Integrand size = 28, antiderivative size = 73 \[ \int \frac {x^{3/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx=\frac {3}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}+\frac {1}{2} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}+\frac {3 \text {arccosh}\left (\sqrt {x}\right )}{4} \]
3/4*arccosh(x^(1/2))+1/2*x^(3/2)*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)+3/4* x^(1/2)*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)
Time = 1.37 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.03 \[ \int \frac {x^{3/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx=\frac {1}{4} \left (\sqrt {\frac {-1+\sqrt {x}}{1+\sqrt {x}}} \sqrt {x} \left (3+3 \sqrt {x}+2 x+2 x^{3/2}\right )+6 \text {arctanh}\left (\sqrt {\frac {-1+\sqrt {x}}{1+\sqrt {x}}}\right )\right ) \]
(Sqrt[(-1 + Sqrt[x])/(1 + Sqrt[x])]*Sqrt[x]*(3 + 3*Sqrt[x] + 2*x + 2*x^(3/ 2)) + 6*ArcTanh[Sqrt[(-1 + Sqrt[x])/(1 + Sqrt[x])]])/4
Time = 0.19 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {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.
\(\displaystyle \int \frac {x^{3/2}}{\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}} \, dx\) |
\(\Big \downarrow \) 845 |
\(\displaystyle \frac {3}{4} \int \frac {\sqrt {x}}{\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}dx+\frac {1}{2} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}\) |
\(\Big \downarrow \) 845 |
\(\displaystyle \frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}}dx+\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}\right )+\frac {1}{2} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}\) |
\(\Big \downarrow \) 852 |
\(\displaystyle \frac {3}{4} \left (\int \frac {1}{\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}d\sqrt {x}+\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}\right )+\frac {1}{2} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}\) |
\(\Big \downarrow \) 43 |
\(\displaystyle \frac {3}{4} \left (\text {arccosh}\left (\sqrt {x}\right )+\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}\right )+\frac {1}{2} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}\) |
(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(3/2))/2 + (3*(Sqrt[-1 + Sqrt[x]]* Sqrt[1 + Sqrt[x]]*Sqrt[x] + ArcCosh[Sqrt[x]]))/4
3.11.12.3.1 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 = 4.75 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(\frac {\sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}\, \left (2 x^{\frac {3}{2}} \sqrt {-1+x}+3 \sqrt {x}\, \sqrt {-1+x}+3 \ln \left (\sqrt {x}+\sqrt {-1+x}\right )\right )}{4 \sqrt {-1+x}}\) | \(55\) |
default | \(\frac {\sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}\, \left (2 x^{\frac {3}{2}} \sqrt {-1+x}+3 \sqrt {x}\, \sqrt {-1+x}+3 \ln \left (\sqrt {x}+\sqrt {-1+x}\right )\right )}{4 \sqrt {-1+x}}\) | \(55\) |
1/4*(x^(1/2)-1)^(1/2)*(x^(1/2)+1)^(1/2)*(2*x^(3/2)*(-1+x)^(1/2)+3*x^(1/2)* (-1+x)^(1/2)+3*ln(x^(1/2)+(-1+x)^(1/2)))/(-1+x)^(1/2)
Time = 0.26 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.71 \[ \int \frac {x^{3/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx=\frac {1}{4} \, {\left (2 \, x + 3\right )} \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - \frac {3}{8} \, \log \left (2 \, \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - 2 \, x + 1\right ) \]
1/4*(2*x + 3)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - 3/8*log(2*sqrt (x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - 2*x + 1)
\[ \int \frac {x^{3/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx=\int \frac {x^{\frac {3}{2}}}{\sqrt {\sqrt {x} - 1} \sqrt {\sqrt {x} + 1}}\, dx \]
Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.51 \[ \int \frac {x^{3/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx=\frac {1}{2} \, \sqrt {x - 1} x^{\frac {3}{2}} + \frac {3}{4} \, \sqrt {x - 1} \sqrt {x} + \frac {3}{4} \, \log \left (2 \, \sqrt {x - 1} + 2 \, \sqrt {x}\right ) \]
