\(\int \frac {-((2 a-b) b^2)+(4 a-b) b x-(2 a+b) x^2+x^3}{(-a+x) \sqrt [4]{(-a+x) (-b+x)^2} (b^2+a d-(2 b+d) x+x^2)} \, dx\) [2217]

Optimal result

Integrand size = 81, antiderivative size = 165 \[ \int \frac {-\left ((2 a-b) b^2\right )+(4 a-b) b x-(2 a+b) x^2+x^3}{(-a+x) \sqrt [4]{(-a+x) (-b+x)^2} \left (b^2+a d-(2 b+d) x+x^2\right )} \, dx=-\frac {4 \left (-a b^2+2 a b x+b^2 x-a x^2-2 b x^2+x^3\right )^{3/4}}{(b-x) (-a+x)}-2 \sqrt [4]{d} \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{b-x}\right )+2 \sqrt [4]{d} \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{b-x}\right ) \] Output:

-4*(-a*b^2+2*a*b*x-a*x^2+b^2*x-2*b*x^2+x^3)^(3/4)/(b-x)/(-a+x)-2*d^(1/4)*a 
rctan(d^(1/4)*(-a*b^2+(2*a*b+b^2)*x+(-a-2*b)*x^2+x^3)^(1/4)/(b-x))+2*d^(1/ 
4)*arctanh(d^(1/4)*(-a*b^2+(2*a*b+b^2)*x+(-a-2*b)*x^2+x^3)^(1/4)/(b-x))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 3.46 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.13 \[ \int \frac {-\left ((2 a-b) b^2\right )+(4 a-b) b x-(2 a+b) x^2+x^3}{(-a+x) \sqrt [4]{(-a+x) (-b+x)^2} \left (b^2+a d-(2 b+d) x+x^2\right )} \, dx=-\frac {4 b-4 x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{a-x} \sqrt {b-x} \arctan \left (\frac {-b+\sqrt {d} \sqrt {a-x}+x}{\sqrt {2} \sqrt [4]{d} \sqrt [4]{a-x} \sqrt {b-x}}\right )-\sqrt {2} \sqrt [4]{d} \sqrt [4]{a-x} \sqrt {b-x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{a-x} \sqrt {b-x}}{b+\sqrt {d} \sqrt {a-x}-x}\right )}{\sqrt [4]{(b-x)^2 (-a+x)}} \] Input:

Integrate[(-((2*a - b)*b^2) + (4*a - b)*b*x - (2*a + b)*x^2 + x^3)/((-a + 
x)*((-a + x)*(-b + x)^2)^(1/4)*(b^2 + a*d - (2*b + d)*x + x^2)),x]
 

Output:

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

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 6.46 (sec) , antiderivative size = 1427, normalized size of antiderivative = 8.65, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {7239, 7270, 25, 2153, 2009}

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.

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

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {(b-x)^4 (-2 a+b+x)}{\left (-\left ((a-x) (b-x)^2\right )\right )^{5/4} \left (a d+b^2-x (2 b+d)+x^2\right )}dx\)

\(\Big \downarrow \) 7270

\(\displaystyle -\frac {\sqrt [4]{a-x} \sqrt {b-x} \int -\frac {(2 a-b-x) (b-x)^{3/2}}{(a-x)^{5/4} \left (b^2+x^2+a d-(2 b+d) x\right )}dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt [4]{a-x} \sqrt {b-x} \int \frac {(2 a-b-x) (b-x)^{3/2}}{(a-x)^{5/4} \left (b^2+x^2+a d-(2 b+d) x\right )}dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\)

