3.28.44 \(\int \frac {(b+x^4)^2}{\sqrt [4]{a b^4+4 a b^3 x^4+b^4 x^4+6 a b^2 x^8+4 b^3 x^8+4 a b x^{12}+6 b^2 x^{12}+a x^{16}+4 b x^{16}+x^{20}}} \, dx\) [2744]

3.28.44.1 Optimal result

Integrand size = 80, antiderivative size = 255 \[ \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{a b^4+4 a b^3 x^4+b^4 x^4+6 a b^2 x^8+4 b^3 x^8+4 a b x^{12}+6 b^2 x^{12}+a x^{16}+4 b x^{16}+x^{20}}} \, dx=\frac {x \left (a b^4+\left (4 a b^3+b^4\right ) x^4+\left (6 a b^2+4 b^3\right ) x^8+\left (4 a b+6 b^2\right ) x^{12}+(a+4 b) x^{16}+x^{20}\right )^{3/4}}{4 \left (b+x^4\right )^3}+\frac {1}{8} (a-4 b) \arctan \left (\frac {\sqrt [4]{a b^4+\left (4 a b^3+b^4\right ) x^4+\left (6 a b^2+4 b^3\right ) x^8+\left (4 a b+6 b^2\right ) x^{12}+(a+4 b) x^{16}+x^{20}}}{x \left (b+x^4\right )}\right )+\frac {1}{8} (-a+4 b) \text {arctanh}\left (\frac {\sqrt [4]{a b^4+\left (4 a b^3+b^4\right ) x^4+\left (6 a b^2+4 b^3\right ) x^8+\left (4 a b+6 b^2\right ) x^{12}+(a+4 b) x^{16}+x^{20}}}{x \left (b+x^4\right )}\right ) \]

output

1/4*x*(a*b^4+(4*a*b^3+b^4)*x^4+(6*a*b^2+4*b^3)*x^8+(4*a*b+6*b^2)*x^12+(a+4 
*b)*x^16+x^20)^(3/4)/(x^4+b)^3+1/8*(a-4*b)*arctan((a*b^4+(4*a*b^3+b^4)*x^4 
+(6*a*b^2+4*b^3)*x^8+(4*a*b+6*b^2)*x^12+(a+4*b)*x^16+x^20)^(1/4)/x/(x^4+b) 
)+1/8*(-a+4*b)*arctanh((a*b^4+(4*a*b^3+b^4)*x^4+(6*a*b^2+4*b^3)*x^8+(4*a*b 
+6*b^2)*x^12+(a+4*b)*x^16+x^20)^(1/4)/x/(x^4+b))
 

3.28.44.2 Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.36 \[ \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{a b^4+4 a b^3 x^4+b^4 x^4+6 a b^2 x^8+4 b^3 x^8+4 a b x^{12}+6 b^2 x^{12}+a x^{16}+4 b x^{16}+x^{20}}} \, dx=\frac {\left (b+x^4\right ) \left (2 x \left (a+x^4\right )-(a-4 b) \sqrt [4]{a+x^4} \arctan \left (\frac {x}{\sqrt [4]{a+x^4}}\right )-(a-4 b) \sqrt [4]{a+x^4} \text {arctanh}\left (\frac {x}{\sqrt [4]{a+x^4}}\right )\right )}{8 \sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}} \]

input

Integrate[(b + x^4)^2/(a*b^4 + 4*a*b^3*x^4 + b^4*x^4 + 6*a*b^2*x^8 + 4*b^3 
*x^8 + 4*a*b*x^12 + 6*b^2*x^12 + a*x^16 + 4*b*x^16 + x^20)^(1/4),x]
 

output

((b + x^4)*(2*x*(a + x^4) - (a - 4*b)*(a + x^4)^(1/4)*ArcTan[x/(a + x^4)^( 
1/4)] - (a - 4*b)*(a + x^4)^(1/4)*ArcTanh[x/(a + x^4)^(1/4)]))/(8*((a + x^ 
4)*(b + x^4)^4)^(1/4))
 

3.28.44.3 Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.35, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {6, 6, 6, 6, 7239, 2058, 913, 770, 756, 216, 219}

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.

