Integrand size = 25, antiderivative size = 99 \[ \int \frac {x^3 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\frac {-b d+2 a e-(2 c d-b e) x^4}{4 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}+\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{2 \left (b^2-4 a c\right )^{3/2}} \] Output:
1/4*(-b*d+2*a*e-(-b*e+2*c*d)*x^4)/(-4*a*c+b^2)/(c*x^8+b*x^4+a)+1/2*(-b*e+2 *c*d)*arctanh((2*c*x^4+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(3/2)
Time = 0.08 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.01 \[ \int \frac {x^3 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\frac {\frac {-b d+2 a e-2 c d x^4+b e x^4}{a+b x^4+c x^8}+\frac {2 (-2 c d+b e) \arctan \left (\frac {b+2 c x^4}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}}{4 \left (b^2-4 a c\right )} \] Input:
Integrate[(x^3*(d + e*x^4))/(a + b*x^4 + c*x^8)^2,x]
Output:
((-(b*d) + 2*a*e - 2*c*d*x^4 + b*e*x^4)/(a + b*x^4 + c*x^8) + (2*(-2*c*d + b*e)*ArcTan[(b + 2*c*x^4)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(4*(b^ 2 - 4*a*c))
Time = 0.26 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1798, 1159, 1083, 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.
\(\displaystyle \int \frac {x^3 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx\) |
\(\Big \downarrow \) 1798 |
\(\displaystyle \frac {1}{4} \int \frac {e x^4+d}{\left (c x^8+b x^4+a\right )^2}dx^4\) |
\(\Big \downarrow \) 1159 |
\(\displaystyle \frac {1}{4} \left (-\frac {(2 c d-b e) \int \frac {1}{c x^8+b x^4+a}dx^4}{b^2-4 a c}-\frac {-2 a e+x^4 (2 c d-b e)+b d}{\left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {1}{4} \left (\frac {2 (2 c d-b e) \int \frac {1}{-x^8+b^2-4 a c}d\left (2 c x^4+b\right )}{b^2-4 a c}-\frac {-2 a e+x^4 (2 c d-b e)+b d}{\left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{4} \left (\frac {2 (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {-2 a e+x^4 (2 c d-b e)+b d}{\left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\right )\) |
Input:
Int[(x^3*(d + e*x^4))/(a + b*x^4 + c*x^8)^2,x]
Output:
(-((b*d - 2*a*e + (2*c*d - b*e)*x^4)/((b^2 - 4*a*c)*(a + b*x^4 + c*x^8))) + (2*(2*c*d - b*e)*ArcTanh[(b + 2*c*x^4)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c) ^(3/2))/4
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])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[((b*d - 2*a*e + (2*c*d - b*e)*x)/((p + 1)*(b^2 - 4*a*c)))*(a + b* x + c*x^2)^(p + 1), x] - Simp[(2*p + 3)*((2*c*d - b*e)/((p + 1)*(b^2 - 4*a* c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] & & LtQ[p, -1] && NeQ[p, -3/2]
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + ( e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Simp[1/n Subst[Int[(d + e*x)^q*(a + b *x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]
Time = 0.12 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.97
method | result | size |
default | \(\frac {b d -2 a e +\left (-e b +2 c d \right ) x^{4}}{4 \left (4 a c -b^{2}\right ) \left (c \,x^{8}+b \,x^{4}+a \right )}+\frac {\left (-e b +2 c d \right ) \arctan \left (\frac {2 c \,x^{4}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \left (4 a c -b^{2}\right )^{\frac {3}{2}}}\) | \(96\) |
risch | \(\frac {-\frac {\left (e b -2 c d \right ) x^{4}}{4 \left (4 a c -b^{2}\right )}-\frac {2 a e -b d}{4 \left (4 a c -b^{2}\right )}}{c \,x^{8}+b \,x^{4}+a}+\frac {\ln \left (\left (-\left (-4 a c +b^{2}\right )^{\frac {3}{2}}+4 a b c -b^{3}\right ) x^{4}+8 a^{2} c -2 b^{2} a \right ) e b}{4 \left (-4 a c +b^{2}\right )^{\frac {3}{2}}}-\frac {\ln \left (\left (-\left (-4 a c +b^{2}\right )^{\frac {3}{2}}+4 a b c -b^{3}\right ) x^{4}+8 a^{2} c -2 b^{2} a \right ) c d}{2 \left (-4 a c +b^{2}\right )^{\frac {3}{2}}}-\frac {\ln \left (\left (-\left (-4 a c +b^{2}\right )^{\frac {3}{2}}-4 a b c +b^{3}\right ) x^{4}-8 a^{2} c +2 b^{2} a \right ) e b}{4 \left (-4 a c +b^{2}\right )^{\frac {3}{2}}}+\frac {\ln \left (\left (-\left (-4 a c +b^{2}\right )^{\frac {3}{2}}-4 a b c +b^{3}\right ) x^{4}-8 a^{2} c +2 b^{2} a \right ) c d}{2 \left (-4 a c +b^{2}\right )^{\frac {3}{2}}}\) | \(282\) |
Input:
int(x^3*(e*x^4+d)/(c*x^8+b*x^4+a)^2,x,method=_RETURNVERBOSE)
Output:
1/4*(b*d-2*a*e+(-b*e+2*c*d)*x^4)/(4*a*c-b^2)/(c*x^8+b*x^4+a)+1/2*(-b*e+2*c *d)/(4*a*c-b^2)^(3/2)*arctan((2*c*x^4+b)/(4*a*c-b^2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (89) = 178\).