\(\Big \downarrow \) 2153

\(\displaystyle \frac {\sqrt [4]{a-x} \sqrt {b-x} \int \left (\frac {\left (-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}-1\right ) (b-x)^{3/2}}{(a-x)^{5/4} \left (-2 b-d+2 x-\sqrt {d} \sqrt {-4 a+4 b+d}\right )}+\frac {\left (\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}-1\right ) (b-x)^{3/2}}{(a-x)^{5/4} \left (-2 b-d+2 x+\sqrt {d} \sqrt {-4 a+4 b+d}\right )}\right )dx}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt [4]{a-x} \sqrt {b-x} \left (-\frac {2 (a-b)^{3/4} \left (4 a-4 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt {-\frac {b-x}{a-b}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{a-x}}{\sqrt [4]{a-b}}\right )\right |-1\right ) \left (\sqrt {d}+\sqrt {-4 a+4 b+d}\right )}{\sqrt {d} \left (2 a-2 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt {b-x}}+\frac {2 (a-b)^{3/4} \left (4 a-4 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt {-\frac {b-x}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a-x}}{\sqrt [4]{a-b}}\right ),-1\right ) \left (\sqrt {d}+\sqrt {-4 a+4 b+d}\right )}{\sqrt {d} \left (2 a-2 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt {b-x}}+\frac {\sqrt {2} \sqrt [4]{a-b} \sqrt {d} \sqrt {-\frac {b-x}{a-b}} \operatorname {EllipticPi}\left (-\frac {\sqrt {2} \sqrt {a-b}}{\sqrt {2 a-2 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}}},\arcsin \left (\frac {\sqrt [4]{a-x}}{\sqrt [4]{a-b}}\right ),-1\right ) \left (\sqrt {d}+\sqrt {-4 a+4 b+d}\right )}{\sqrt {2 a-2 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}} \sqrt {b-x}}-\frac {\sqrt {2} \sqrt [4]{a-b} \sqrt {d} \sqrt {-\frac {b-x}{a-b}} \operatorname {EllipticPi}\left (\frac {\sqrt {2} \sqrt {a-b}}{\sqrt {2 a-2 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}}},\arcsin \left (\frac {\sqrt [4]{a-x}}{\sqrt [4]{a-b}}\right ),-1\right ) \left (\sqrt {d}+\sqrt {-4 a+4 b+d}\right )}{\sqrt {2 a-2 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}} \sqrt {b-x}}+\frac {4 (a-b) \sqrt {b-x} \left (\sqrt {d}+\sqrt {-4 a+4 b+d}\right )}{\sqrt {d} \left (2 a-2 b-d-\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{a-x}}-\frac {2 (a-b)^{3/4} \left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (4 a-4 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt {-\frac {b-x}{a-b}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{a-x}}{\sqrt [4]{a-b}}\right )\right |-1\right )}{\left (2 a-2 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt {b-x}}+\frac {2 (a-b)^{3/4} \left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \left (4 a-4 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt {-\frac {b-x}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a-x}}{\sqrt [4]{a-b}}\right ),-1\right )}{\left (2 a-2 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt {b-x}}+\frac {\sqrt {2} \sqrt [4]{a-b} d \left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt {-\frac {b-x}{a-b}} \operatorname {EllipticPi}\left (-\frac {\sqrt {2} \sqrt {a-b}}{\sqrt {2 a-2 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}}},\arcsin \left (\frac {\sqrt [4]{a-x}}{\sqrt [4]{a-b}}\right ),-1\right )}{\sqrt {2 a-2 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}} \sqrt {b-x}}-\frac {\sqrt {2} \sqrt [4]{a-b} d \left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt {-\frac {b-x}{a-b}} \operatorname {EllipticPi}\left (\frac {\sqrt {2} \sqrt {a-b}}{\sqrt {2 a-2 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}}},\arcsin \left (\frac {\sqrt [4]{a-x}}{\sqrt [4]{a-b}}\right ),-1\right )}{\sqrt {2 a-2 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}} \sqrt {b-x}}+\frac {4 (a-b) \left (1-\frac {\sqrt {-4 a+4 b+d}}{\sqrt {d}}\right ) \sqrt {b-x}}{\left (2 a-2 b-d+\sqrt {d} \sqrt {-4 a+4 b+d}\right ) \sqrt [4]{a-x}}\right )}{\sqrt [4]{-\left ((a-x) (b-x)^2\right )}}\)