\(\displaystyle \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{a b^4+4 a b^3 x^4+6 a b^2 x^8+4 a b x^{12}+a x^{16}+b^4 x^4+4 b^3 x^8+6 b^2 x^{12}+4 b x^{16}+x^{20}}} \, dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{a b^4+6 a b^2 x^8+x^4 \left (4 a b^3+b^4\right )+4 a b x^{12}+a x^{16}+4 b^3 x^8+6 b^2 x^{12}+4 b x^{16}+x^{20}}}dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{a b^4+x^4 \left (4 a b^3+b^4\right )+x^8 \left (6 a b^2+4 b^3\right )+4 a b x^{12}+a x^{16}+6 b^2 x^{12}+4 b x^{16}+x^{20}}}dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{a b^4+x^{12} \left (4 a b+6 b^2\right )+x^4 \left (4 a b^3+b^4\right )+x^8 \left (6 a b^2+4 b^3\right )+a x^{16}+4 b x^{16}+x^{20}}}dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{a b^4+x^{12} \left (4 a b+6 b^2\right )+x^4 \left (4 a b^3+b^4\right )+x^8 \left (6 a b^2+4 b^3\right )+x^{16} (a+4 b)+x^{20}}}dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}}dx\)

\(\Big \downarrow \) 2058

\(\displaystyle \frac {\sqrt [4]{a+x^4} \left (b+x^4\right ) \int \frac {x^4+b}{\sqrt [4]{x^4+a}}dx}{\sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}}\)

\(\Big \downarrow \) 913

\(\displaystyle \frac {\sqrt [4]{a+x^4} \left (b+x^4\right ) \left (\frac {1}{4} x \left (a+x^4\right )^{3/4}-\frac {1}{4} (a-4 b) \int \frac {1}{\sqrt [4]{x^4+a}}dx\right )}{\sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}}\)

\(\Big \downarrow \) 770

\(\displaystyle \frac {\sqrt [4]{a+x^4} \left (b+x^4\right ) \left (\frac {1}{4} x \left (a+x^4\right )^{3/4}-\frac {1}{4} (a-4 b) \int \frac {1}{1-\frac {x^4}{x^4+a}}d\frac {x}{\sqrt [4]{x^4+a}}\right )}{\sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {\sqrt [4]{a+x^4} \left (b+x^4\right ) \left (\frac {1}{4} x \left (a+x^4\right )^{3/4}-\frac {1}{4} (a-4 b) \left (\frac {1}{2} \int \frac {1}{1-\frac {x^2}{\sqrt {x^4+a}}}d\frac {x}{\sqrt [4]{x^4+a}}+\frac {1}{2} \int \frac {1}{\frac {x^2}{\sqrt {x^4+a}}+1}d\frac {x}{\sqrt [4]{x^4+a}}\right )\right )}{\sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\sqrt [4]{a+x^4} \left (b+x^4\right ) \left (\frac {1}{4} x \left (a+x^4\right )^{3/4}-\frac {1}{4} (a-4 b) \left (\frac {1}{2} \int \frac {1}{1-\frac {x^2}{\sqrt {x^4+a}}}d\frac {x}{\sqrt [4]{x^4+a}}+\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{a+x^4}}\right )\right )\right )}{\sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt [4]{a+x^4} \left (b+x^4\right ) \left (\frac {1}{4} x \left (a+x^4\right )^{3/4}-\frac {1}{4} (a-4 b) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{a+x^4}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{a+x^4}}\right )\right )\right )}{\sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}}\)

input

Int[(b + x^4)^2/(a*b^4 + 4*a*b^3*x^4 + b^4*x^4 + 6*a*b^2*x^8 + 4*b^3*x^8 + 
 4*a*b*x^12 + 6*b^2*x^12 + a*x^16 + 4*b*x^16 + x^20)^(1/4),x]
 

output

((a + x^4)^(1/4)*(b + x^4)*((x*(a + x^4)^(3/4))/4 - ((a - 4*b)*(ArcTan[x/( 
a + x^4)^(1/4)]/2 + ArcTanh[x/(a + x^4)^(1/4)]/2))/4))/((a + x^4)*(b + x^4 
)^4)^(1/4)
 