Time = 0.10 (sec) , antiderivative size = 478, normalized size of antiderivative = 4.83 \[ \int \frac {x^3 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\left [-\frac {{\left (2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d - {\left (b^{3} - 4 \, a b c\right )} e\right )} x^{4} - {\left ({\left (2 \, c^{2} d - b c e\right )} x^{8} + {\left (2 \, b c d - b^{2} e\right )} x^{4} + 2 \, a c d - a b e\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{8} + 2 \, b c x^{4} + b^{2} - 2 \, a c + {\left (2 \, c x^{4} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{8} + b x^{4} + a}\right ) + {\left (b^{3} - 4 \, a b c\right )} d - 2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} e}{4 \, {\left ({\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{8} + a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{4}\right )}}, -\frac {{\left (2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d - {\left (b^{3} - 4 \, a b c\right )} e\right )} x^{4} - 2 \, {\left ({\left (2 \, c^{2} d - b c e\right )} x^{8} + {\left (2 \, b c d - b^{2} e\right )} x^{4} + 2 \, a c d - a b e\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c x^{4} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) + {\left (b^{3} - 4 \, a b c\right )} d - 2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} e}{4 \, {\left ({\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{8} + a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{4}\right )}}\right ] \] Input:
integrate(x^3*(e*x^4+d)/(c*x^8+b*x^4+a)^2,x, algorithm="fricas")
Output:
[-1/4*((2*(b^2*c - 4*a*c^2)*d - (b^3 - 4*a*b*c)*e)*x^4 - ((2*c^2*d - b*c*e )*x^8 + (2*b*c*d - b^2*e)*x^4 + 2*a*c*d - a*b*e)*sqrt(b^2 - 4*a*c)*log((2* c^2*x^8 + 2*b*c*x^4 + b^2 - 2*a*c + (2*c*x^4 + b)*sqrt(b^2 - 4*a*c))/(c*x^ 8 + b*x^4 + a)) + (b^3 - 4*a*b*c)*d - 2*(a*b^2 - 4*a^2*c)*e)/((b^4*c - 8*a *b^2*c^2 + 16*a^2*c^3)*x^8 + a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2 + (b^5 - 8*a *b^3*c + 16*a^2*b*c^2)*x^4), -1/4*((2*(b^2*c - 4*a*c^2)*d - (b^3 - 4*a*b*c )*e)*x^4 - 2*((2*c^2*d - b*c*e)*x^8 + (2*b*c*d - b^2*e)*x^4 + 2*a*c*d - a* b*e)*sqrt(-b^2 + 4*a*c)*arctan(-(2*c*x^4 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4* a*c)) + (b^3 - 4*a*b*c)*d - 2*(a*b^2 - 4*a^2*c)*e)/((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*x^8 + a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2 + (b^5 - 8*a*b^3*c + 1 6*a^2*b*c^2)*x^4)]
Leaf count of result is larger than twice the leaf count of optimal. 374 vs. \(2 (85) = 170\).
