\(\int \frac {x^{3/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\) [244]

Optimal result

Integrand size = 28, antiderivative size = 73 \[ \int \frac {x^{3/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx=\frac {3}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}+\frac {1}{2} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}+\frac {3 \text {arccosh}\left (\sqrt {x}\right )}{4} \] Output:

3/4*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)*x^(1/2)+1/2*(-1+x^(1/2))^(1/2)*(1 
+x^(1/2))^(1/2)*x^(3/2)+3/4*arccosh(x^(1/2))
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(406\) vs. \(2(73)=146\).

Time = 1.52 (sec) , antiderivative size = 406, normalized size of antiderivative = 5.56 \[ \int \frac {x^{3/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx=\frac {-4 \sqrt {1+\sqrt {x}} \left (-29568+50496 \sqrt {x}+98112 x+21840 x^{3/2}-2264 x^2-3368 x^{5/2}-4752 x^3-1136 x^{7/2}\right )-4 \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \left (51216+120600 \sqrt {x}+56904 x-4016 x^{3/2}-6344 x^2-6467 x^{5/2}-3120 x^3-194 x^{7/2}\right )+\sqrt {3} \left (-4 \sqrt {-1+\sqrt {x}} \left (-29568-84416 \sqrt {x}-64000 x-7152 x^{3/2}+5624 x^2+5144 x^{5/2}+3408 x^3+656 x^{7/2}\right )-4 \left (17072-20632 \sqrt {x}-73312 x-36244 x^{3/2}-510 x^2+2452 x^{5/2}+3640 x^3+1800 x^{7/2}+112 x^4\right )\right )}{-12416+13312 \sqrt {x}+49408 x+24960 x^{3/2}+1552 x^2+\sqrt {3} \sqrt {1+\sqrt {x}} \left (7168-11264 \sqrt {x}-22016 x-5248 x^{3/2}\right )+\sqrt {-1+\sqrt {x}} \left (21504+60416 \sqrt {x}+47104 x+9088 x^{3/2}+\sqrt {3} \sqrt {1+\sqrt {x}} \left (-12416-28672 \sqrt {x}-14400 x-896 x^{3/2}\right )\right )}-3 \text {arctanh}\left (\frac {-1+\sqrt {-1+\sqrt {x}}}{\sqrt {3}-\sqrt {1+\sqrt {x}}}\right ) \] Input:

Integrate[x^(3/2)/(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]),x]
 

Output:

(-4*Sqrt[1 + Sqrt[x]]*(-29568 + 50496*Sqrt[x] + 98112*x + 21840*x^(3/2) - 
2264*x^2 - 3368*x^(5/2) - 4752*x^3 - 1136*x^(7/2)) - 4*Sqrt[-1 + Sqrt[x]]* 
Sqrt[1 + Sqrt[x]]*(51216 + 120600*Sqrt[x] + 56904*x - 4016*x^(3/2) - 6344* 
x^2 - 6467*x^(5/2) - 3120*x^3 - 194*x^(7/2)) + Sqrt[3]*(-4*Sqrt[-1 + Sqrt[ 
x]]*(-29568 - 84416*Sqrt[x] - 64000*x - 7152*x^(3/2) + 5624*x^2 + 5144*x^( 
5/2) + 3408*x^3 + 656*x^(7/2)) - 4*(17072 - 20632*Sqrt[x] - 73312*x - 3624 
4*x^(3/2) - 510*x^2 + 2452*x^(5/2) + 3640*x^3 + 1800*x^(7/2) + 112*x^4)))/ 
(-12416 + 13312*Sqrt[x] + 49408*x + 24960*x^(3/2) + 1552*x^2 + Sqrt[3]*Sqr 
t[1 + Sqrt[x]]*(7168 - 11264*Sqrt[x] - 22016*x - 5248*x^(3/2)) + Sqrt[-1 + 
 Sqrt[x]]*(21504 + 60416*Sqrt[x] + 47104*x + 9088*x^(3/2) + Sqrt[3]*Sqrt[1 
 + Sqrt[x]]*(-12416 - 28672*Sqrt[x] - 14400*x - 896*x^(3/2)))) - 3*ArcTanh 
[(-1 + Sqrt[-1 + Sqrt[x]])/(Sqrt[3] - Sqrt[1 + Sqrt[x]])]
 

Rubi [A] (verified)

Time = 0.32 (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

\(\displaystyle \frac {3}{4} \left (\text {arccosh}\left (\sqrt {x}\right )+\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}\right )+\frac {1}{2} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}\)

Input:

Int[x^(3/2)/(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]),x]
 

Output:

(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
 

Defintions of rubi rules used

rule 43

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]
 

rule 845

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]
 

rule 852

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]
 