3.28.44.3.1 Defintions of rubi rules used

rule 6

Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v 
+ (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] &&  !FreeQ[Fx, x]
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ[a 
/b, 0]
 

rule 770

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n)   Subst[In 
t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, 
 b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1 
/n]
 

rule 913

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si 
mp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(p + 1) + 1))), x] - Simp[(a*d - b*c*(n*( 
p + 1) + 1))/(b*(n*(p + 1) + 1))   Int[(a + b*x^n)^p, x], x] /; FreeQ[{a, b 
, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]
 

rule 2058

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ 
(r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + 
 b*x^n)^(p*q)*(c + d*x^n)^(p*r))]   Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* 
r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

3.28.44.4 Maple [F]

\[\int \frac {\left (x^{4}+b \right )^{2}}{\left (x^{20}+a \,x^{16}+4 x^{16} b +4 a b \,x^{12}+6 b^{2} x^{12}+6 a \,b^{2} x^{8}+4 b^{3} x^{8}+4 b^{3} x^{4} a +b^{4} x^{4}+a \,b^{4}\right )^{\frac {1}{4}}}d x\]

input

int((x^4+b)^2/(x^20+a*x^16+4*b*x^16+4*a*b*x^12+6*b^2*x^12+6*a*b^2*x^8+4*b^ 
3*x^8+4*a*b^3*x^4+b^4*x^4+a*b^4)^(1/4),x)
 

output

int((x^4+b)^2/(x^20+a*x^16+4*b*x^16+4*a*b*x^12+6*b^2*x^12+6*a*b^2*x^8+4*b^ 
3*x^8+4*a*b^3*x^4+b^4*x^4+a*b^4)^(1/4),x)
 

3.28.44.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 498 vs. \(2 (247) = 494\).

Time = 0.26 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.95 \[ \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{a b^4+4 a b^3 x^4+b^4 x^4+6 a b^2 x^8+4 b^3 x^8+4 a b x^{12}+6 b^2 x^{12}+a x^{16}+4 b x^{16}+x^{20}}} \, dx=\frac {2 \, {\left ({\left (a - 4 \, b\right )} x^{12} + 3 \, {\left (a b - 4 \, b^{2}\right )} x^{8} + 3 \, {\left (a b^{2} - 4 \, b^{3}\right )} x^{4} + a b^{3} - 4 \, b^{4}\right )} \arctan \left (\frac {{\left (x^{20} + {\left (a + 4 \, b\right )} x^{16} + 2 \, {\left (2 \, a b + 3 \, b^{2}\right )} x^{12} + 2 \, {\left (3 \, a b^{2} + 2 \, b^{3}\right )} x^{8} + a b^{4} + {\left (4 \, a b^{3} + b^{4}\right )} x^{4}\right )}^{\frac {1}{4}}}{x^{5} + b x}\right ) - {\left ({\left (a - 4 \, b\right )} x^{12} + 3 \, {\left (a b - 4 \, b^{2}\right )} x^{8} + 3 \, {\left (a b^{2} - 4 \, b^{3}\right )} x^{4} + a b^{3} - 4 \, b^{4}\right )} \log \left (\frac {x^{5} + b x + {\left (x^{20} + {\left (a + 4 \, b\right )} x^{16} + 2 \, {\left (2 \, a b + 3 \, b^{2}\right )} x^{12} + 2 \, {\left (3 \, a b^{2} + 2 \, b^{3}\right )} x^{8} + a b^{4} + {\left (4 \, a b^{3} + b^{4}\right )} x^{4}\right )}^{\frac {1}{4}}}{x^{5} + b x}\right ) + {\left ({\left (a - 4 \, b\right )} x^{12} + 3 \, {\left (a b - 4 \, b^{2}\right )} x^{8} + 3 \, {\left (a b^{2} - 4 \, b^{3}\right )} x^{4} + a b^{3} - 4 \, b^{4}\right )} \log \left (-\frac {x^{5} + b x - {\left (x^{20} + {\left (a + 4 \, b\right )} x^{16} + 2 \, {\left (2 \, a b + 3 \, b^{2}\right )} x^{12} + 2 \, {\left (3 \, a b^{2} + 2 \, b^{3}\right )} x^{8} + a b^{4} + {\left (4 \, a b^{3} + b^{4}\right )} x^{4}\right )}^{\frac {1}{4}}}{x^{5} + b x}\right ) + 4 \, {\left (x^{20} + {\left (a + 4 \, b\right )} x^{16} + 2 \, {\left (2 \, a b + 3 \, b^{2}\right )} x^{12} + 2 \, {\left (3 \, a b^{2} + 2 \, b^{3}\right )} x^{8} + a b^{4} + {\left (4 \, a b^{3} + b^{4}\right )} x^{4}\right )}^{\frac {3}{4}} x}{16 \, {\left (x^{12} + 3 \, b x^{8} + 3 \, b^{2} x^{4} + b^{3}\right )}} \]

