Integrand size = 28, antiderivative size = 135 \[ \int \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2} \, dx=-\frac {5}{64} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {5}{96} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {1}{24} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {1}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{7/2}-\frac {5 \text {arccosh}\left (\sqrt {x}\right )}{64} \]
-5/64*arccosh(x^(1/2))-5/96*x^(3/2)*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)-1 /24*x^(5/2)*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)+1/4*x^(7/2)*(-1+x^(1/2))^ (1/2)*(1+x^(1/2))^(1/2)-5/64*x^(1/2)*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)
Time = 7.24 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.73 \[ \int \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2} \, dx=\frac {1}{192} \left (\sqrt {\frac {-1+\sqrt {x}}{1+\sqrt {x}}} \sqrt {x} \left (-15-15 \sqrt {x}-10 x-10 x^{3/2}-8 x^2-8 x^{5/2}+48 x^3+48 x^{7/2}\right )-30 \text {arctanh}\left (\sqrt {\frac {-1+\sqrt {x}}{1+\sqrt {x}}}\right )\right ) \]
(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) + 48*x^3 + 48*x^(7/2)) - 30*ArcTanh[Sqrt[(-1 + Sqrt[x])/(1 + Sqrt[x])]])/192
Time = 0.26 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {812, 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.
| \(\displaystyle \int \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{5/2} \, dx\) |
| \(\Big \downarrow \) 812 |
| \(\displaystyle \frac {1}{4} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{7/2}-\frac {1}{8} \int \frac {x^{5/2}}{\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}dx\) |
| \(\Big \downarrow \) 845 |
| \(\displaystyle \frac {1}{8} \left (-\frac {5}{6} \int \frac {x^{3/2}}{\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}dx-\frac {1}{3} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{5/2}\right )+\frac {1}{4} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{7/2}\) |
| \(\Big \downarrow \) 845 |
| \(\displaystyle \frac {1}{8} \left (-\frac {5}{6} \left (\frac {3}{4} \int \frac {\sqrt {x}}{\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}dx+\frac {1}{2} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}\right )-\frac {1}{3} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{5/2}\right )+\frac {1}{4} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{7/2}\) |
| \(\Big \downarrow \) 845 |
| \(\displaystyle \frac {1}{8} \left (-\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}}dx+\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}\right )+\frac {1}{2} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}\right )-\frac {1}{3} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{5/2}\right )+\frac {1}{4} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{7/2}\) |
| \(\Big \downarrow \) 852 |
| \(\displaystyle \frac {1}{8} \left (-\frac {5}{6} \left (\frac {3}{4} \left (\int \frac {1}{\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}d\sqrt {x}+\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}\right )+\frac {1}{2} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}\right )-\frac {1}{3} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{5/2}\right )+\frac {1}{4} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{7/2}\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle \frac {1}{8} \left (-\frac {5}{6} \left (\frac {3}{4} \left (\text {arccosh}\left (\sqrt {x}\right )+\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}\right )+\frac {1}{2} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}\right )-\frac {1}{3} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{5/2}\right )+\frac {1}{4} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{7/2}\) |
