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

Optimal result

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

[Out]

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))

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.54, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6, 6820, 1986, 396, 246, 218, 212, 209} \[ \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{a b^4+4 a b^3 x^4+b^4 x^4+6 a b^2 x^8+4 b^3 x^8+4 a b x^{12}+6 b^2 x^{12}+a x^{16}+4 b x^{16}+x^{20}}} \, dx=-\frac {(a-4 b) \sqrt [4]{a+x^4} \left (b+x^4\right ) \arctan \left (\frac {x}{\sqrt [4]{a+x^4}}\right )}{8 \sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}}-\frac {(a-4 b) \sqrt [4]{a+x^4} \left (b+x^4\right ) \text {arctanh}\left (\frac {x}{\sqrt [4]{a+x^4}}\right )}{8 \sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}}+\frac {x \left (a+x^4\right ) \left (b+x^4\right )}{4 \sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}} \]

[In]

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]

[Out]

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

Rule 6

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

Rule 209

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

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 218

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

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[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 396

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

Rule 1986

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^(r_.))^(p_), x_Symbol] :> Dist[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 6820

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{a b^4+\left (4 a b^3+b^4\right ) 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]{a b^4+\left (4 a b^3+b^4\right ) x^4+\left (6 a b^2+4 b^3\right ) 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]{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 x^{16}+4 b x^{16}+x^{20}}} \, dx \\ & = \int \frac {\left (b+x^4\right )^2}{\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}}} \, dx \\ & = \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}} \, dx \\ & = \frac {\left (\sqrt [4]{a+x^4} \left (b+x^4\right )\right ) \int \frac {b+x^4}{\sqrt [4]{a+x^4}} \, dx}{\sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}} \\ & = \frac {x \left (a+x^4\right ) \left (b+x^4\right )}{4 \sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}}-\frac {\left ((a-4 b) \sqrt [4]{a+x^4} \left (b+x^4\right )\right ) \int \frac {1}{\sqrt [4]{a+x^4}} \, dx}{4 \sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}} \\ & = \frac {x \left (a+x^4\right ) \left (b+x^4\right )}{4 \sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}}-\frac {\left ((a-4 b) \sqrt [4]{a+x^4} \left (b+x^4\right )\right ) \text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{a+x^4}}\right )}{4 \sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}} \\ & = \frac {x \left (a+x^4\right ) \left (b+x^4\right )}{4 \sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}}-\frac {\left ((a-4 b) \sqrt [4]{a+x^4} \left (b+x^4\right )\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{a+x^4}}\right )}{8 \sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}}-\frac {\left ((a-4 b) \sqrt [4]{a+x^4} \left (b+x^4\right )\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{a+x^4}}\right )}{8 \sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}} \\ & = \frac {x \left (a+x^4\right ) \left (b+x^4\right )}{4 \sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}}-\frac {(a-4 b) \sqrt [4]{a+x^4} \left (b+x^4\right ) \arctan \left (\frac {x}{\sqrt [4]{a+x^4}}\right )}{8 \sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}}-\frac {(a-4 b) \sqrt [4]{a+x^4} \left (b+x^4\right ) \text {arctanh}\left (\frac {x}{\sqrt [4]{a+x^4}}\right )}{8 \sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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]

[Out]

((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)*Ar
cTanh[x/(a + x^4)^(1/4)]))/(8*((a + x^4)*(b + x^4)^4)^(1/4))

Maple [F]

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

[In]

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)

[Out]

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)

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 \[ \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{a b^4+4 a b^3 x^4+b^4 x^4+6 a b^2 x^8+4 b^3 x^8+4 a b x^{12}+6 b^2 x^{12}+a x^{16}+4 b x^{16}+x^{20}}} \, dx=\frac {2 \, {\left ({\left (a - 4 \, b\right )} x^{12} + 3 \, {\left (a b - 4 \, b^{2}\right )} x^{8} + 3 \, {\left (a b^{2} - 4 \, b^{3}\right )} x^{4} + a b^{3} - 4 \, b^{4}\right )} \arctan \left (\frac {{\left (x^{20} + {\left (a + 4 \, b\right )} x^{16} + 2 \, {\left (2 \, a b + 3 \, b^{2}\right )} x^{12} + 2 \, {\left (3 \, a b^{2} + 2 \, b^{3}\right )} x^{8} + a b^{4} + {\left (4 \, a b^{3} + b^{4}\right )} x^{4}\right )}^{\frac {1}{4}}}{x^{5} + b x}\right ) - {\left ({\left (a - 4 \, b\right )} x^{12} + 3 \, {\left (a b - 4 \, b^{2}\right )} x^{8} + 3 \, {\left (a b^{2} - 4 \, b^{3}\right )} x^{4} + a b^{3} - 4 \, b^{4}\right )} \log \left (\frac {x^{5} + b x + {\left (x^{20} + {\left (a + 4 \, b\right )} x^{16} + 2 \, {\left (2 \, a b + 3 \, b^{2}\right )} x^{12} + 2 \, {\left (3 \, a b^{2} + 2 \, b^{3}\right )} x^{8} + a b^{4} + {\left (4 \, a b^{3} + b^{4}\right )} x^{4}\right )}^{\frac {1}{4}}}{x^{5} + b x}\right ) + {\left ({\left (a - 4 \, b\right )} x^{12} + 3 \, {\left (a b - 4 \, b^{2}\right )} x^{8} + 3 \, {\left (a b^{2} - 4 \, b^{3}\right )} x^{4} + a b^{3} - 4 \, b^{4}\right )} \log \left (-\frac {x^{5} + b x - {\left (x^{20} + {\left (a + 4 \, b\right )} x^{16} + 2 \, {\left (2 \, a b + 3 \, b^{2}\right )} x^{12} + 2 \, {\left (3 \, a b^{2} + 2 \, b^{3}\right )} x^{8} + a b^{4} + {\left (4 \, a b^{3} + b^{4}\right )} x^{4}\right )}^{\frac {1}{4}}}{x^{5} + b x}\right ) + 4 \, {\left (x^{20} + {\left (a + 4 \, b\right )} x^{16} + 2 \, {\left (2 \, a b + 3 \, b^{2}\right )} x^{12} + 2 \, {\left (3 \, a b^{2} + 2 \, b^{3}\right )} x^{8} + a b^{4} + {\left (4 \, a b^{3} + b^{4}\right )} x^{4}\right )}^{\frac {3}{4}} x}{16 \, {\left (x^{12} + 3 \, b x^{8} + 3 \, b^{2} x^{4} + b^{3}\right )}} \]

[In]

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")

[Out]

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)

Sympy [F]

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

[In]

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)

[Out]

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

Maxima [F]

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

[In]

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")

[Out]

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)

Giac [F]

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

[In]

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")

[Out]

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)

Mupad [F(-1)]

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

[In]

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)

[Out]

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)