input

integrate((x^4+b)^2/(x^20+a*x^16+4*b*x^16+4*a*b*x^12+6*b^2*x^12+6*a*b^2*x^ 
8+4*b^3*x^8+4*a*b^3*x^4+b^4*x^4+a*b^4)^(1/4),x, algorithm="fricas")
 

output

1/16*(2*((a - 4*b)*x^12 + 3*(a*b - 4*b^2)*x^8 + 3*(a*b^2 - 4*b^3)*x^4 + a* 
b^3 - 4*b^4)*arctan((x^20 + (a + 4*b)*x^16 + 2*(2*a*b + 3*b^2)*x^12 + 2*(3 
*a*b^2 + 2*b^3)*x^8 + a*b^4 + (4*a*b^3 + b^4)*x^4)^(1/4)/(x^5 + b*x)) - (( 
a - 4*b)*x^12 + 3*(a*b - 4*b^2)*x^8 + 3*(a*b^2 - 4*b^3)*x^4 + a*b^3 - 4*b^ 
4)*log((x^5 + b*x + (x^20 + (a + 4*b)*x^16 + 2*(2*a*b + 3*b^2)*x^12 + 2*(3 
*a*b^2 + 2*b^3)*x^8 + a*b^4 + (4*a*b^3 + b^4)*x^4)^(1/4))/(x^5 + b*x)) + ( 
(a - 4*b)*x^12 + 3*(a*b - 4*b^2)*x^8 + 3*(a*b^2 - 4*b^3)*x^4 + a*b^3 - 4*b 
^4)*log(-(x^5 + b*x - (x^20 + (a + 4*b)*x^16 + 2*(2*a*b + 3*b^2)*x^12 + 2* 
(3*a*b^2 + 2*b^3)*x^8 + a*b^4 + (4*a*b^3 + b^4)*x^4)^(1/4))/(x^5 + b*x)) + 
 4*(x^20 + (a + 4*b)*x^16 + 2*(2*a*b + 3*b^2)*x^12 + 2*(3*a*b^2 + 2*b^3)*x 
^8 + a*b^4 + (4*a*b^3 + b^4)*x^4)^(3/4)*x)/(x^12 + 3*b*x^8 + 3*b^2*x^4 + b 
^3)
 

3.28.44.6 Sympy [F]

\[ \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{a b^4+4 a b^3 x^4+b^4 x^4+6 a b^2 x^8+4 b^3 x^8+4 a b x^{12}+6 b^2 x^{12}+a x^{16}+4 b x^{16}+x^{20}}} \, dx=\int \frac {\left (b + x^{4}\right )^{2}}{\sqrt [4]{\left (a + x^{4}\right ) \left (b + x^{4}\right )^{4}}}\, dx \]

input

integrate((x**4+b)**2/(x**20+a*x**16+4*b*x**16+4*a*b*x**12+6*b**2*x**12+6* 
a*b**2*x**8+4*b**3*x**8+4*a*b**3*x**4+b**4*x**4+a*b**4)**(1/4),x)
 

output

Integral((b + x**4)**2/((a + x**4)*(b + x**4)**4)**(1/4), x)
 