(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(7/2))/4 + (-1/3*(Sqrt[-1 + Sqrt[x ]]*Sqrt[1 + Sqrt[x]]*x^(5/2)) - (5*((Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]* x^(3/2))/2 + (3*(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x] + ArcCosh[Sq rt[x]]))/4))/6)/8
3.11.2.3.1 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*x)^(m + 1)*(a1 + b1*x^n)^p*((a2 + b2*x^n) ^p/(c*(m + 2*n*p + 1))), x] + Simp[2*a1*a2*n*(p/(m + 2*n*p + 1)) Int[(c*x )^m*(a1 + b1*x^n)^(p - 1)*(a2 + b2*x^n)^(p - 1), x], x] /; FreeQ[{a1, b1, a 2, b2, c, m}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && GtQ[p, 0] && N eQ[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] :> 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 = 4.58 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.56
| method | result | size |
| derivativedivides | \(-\frac {\sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}\, \left (-48 \sqrt {-1+x}\, x^{\frac {7}{2}}+8 x^{\frac {5}{2}} \sqrt {-1+x}+10 x^{\frac {3}{2}} \sqrt {-1+x}+15 \sqrt {x}\, \sqrt {-1+x}+15 \ln \left (\sqrt {x}+\sqrt {-1+x}\right )\right )}{192 \sqrt {-1+x}}\) | \(75\) |
| default | \(-\frac {\sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}\, \left (-48 \sqrt {-1+x}\, x^{\frac {7}{2}}+8 x^{\frac {5}{2}} \sqrt {-1+x}+10 x^{\frac {3}{2}} \sqrt {-1+x}+15 \sqrt {x}\, \sqrt {-1+x}+15 \ln \left (\sqrt {x}+\sqrt {-1+x}\right )\right )}{192 \sqrt {-1+x}}\) | \(75\) |
-1/192*(x^(1/2)-1)^(1/2)*(x^(1/2)+1)^(1/2)*(-48*(-1+x)^(1/2)*x^(7/2)+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 = 0.31 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.46 \[ \int \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2} \, dx=\frac {1}{192} \, {\left (48 \, x^{3} - 8 \, x^{2} - 10 \, x - 15\right )} \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \frac {5}{128} \, \log \left (2 \, \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - 2 \, x + 1\right ) \]
1/192*(48*x^3 - 8*x^2 - 10*x - 15)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) + 5/128*log(2*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - 2*x + 1)
\[ \int \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2} \, dx=\int x^{\frac {5}{2}} \sqrt {\sqrt {x} - 1} \sqrt {\sqrt {x} + 1}\, dx \]
Time = 0.19 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.42 \[ \int \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2} \, dx=\frac {1}{4} \, {\left (x - 1\right )}^{\frac {3}{2}} x^{\frac {5}{2}} + \frac {5}{24} \, {\left (x - 1\right )}^{\frac {3}{2}} x^{\frac {3}{2}} + \frac {5}{32} \, {\left (x - 1\right )}^{\frac {3}{2}} \sqrt {x} + \frac {5}{64} \, \sqrt {x - 1} \sqrt {x} - \frac {5}{64} \, \log \left (2 \, \sqrt {x - 1} + 2 \, \sqrt {x}\right ) \]
1/4*(x - 1)^(3/2)*x^(5/2) + 5/24*(x - 1)^(3/2)*x^(3/2) + 5/32*(x - 1)^(3/2 )*sqrt(x) + 5/64*sqrt(x - 1)*sqrt(x) - 5/64*log(2*sqrt(x - 1) + 2*sqrt(x))
