/(1 + Sqrt [x])]])/4
Time = 0.23 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {845, 845, 845, 852, 43}
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.
| \(\displaystyle \int \frac {x^{5/2}}{\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}} \, dx\) |
| \(\Big \downarrow \) 845 |
| \(\displaystyle \frac {5}{6} \int \frac {x^{3/2}}{\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}dx+\frac {1}{3} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{5/2}\) |
| \(\Big \downarrow \) 845 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{4} \int \frac {\sqrt {x}}{\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}dx+\frac {1}{2} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}\right )+\frac {1}{3} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{5/2}\) |
| \(\Big \downarrow \) 845 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}}dx+\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}\right )+\frac {1}{2} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}\right )+\frac {1}{3} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{5/2}\) |
| \(\Big \downarrow \) 852 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{4} \left (\int \frac {1}{\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}d\sqrt {x}+\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}\right )+\frac {1}{2} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}\right )+\frac {1}{3} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{5/2}\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{4} \left (\text {arccosh}\left (\sqrt {x}\right )+\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}\right )+\frac {1}{2} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}\right )+\frac {1}{3} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{5/2}\) |
(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(5/2))/3 + (5*((Sqrt[-1 + Sqrt[x]] *Sqrt[1 + Sqrt[x]]*x^(3/2))/2 + (3*(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*S qrt[x] + ArcCosh[Sqrt[x]]))/4))/6
3.11.11.3.1 Defintions of rubi rules used
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ ArcCosh[b*(x/a)]/(b*Sqrt[d/b]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a *d, 0] && GtQ[a, 0] && GtQ[d/b, 0]
Int[((c_.)*(x_))^(m_)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^ (n_))^(p_), x_Symbol] :> Simp[c^(2*n - 1)*(c*x)^(m - 2*n + 1)*(a1 + b1*x^n) ^(p + 1)*((a2 + b2*x^n)^(p + 1)/(b1*b2*(m + 2*n*p + 1))), x] - Simp[a1*a2*c ^(2*n)*((m - 2*n + 1)/(b1*b2*(m + 2*n*p + 1))) Int[(c*x)^(m - 2*n)*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, p}, x] && Eq Q[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && GtQ[m, 2*n - 1] && NeQ[m + 2*n*p + 1 , 0] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]
Int[((c_.)*(x_))^(m_)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^ (n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/c Subst[Int[x^ (k*(m + 1) - 1)*(a1 + b1*(x^(k*n)/c^n))^p*(a2 + b2*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a1, b1, a2, b2, c, p}, x] && EqQ[a2*b1 + a1*b2 , 0] && IGtQ[2*n, 0] && FractionQ[m] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]
Time = 4.82 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.62
| method | result | size |
| derivativedivides | \(\frac {\sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}\, \left (8 x^{\frac {5}{2}} \sqrt {-1+x}+10 x^{\frac {3}{2}} \sqrt {-1+x}+15 \sqrt {x}\, \sqrt {-1+x}+15 \ln \left (\sqrt {x}+\sqrt {-1+x}\right )\right )}{24 \sqrt {-1+x}}\) | \(65\) |
| default | \(\frac {\sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}\, \left (8 x^{\frac {5}{2}} \sqrt {-1+x}+10 x^{\frac {3}{2}} \sqrt {-1+x}+15 \sqrt {x}\, \sqrt {-1+x}+15 \ln \left (\sqrt {x}+\sqrt {-1+x}\right )\right )}{24 \sqrt {-1+x}}\) | \(65\) |
1/24*(x^(1/2)-1)^(1/2)*(x^(1/2)+1)^(1/2)*(8*x^(5/2)*(-1+x)^(1/2)+10*x^(3/2 )*(-1+x)^(1/2)+15*x^(1/2)*(-1+x)^(1/2)+15*ln(x^(1/2)+(-1+x)^(1/2)))/(-1+x) ^(1/2)
Time = 0.26 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.55 \[ \int \frac {x^{5/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx=\frac {1}{24} \, {\left (8 \, x^{2} + 10 \, x + 15\right )} \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - \frac {5}{16} \, \log \left (2 \, \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - 2 \, x + 1\right ) \]
1/24*(8*x^2 + 10*x + 15)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - 5/1 6*log(2*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - 2*x + 1)
\[ \int \frac {x^{5/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx=\int \frac {x^{\frac {5}{2}}}{\sqrt {\sqrt {x} - 1} \sqrt {\sqrt {x} + 1}}\, dx \]
Time = 0.19 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.45 \[ \int \frac {x^{5/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx=\frac {1}{3} \, \sqrt {x - 1} x^{\frac {5}{2}} + \frac {5}{12} \, \sqrt {x - 1} x^{\frac {3}{2}} + \frac {5}{8} \, \sqrt {x - 1} \sqrt {x} + \frac {5}{8} \, \log \left (2 \, \sqrt {x - 1} + 2 \, \sqrt {x}\right ) \]
1/3*sqrt(x - 1)*x^(5/2) + 5/12*sqrt(x - 1)*x^(3/2) + 5/8*sqrt(x - 1)*sqrt( x) + 5/8*log(2*sqrt(x - 1) + 2*sqrt(x))
Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.73 \[ \int \frac {x^{5/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx=\frac {1}{24} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (\sqrt {x} + 1\right )} {\left (\sqrt {x} - 4\right )} + 45\right )} {\left (\sqrt {x} + 1\right )} - 55\right )} {\left (\sqrt {x} + 1\right )} + 85\right )} {\left (\sqrt {x} + 1\right )} - 33\right )} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - \frac {5}{4} \, \log \left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right ) \]
1/24*((2*((4*(sqrt(x) + 1)*(sqrt(x) - 4) + 45)*(sqrt(x) + 1) - 55)*(sqrt(x ) + 1) + 85)*(sqrt(x) + 1) - 33)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - 5/4 *log(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))
Time = 38.44 (sec) , antiderivative size = 632, normalized size of antiderivative = 6.08 \[ \int \frac {x^{5/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx=\frac {5\,\mathrm {atanh}\left (\frac {\sqrt {\sqrt {x}-1}-\mathrm {i}}{\sqrt {\sqrt {x}+1}-1}\right )}{2}-\frac {-\frac {175\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^3}{6\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^3}+\frac {311\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^5}{2\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^5}+\frac {8361\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^7}{2\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^7}+\frac {42259\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^9}{3\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^9}+\frac {25295\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{11}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{11}}+\frac {25295\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{13}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{13}}+\frac {42259\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{15}}{3\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{15}}+\frac {8361\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{17}}{2\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{17}}+\frac {311\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{19}}{2\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{19}}-\frac {175\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{21}}{6\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{21}}+\frac {5\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{23}}{2\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{23}}+\frac {5\,\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}{2\,\left (\sqrt {\sqrt {x}+1}-1\right )}}{1+\frac {66\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^4}-\frac {220\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^6}+\frac {495\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^8}-\frac {792\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{10}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{10}}+\frac {924\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{12}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{12}}-\frac {792\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{14}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{14}}+\frac {495\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{16}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{16}}-\frac {220\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{18}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{18}}+\frac {66\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{20}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{20}}-\frac {12\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{22}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{22}}+\frac {{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{24}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{24}}-\frac {12\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^2}} \]
(5*atanh(((x^(1/2) - 1)^(1/2) - 1i)/((x^(1/2) + 1)^(1/2) - 1)))/2 - ((311* ((x^(1/2) - 1)^(1/2) - 1i)^5)/(2*((x^(1/2) + 1)^(1/2) - 1)^5) - (175*((x^( 1/2) - 1)^(1/2) - 1i)^3)/(6*((x^(1/2) + 1)^(1/2) - 1)^3) + (8361*((x^(1/2) - 1)^(1/2) - 1i)^7)/(2*((x^(1/2) + 1)^(1/2) - 1)^7) + (42259*((x^(1/2) - 1)^(1/2) - 1i)^9)/(3*((x^(1/2) + 1)^(1/2) - 1)^9) + (25295*((x^(1/2) - 1)^ (1/2) - 1i)^11)/((x^(1/2) + 1)^(1/2) - 1)^11 + (25295*((x^(1/2) - 1)^(1/2) - 1i)^13)/((x^(1/2) + 1)^(1/2) - 1)^13 + (42259*((x^(1/2) - 1)^(1/2) - 1i )^15)/(3*((x^(1/2) + 1)^(1/2) - 1)^15) + (8361*((x^(1/2) - 1)^(1/2) - 1i)^ 17)/(2*((x^(1/2) + 1)^(1/2) - 1)^17) + (311*((x^(1/2) - 1)^(1/2) - 1i)^19) /(2*((x^(1/2) + 1)^(1/2) - 1)^19) - (175*((x^(1/2) - 1)^(1/2) - 1i)^21)/(6 *((x^(1/2) + 1)^(1/2) - 1)^21) + (5*((x^(1/2) - 1)^(1/2) - 1i)^23)/(2*((x^ (1/2) + 1)^(1/2) - 1)^23) + (5*((x^(1/2) - 1)^(1/2) - 1i))/(2*((x^(1/2) + 1)^(1/2) - 1)))/((66*((x^(1/2) - 1)^(1/2) - 1i)^4)/((x^(1/2) + 1)^(1/2) - 1)^4 - (12*((x^(1/2) - 1)^(1/2) - 1i)^2)/((x^(1/2) + 1)^(1/2) - 1)^2 - (22 0*((x^(1/2) - 1)^(1/2) - 1i)^6)/((x^(1/2) + 1)^(1/2) - 1)^6 + (495*((x^(1/ 2) - 1)^(1/2) - 1i)^8)/((x^(1/2) + 1)^(1/2) - 1)^8 - (792*((x^(1/2) - 1)^( 1/2) - 1i)^10)/((x^(1/2) + 1)^(1/2) - 1)^10 + (924*((x^(1/2) - 1)^(1/2) - 1i)^12)/((x^(1/2) + 1)^(1/2) - 1)^12 - (792*((x^(1/2) - 1)^(1/2) - 1i)^14) /((x^(1/2) + 1)^(1/2) - 1)^14 + (495*((x^(1/2) - 1)^(1/2) - 1i)^16)/((x^(1 /2) + 1)^(1/2) - 1)^16 - (220*((x^(1/2) - 1)^(1/2) - 1i)^18)/((x^(1/2) ...