3.28.44.7 Maxima [F]

\[ \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{a b^4+4 a b^3 x^4+b^4 x^4+6 a b^2 x^8+4 b^3 x^8+4 a b x^{12}+6 b^2 x^{12}+a x^{16}+4 b x^{16}+x^{20}}} \, dx=\int { \frac {{\left (x^{4} + b\right )}^{2}}{{\left (x^{20} + a x^{16} + 4 \, b x^{16} + 4 \, a b x^{12} + 6 \, b^{2} x^{12} + 6 \, a b^{2} x^{8} + 4 \, b^{3} x^{8} + 4 \, a b^{3} x^{4} + b^{4} x^{4} + a b^{4}\right )}^{\frac {1}{4}}} \,d x } \]

input

integrate((x^4+b)^2/(x^20+a*x^16+4*b*x^16+4*a*b*x^12+6*b^2*x^12+6*a*b^2*x^ 
8+4*b^3*x^8+4*a*b^3*x^4+b^4*x^4+a*b^4)^(1/4),x, algorithm="maxima")
 

output

integrate((x^4 + b)^2/(x^20 + a*x^16 + 4*b*x^16 + 4*a*b*x^12 + 6*b^2*x^12 
+ 6*a*b^2*x^8 + 4*b^3*x^8 + 4*a*b^3*x^4 + b^4*x^4 + a*b^4)^(1/4), x)
 

3.28.44.8 Giac [F]

\[ \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{a b^4+4 a b^3 x^4+b^4 x^4+6 a b^2 x^8+4 b^3 x^8+4 a b x^{12}+6 b^2 x^{12}+a x^{16}+4 b x^{16}+x^{20}}} \, dx=\int { \frac {{\left (x^{4} + b\right )}^{2}}{{\left (x^{20} + a x^{16} + 4 \, b x^{16} + 4 \, a b x^{12} + 6 \, b^{2} x^{12} + 6 \, a b^{2} x^{8} + 4 \, b^{3} x^{8} + 4 \, a b^{3} x^{4} + b^{4} x^{4} + a b^{4}\right )}^{\frac {1}{4}}} \,d x } \]

input

integrate((x^4+b)^2/(x^20+a*x^16+4*b*x^16+4*a*b*x^12+6*b^2*x^12+6*a*b^2*x^ 
8+4*b^3*x^8+4*a*b^3*x^4+b^4*x^4+a*b^4)^(1/4),x, algorithm="giac")
 

output

integrate((x^4 + b)^2/(x^20 + a*x^16 + 4*b*x^16 + 4*a*b*x^12 + 6*b^2*x^12 
+ 6*a*b^2*x^8 + 4*b^3*x^8 + 4*a*b^3*x^4 + b^4*x^4 + a*b^4)^(1/4), x)
 

3.28.44.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{a b^4+4 a b^3 x^4+b^4 x^4+6 a b^2 x^8+4 b^3 x^8+4 a b x^{12}+6 b^2 x^{12}+a x^{16}+4 b x^{16}+x^{20}}} \, dx=\int \frac {{\left (x^4+b\right )}^2}{{\left (b^4\,x^4+a\,b^4+4\,b^3\,x^8+4\,a\,b^3\,x^4+6\,b^2\,x^{12}+6\,a\,b^2\,x^8+4\,b\,x^{16}+4\,a\,b\,x^{12}+x^{20}+a\,x^{16}\right )}^{1/4}} \,d x \]

input

int((b + x^4)^2/(a*b^4 + a*x^16 + 4*b*x^16 + x^20 + b^4*x^4 + 4*b^3*x^8 + 
6*b^2*x^12 + 4*a*b^3*x^4 + 6*a*b^2*x^8 + 4*a*b*x^12)^(1/4),x)
 

output

int((b + x^4)^2/(a*b^4 + a*x^16 + 4*b*x^16 + x^20 + b^4*x^4 + 4*b^3*x^8 + 
6*b^2*x^12 + 4*a*b^3*x^4 + 6*a*b^2*x^8 + 4*a*b*x^12)^(1/4), x)