Time = 0.29 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.20 \[ \int \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2} \, dx=\frac {1}{6720} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, {\left (7 \, \sqrt {x} - 50\right )} {\left (\sqrt {x} + 1\right )} + 1219\right )} {\left (\sqrt {x} + 1\right )} - 12463\right )} {\left (\sqrt {x} + 1\right )} + 64233\right )} {\left (\sqrt {x} + 1\right )} - 53963\right )} {\left (\sqrt {x} + 1\right )} + 59465\right )} {\left (\sqrt {x} + 1\right )} - 23205\right )} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \frac {1}{840} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, \sqrt {x} - 37\right )} {\left (\sqrt {x} + 1\right )} + 661\right )} {\left (\sqrt {x} + 1\right )} - 4551\right )} {\left (\sqrt {x} + 1\right )} + 4781\right )} {\left (\sqrt {x} + 1\right )} - 6335\right )} {\left (\sqrt {x} + 1\right )} + 2835\right )} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \frac {5}{32} \, \log \left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right ) \]
1/6720*((2*((4*(5*(6*(7*sqrt(x) - 50)*(sqrt(x) + 1) + 1219)*(sqrt(x) + 1) - 12463)*(sqrt(x) + 1) + 64233)*(sqrt(x) + 1) - 53963)*(sqrt(x) + 1) + 594 65)*(sqrt(x) + 1) - 23205)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) + 1/840*((2 *((4*(5*(6*sqrt(x) - 37)*(sqrt(x) + 1) + 661)*(sqrt(x) + 1) - 4551)*(sqrt( x) + 1) + 4781)*(sqrt(x) + 1) - 6335)*(sqrt(x) + 1) + 2835)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) + 5/32*log(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))
Time = 75.08 (sec) , antiderivative size = 831, normalized size of antiderivative = 6.16 \[ \int \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2} \, dx=\text {Too large to display} \]
((1723*((x^(1/2) - 1)^(1/2) - 1i)^5)/(48*((x^(1/2) + 1)^(1/2) - 1)^5) - (2 35*((x^(1/2) - 1)^(1/2) - 1i)^3)/(48*((x^(1/2) + 1)^(1/2) - 1)^3) + (72283 *((x^(1/2) - 1)^(1/2) - 1i)^7)/(16*((x^(1/2) + 1)^(1/2) - 1)^7) + (848801* ((x^(1/2) - 1)^(1/2) - 1i)^9)/(16*((x^(1/2) + 1)^(1/2) - 1)^9) + (4181067* ((x^(1/2) - 1)^(1/2) - 1i)^11)/(16*((x^(1/2) + 1)^(1/2) - 1)^11) + (109941 81*((x^(1/2) - 1)^(1/2) - 1i)^13)/(16*((x^(1/2) + 1)^(1/2) - 1)^13) + (174 57599*((x^(1/2) - 1)^(1/2) - 1i)^15)/(16*((x^(1/2) + 1)^(1/2) - 1)^15) + ( 17457599*((x^(1/2) - 1)^(1/2) - 1i)^17)/(16*((x^(1/2) + 1)^(1/2) - 1)^17) + (10994181*((x^(1/2) - 1)^(1/2) - 1i)^19)/(16*((x^(1/2) + 1)^(1/2) - 1)^1 9) + (4181067*((x^(1/2) - 1)^(1/2) - 1i)^21)/(16*((x^(1/2) + 1)^(1/2) - 1) ^21) + (848801*((x^(1/2) - 1)^(1/2) - 1i)^23)/(16*((x^(1/2) + 1)^(1/2) - 1 )^23) + (72283*((x^(1/2) - 1)^(1/2) - 1i)^25)/(16*((x^(1/2) + 1)^(1/2) - 1 )^25) + (1723*((x^(1/2) - 1)^(1/2) - 1i)^27)/(48*((x^(1/2) + 1)^(1/2) - 1) ^27) - (235*((x^(1/2) - 1)^(1/2) - 1i)^29)/(48*((x^(1/2) + 1)^(1/2) - 1)^2 9) + (5*((x^(1/2) - 1)^(1/2) - 1i)^31)/(16*((x^(1/2) + 1)^(1/2) - 1)^31) + (5*((x^(1/2) - 1)^(1/2) - 1i))/(16*((x^(1/2) + 1)^(1/2) - 1)))/((120*((x^ (1/2) - 1)^(1/2) - 1i)^4)/((x^(1/2) + 1)^(1/2) - 1)^4 - (16*((x^(1/2) - 1) ^(1/2) - 1i)^2)/((x^(1/2) + 1)^(1/2) - 1)^2 - (560*((x^(1/2) - 1)^(1/2) - 1i)^6)/((x^(1/2) + 1)^(1/2) - 1)^6 + (1820*((x^(1/2) - 1)^(1/2) - 1i)^8)/( (x^(1/2) + 1)^(1/2) - 1)^8 - (4368*((x^(1/2) - 1)^(1/2) - 1i)^10)/((x^(...