Time = 0.30 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.81 \[ \int \frac {x^{3/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx=\frac {1}{4} \, {\left ({\left (2 \, {\left (\sqrt {x} + 1\right )} {\left (\sqrt {x} - 2\right )} + 9\right )} {\left (\sqrt {x} + 1\right )} - 5\right )} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - \frac {3}{2} \, \log \left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right ) \]
1/4*((2*(sqrt(x) + 1)*(sqrt(x) - 2) + 9)*(sqrt(x) + 1) - 5)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - 3/2*log(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))
Time = 26.04 (sec) , antiderivative size = 429, normalized size of antiderivative = 5.88 \[ \int \frac {x^{3/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx=3\,\mathrm {atanh}\left (\frac {\sqrt {\sqrt {x}-1}-\mathrm {i}}{\sqrt {\sqrt {x}+1}-1}\right )+\frac {\frac {23\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^3}+\frac {333\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^5}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^5}+\frac {671\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^7}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^7}+\frac {671\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^9}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^9}+\frac {333\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{11}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{11}}+\frac {23\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{13}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{13}}-\frac {3\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{15}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{15}}-\frac {3\,\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}{\sqrt {\sqrt {x}+1}-1}}{1+\frac {28\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^4}-\frac {56\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^6}+\frac {70\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^8}-\frac {56\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{10}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{10}}+\frac {28\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{12}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{12}}-\frac {8\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{14}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{14}}+\frac {{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{16}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{16}}-\frac {8\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^2}} \]
3*atanh(((x^(1/2) - 1)^(1/2) - 1i)/((x^(1/2) + 1)^(1/2) - 1)) + ((23*((x^( 1/2) - 1)^(1/2) - 1i)^3)/((x^(1/2) + 1)^(1/2) - 1)^3 + (333*((x^(1/2) - 1) ^(1/2) - 1i)^5)/((x^(1/2) + 1)^(1/2) - 1)^5 + (671*((x^(1/2) - 1)^(1/2) - 1i)^7)/((x^(1/2) + 1)^(1/2) - 1)^7 + (671*((x^(1/2) - 1)^(1/2) - 1i)^9)/(( x^(1/2) + 1)^(1/2) - 1)^9 + (333*((x^(1/2) - 1)^(1/2) - 1i)^11)/((x^(1/2) + 1)^(1/2) - 1)^11 + (23*((x^(1/2) - 1)^(1/2) - 1i)^13)/((x^(1/2) + 1)^(1/ 2) - 1)^13 - (3*((x^(1/2) - 1)^(1/2) - 1i)^15)/((x^(1/2) + 1)^(1/2) - 1)^1 5 - (3*((x^(1/2) - 1)^(1/2) - 1i))/((x^(1/2) + 1)^(1/2) - 1))/((28*((x^(1/ 2) - 1)^(1/2) - 1i)^4)/((x^(1/2) + 1)^(1/2) - 1)^4 - (8*((x^(1/2) - 1)^(1/ 2) - 1i)^2)/((x^(1/2) + 1)^(1/2) - 1)^2 - (56*((x^(1/2) - 1)^(1/2) - 1i)^6 )/((x^(1/2) + 1)^(1/2) - 1)^6 + (70*((x^(1/2) - 1)^(1/2) - 1i)^8)/((x^(1/2 ) + 1)^(1/2) - 1)^8 - (56*((x^(1/2) - 1)^(1/2) - 1i)^10)/((x^(1/2) + 1)^(1 /2) - 1)^10 + (28*((x^(1/2) - 1)^(1/2) - 1i)^12)/((x^(1/2) + 1)^(1/2) - 1) ^12 - (8*((x^(1/2) - 1)^(1/2) - 1i)^14)/((x^(1/2) + 1)^(1/2) - 1)^14 + ((x ^(1/2) - 1)^(1/2) - 1i)^16/((x^(1/2) + 1)^(1/2) - 1)^16 + 1)