Time = 15.18 (sec) , antiderivative size = 374, normalized size of antiderivative = 3.78 \[ \int \frac {x^3 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\frac {\sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) \log {\left (x^{4} + \frac {- 16 a^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) + 8 a b^{2} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) - b^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) + b^{2} e - 2 b c d}{2 b c e - 4 c^{2} d} \right )}}{4} - \frac {\sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) \log {\left (x^{4} + \frac {16 a^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) - 8 a b^{2} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) + b^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) + b^{2} e - 2 b c d}{2 b c e - 4 c^{2} d} \right )}}{4} + \frac {- 2 a e + b d + x^{4} \left (- b e + 2 c d\right )}{16 a^{2} c - 4 a b^{2} + x^{8} \cdot \left (16 a c^{2} - 4 b^{2} c\right ) + x^{4} \cdot \left (16 a b c - 4 b^{3}\right )} \] Input:
integrate(x**3*(e*x**4+d)/(c*x**8+b*x**4+a)**2,x)
Output:
sqrt(-1/(4*a*c - b**2)**3)*(b*e - 2*c*d)*log(x**4 + (-16*a**2*c**2*sqrt(-1 /(4*a*c - b**2)**3)*(b*e - 2*c*d) + 8*a*b**2*c*sqrt(-1/(4*a*c - b**2)**3)* (b*e - 2*c*d) - b**4*sqrt(-1/(4*a*c - b**2)**3)*(b*e - 2*c*d) + b**2*e - 2 *b*c*d)/(2*b*c*e - 4*c**2*d))/4 - sqrt(-1/(4*a*c - b**2)**3)*(b*e - 2*c*d) *log(x**4 + (16*a**2*c**2*sqrt(-1/(4*a*c - b**2)**3)*(b*e - 2*c*d) - 8*a*b **2*c*sqrt(-1/(4*a*c - b**2)**3)*(b*e - 2*c*d) + b**4*sqrt(-1/(4*a*c - b** 2)**3)*(b*e - 2*c*d) + b**2*e - 2*b*c*d)/(2*b*c*e - 4*c**2*d))/4 + (-2*a*e + b*d + x**4*(-b*e + 2*c*d))/(16*a**2*c - 4*a*b**2 + x**8*(16*a*c**2 - 4* b**2*c) + x**4*(16*a*b*c - 4*b**3))
Exception generated. \[ \int \frac {x^3 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^3*(e*x^4+d)/(c*x^8+b*x^4+a)^2,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Time = 4.79 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.05 \[ \int \frac {x^3 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=-\frac {{\left (2 \, c d - b e\right )} \arctan \left (\frac {2 \, c x^{4} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (b^{2} - 4 \, a c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, c d x^{4} - b e x^{4} + b d - 2 \, a e}{4 \, {\left (c x^{8} + b x^{4} + a\right )} {\left (b^{2} - 4 \, a c\right )}} \] Input:
integrate(x^3*(e*x^4+d)/(c*x^8+b*x^4+a)^2,x, algorithm="giac")
Output:
-1/2*(2*c*d - b*e)*arctan((2*c*x^4 + b)/sqrt(-b^2 + 4*a*c))/((b^2 - 4*a*c) *sqrt(-b^2 + 4*a*c)) - 1/4*(2*c*d*x^4 - b*e*x^4 + b*d - 2*a*e)/((c*x^8 + b *x^4 + a)*(b^2 - 4*a*c))
Time = 25.57 (sec) , antiderivative size = 1036, normalized size of antiderivative = 10.46 \[ \int \frac {x^3 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx =\text {Too large to display} \] Input:
int((x^3*(d + e*x^4))/(a + b*x^4 + c*x^8)^2,x)
Output:
-(2*a*e*(4*a*c - b^2)^(1/2) - b*d*(4*a*c - b^2)^(1/2) + b*e*x^4*(4*a*c - b ^2)^(1/2) - 2*c*d*x^4*(4*a*c - b^2)^(1/2) - 2*b^2*e*x^4*atan((b^7 - 64*a^3 *b*c^3 + 2*b^6*c*x^4 + 48*a^2*b^3*c^2 - 128*a^3*c^4*x^4 - 12*a*b^5*c - 24* a*b^4*c^2*x^4 + 96*a^2*b^2*c^3*x^4)/(4*a*c - b^2)^(7/2)) + 4*c^2*d*x^8*ata n((b^7 - 64*a^3*b*c^3 + 2*b^6*c*x^4 + 48*a^2*b^3*c^2 - 128*a^3*c^4*x^4 - 1 2*a*b^5*c - 24*a*b^4*c^2*x^4 + 96*a^2*b^2*c^3*x^4)/(4*a*c - b^2)^(7/2)) - 2*b^2*e*x^4*atan((b^3 - 3*a*b*c)/(b^2*(4*a*c - b^2)^(1/2) - a*c*(4*a*c - b ^2)^(1/2))) + 4*c^2*d*x^8*atan((b^3 - 3*a*b*c)/(b^2*(4*a*c - b^2)^(1/2) - a*c*(4*a*c - b^2)^(1/2))) - 2*a*b*e*atan((b^7 - 64*a^3*b*c^3 + 2*b^6*c*x^4 + 48*a^2*b^3*c^2 - 128*a^3*c^4*x^4 - 12*a*b^5*c - 24*a*b^4*c^2*x^4 + 96*a ^2*b^2*c^3*x^4)/(4*a*c - b^2)^(7/2)) + 4*a*c*d*atan((b^7 - 64*a^3*b*c^3 + 2*b^6*c*x^4 + 48*a^2*b^3*c^2 - 128*a^3*c^4*x^4 - 12*a*b^5*c - 24*a*b^4*c^2 *x^4 + 96*a^2*b^2*c^3*x^4)/(4*a*c - b^2)^(7/2)) - 2*a*b*e*atan((b^3 - 3*a* b*c)/(b^2*(4*a*c - b^2)^(1/2) - a*c*(4*a*c - b^2)^(1/2))) + 4*a*c*d*atan(( b^3 - 3*a*b*c)/(b^2*(4*a*c - b^2)^(1/2) - a*c*(4*a*c - b^2)^(1/2))) + 4*b* c*d*x^4*atan((b^7 - 64*a^3*b*c^3 + 2*b^6*c*x^4 + 48*a^2*b^3*c^2 - 128*a^3* c^4*x^4 - 12*a*b^5*c - 24*a*b^4*c^2*x^4 + 96*a^2*b^2*c^3*x^4)/(4*a*c - b^2 )^(7/2)) - 2*b*c*e*x^8*atan((b^7 - 64*a^3*b*c^3 + 2*b^6*c*x^4 + 48*a^2*b^3 *c^2 - 128*a^3*c^4*x^4 - 12*a*b^5*c - 24*a*b^4*c^2*x^4 + 96*a^2*b^2*c^3*x^ 4)/(4*a*c - b^2)^(7/2)) + 4*b*c*d*x^4*atan((b^3 - 3*a*b*c)/(b^2*(4*a*c ...