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.75

method	result	size
derivativedivides	\(\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\, \left (2 x^{\frac {3}{2}} \sqrt {x -1}+3 \sqrt {x}\, \sqrt {x -1}+3 \ln \left (\sqrt {x}+\sqrt {x -1}\right )\right )}{4 \sqrt {x -1}}\)	\(55\)
default	\(\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\, \left (2 x^{\frac {3}{2}} \sqrt {x -1}+3 \sqrt {x}\, \sqrt {x -1}+3 \ln \left (\sqrt {x}+\sqrt {x -1}\right )\right )}{4 \sqrt {x -1}}\)	\(55\)

Input:

int(x^(3/2)/(-1+x^(1/2))^(1/2)/(1+x^(1/2))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/4*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)*(2*x^(3/2)*(x-1)^(1/2)+3*x^(1/2)* 
(x-1)^(1/2)+3*ln(x^(1/2)+(x-1)^(1/2)))/(x-1)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.71 \[ \int \frac {x^{3/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx=\frac {1}{4} \, {\left (2 \, x + 3\right )} \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - \frac {3}{8} \, \log \left (2 \, \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - 2 \, x + 1\right ) \] Input:

integrate(x^(3/2)/(-1+x^(1/2))^(1/2)/(1+x^(1/2))^(1/2),x, algorithm="frica 
s")
 

Output:

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)
 

Sympy [F]

\[ \int \frac {x^{3/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx=\int \frac {x^{\frac {3}{2}}}{\sqrt {\sqrt {x} - 1} \sqrt {\sqrt {x} + 1}}\, dx \] Input:

integrate(x**(3/2)/(-1+x**(1/2))**(1/2)/(1+x**(1/2))**(1/2),x)
 

Output:

Integral(x**(3/2)/(sqrt(sqrt(x) - 1)*sqrt(sqrt(x) + 1)), x)
 

Maxima [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.51 \[ \int \frac {x^{3/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx=\frac {1}{2} \, \sqrt {x - 1} x^{\frac {3}{2}} + \frac {3}{4} \, \sqrt {x - 1} \sqrt {x} + \frac {3}{4} \, \log \left (2 \, \sqrt {x - 1} + 2 \, \sqrt {x}\right ) \] Input:

integrate(x^(3/2)/(-1+x^(1/2))^(1/2)/(1+x^(1/2))^(1/2),x, algorithm="maxim 
a")
 

Output:

1/2*sqrt(x - 1)*x^(3/2) + 3/4*sqrt(x - 1)*sqrt(x) + 3/4*log(2*sqrt(x - 1) 
+ 2*sqrt(x))
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.81 \[ \int \frac {x^{3/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx=\frac {1}{4} \, {\left ({\left (2 \, {\left (\sqrt {x} + 1\right )} {\left (\sqrt {x} - 2\right )} + 9\right )} {\left (\sqrt {x} + 1\right )} - 5\right )} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - \frac {3}{2} \, \log \left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right ) \] Input:

integrate(x^(3/2)/(-1+x^(1/2))^(1/2)/(1+x^(1/2))^(1/2),x, algorithm="giac" 
)
 

Output:

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))
 

Mupad [B] (verification not implemented)

Time = 16.82 (sec) , antiderivative size = 429, normalized size of antiderivative = 5.88 \[ \int \frac {x^{3/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx=3\,\mathrm {atanh}\left (\frac {\sqrt {\sqrt {x}-1}-\mathrm {i}}{\sqrt {\sqrt {x}+1}-1}\right )+\frac {\frac {23\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^3}+\frac {333\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^5}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^5}+\frac {671\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^7}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^7}+\frac {671\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^9}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^9}+\frac {333\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{11}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{11}}+\frac {23\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{13}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{13}}-\frac {3\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{15}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{15}}-\frac {3\,\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}{\sqrt {\sqrt {x}+1}-1}}{1+\frac {28\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^4}-\frac {56\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^6}+\frac {70\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^8}-\frac {56\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{10}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{10}}+\frac {28\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{12}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{12}}-\frac {8\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{14}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{14}}+\frac {{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{16}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{16}}-\frac {8\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^2}} \] Input:

int(x^(3/2)/((x^(1/2) - 1)^(1/2)*(x^(1/2) + 1)^(1/2)),x)
 

Output:

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)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.67 \[ \int \frac {x^{3/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx=\frac {\sqrt {x}\, \sqrt {\sqrt {x}+1}\, \sqrt {\sqrt {x}-1}\, x}{2}+\frac {3 \sqrt {x}\, \sqrt {\sqrt {x}+1}\, \sqrt {\sqrt {x}-1}}{4}+\frac {3 \,\mathrm {log}\left (\frac {\sqrt {\sqrt {x}-1}+\sqrt {\sqrt {x}+1}}{\sqrt {2}}\right )}{2} \] Input:

int(x^(3/2)/(-1+x^(1/2))^(1/2)/(1+x^(1/2))^(1/2),x)
 

Output:

(2*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)*x + 3*sqrt(x)*sqrt(sqrt(x) 
+ 1)*sqrt(sqrt(x) - 1) + 6*log((sqrt(sqrt(x) - 1) + sqrt(sqrt(x) + 1))/sqr 
t(2)))/4