Time = 7.28 (sec) , antiderivative size = 3019, normalized size of antiderivative = 30.49 \[ \int \frac {x^3 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx =\text {Too large to display} \] Input:
int(x^3*(e*x^4+d)/(c*x^8+b*x^4+a)^2,x)
Output:
( - 2*sqrt(2*sqrt(c)*sqrt(a) - b)*sqrt(sqrt(2*sqrt(c)*sqrt(a) - b) + 2*c** (1/4)*a**(1/4))*sqrt( - sqrt(2*sqrt(c)*sqrt(a) - b) + 2*c**(1/4)*a**(1/4)) *atan((sqrt( - sqrt(2*sqrt(c)*sqrt(a) - b) + 2*c**(1/4)*a**(1/4)) - 2*c**( 1/4)*x)/sqrt(sqrt(2*sqrt(c)*sqrt(a) - b) + 2*c**(1/4)*a**(1/4)))*a*b**2*e + 4*sqrt(2*sqrt(c)*sqrt(a) - b)*sqrt(sqrt(2*sqrt(c)*sqrt(a) - b) + 2*c**(1 /4)*a**(1/4))*sqrt( - sqrt(2*sqrt(c)*sqrt(a) - b) + 2*c**(1/4)*a**(1/4))*a tan((sqrt( - sqrt(2*sqrt(c)*sqrt(a) - b) + 2*c**(1/4)*a**(1/4)) - 2*c**(1/ 4)*x)/sqrt(sqrt(2*sqrt(c)*sqrt(a) - b) + 2*c**(1/4)*a**(1/4)))*a*b*c*d - 2 *sqrt(2*sqrt(c)*sqrt(a) - b)*sqrt(sqrt(2*sqrt(c)*sqrt(a) - b) + 2*c**(1/4) *a**(1/4))*sqrt( - sqrt(2*sqrt(c)*sqrt(a) - b) + 2*c**(1/4)*a**(1/4))*atan ((sqrt( - sqrt(2*sqrt(c)*sqrt(a) - b) + 2*c**(1/4)*a**(1/4)) - 2*c**(1/4)* x)/sqrt(sqrt(2*sqrt(c)*sqrt(a) - b) + 2*c**(1/4)*a**(1/4)))*b**3*e*x**4 + 4*sqrt(2*sqrt(c)*sqrt(a) - b)*sqrt(sqrt(2*sqrt(c)*sqrt(a) - b) + 2*c**(1/4 )*a**(1/4))*sqrt( - sqrt(2*sqrt(c)*sqrt(a) - b) + 2*c**(1/4)*a**(1/4))*ata n((sqrt( - sqrt(2*sqrt(c)*sqrt(a) - b) + 2*c**(1/4)*a**(1/4)) - 2*c**(1/4) *x)/sqrt(sqrt(2*sqrt(c)*sqrt(a) - b) + 2*c**(1/4)*a**(1/4)))*b**2*c*d*x**4 - 2*sqrt(2*sqrt(c)*sqrt(a) - b)*sqrt(sqrt(2*sqrt(c)*sqrt(a) - b) + 2*c**( 1/4)*a**(1/4))*sqrt( - sqrt(2*sqrt(c)*sqrt(a) - b) + 2*c**(1/4)*a**(1/4))* atan((sqrt( - sqrt(2*sqrt(c)*sqrt(a) - b) + 2*c**(1/4)*a**(1/4)) - 2*c**(1 /4)*x)/sqrt(sqrt(2*sqrt(c)*sqrt(a) - b) + 2*c**(1/4)*a**(1/4)))*b**2*c*...