Input:

Int[(-((2*a - b)*b^2) + (4*a - b)*b*x - (2*a + b)*x^2 + x^3)/((-a + x)*((- 
a + x)*(-b + x)^2)^(1/4)*(b^2 + a*d - (2*b + d)*x + x^2)),x]
 

Output:

((a - x)^(1/4)*Sqrt[b - x]*((4*(a - b)*(Sqrt[d] + Sqrt[-4*a + 4*b + d])*Sq 
rt[b - x])/(Sqrt[d]*(2*a - 2*b - d - Sqrt[d]*Sqrt[-4*a + 4*b + d])*(a - x) 
^(1/4)) + (4*(a - b)*(1 - Sqrt[-4*a + 4*b + d]/Sqrt[d])*Sqrt[b - x])/((2*a 
 - 2*b - d + Sqrt[d]*Sqrt[-4*a + 4*b + d])*(a - x)^(1/4)) - (2*(a - b)^(3/ 
4)*(Sqrt[d] + Sqrt[-4*a + 4*b + d])*(4*a - 4*b - d - Sqrt[d]*Sqrt[-4*a + 4 
*b + d])*Sqrt[-((b - x)/(a - b))]*EllipticE[ArcSin[(a - x)^(1/4)/(a - b)^( 
1/4)], -1])/(Sqrt[d]*(2*a - 2*b - d - Sqrt[d]*Sqrt[-4*a + 4*b + d])*Sqrt[b 
 - x]) - (2*(a - b)^(3/4)*(1 - Sqrt[-4*a + 4*b + d]/Sqrt[d])*(4*a - 4*b - 
d + Sqrt[d]*Sqrt[-4*a + 4*b + d])*Sqrt[-((b - x)/(a - b))]*EllipticE[ArcSi 
n[(a - x)^(1/4)/(a - b)^(1/4)], -1])/((2*a - 2*b - d + Sqrt[d]*Sqrt[-4*a + 
 4*b + d])*Sqrt[b - x]) + (2*(a - b)^(3/4)*(Sqrt[d] + Sqrt[-4*a + 4*b + d] 
)*(4*a - 4*b - d - Sqrt[d]*Sqrt[-4*a + 4*b + d])*Sqrt[-((b - x)/(a - b))]* 
EllipticF[ArcSin[(a - x)^(1/4)/(a - b)^(1/4)], -1])/(Sqrt[d]*(2*a - 2*b - 
d - Sqrt[d]*Sqrt[-4*a + 4*b + d])*Sqrt[b - x]) + (2*(a - b)^(3/4)*(1 - Sqr 
t[-4*a + 4*b + d]/Sqrt[d])*(4*a - 4*b - d + Sqrt[d]*Sqrt[-4*a + 4*b + d])* 
Sqrt[-((b - x)/(a - b))]*EllipticF[ArcSin[(a - x)^(1/4)/(a - b)^(1/4)], -1 
])/((2*a - 2*b - d + Sqrt[d]*Sqrt[-4*a + 4*b + d])*Sqrt[b - x]) + (Sqrt[2] 
*(a - b)^(1/4)*Sqrt[d]*(Sqrt[d] + Sqrt[-4*a + 4*b + d])*Sqrt[-((b - x)/(a 
- b))]*EllipticPi[-((Sqrt[2]*Sqrt[a - b])/Sqrt[2*a - 2*b - d - Sqrt[d]*Sqr 
t[-4*a + 4*b + d]]), ArcSin[(a - x)^(1/4)/(a - b)^(1/4)], -1])/(Sqrt[2*...
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2153

Int[(Px_)*((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b 
_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(d + e* 
x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m, n, p}, x] && PolyQ[Px, x] && (IntegerQ[p] || (IntegerQ[2*p] && IntegerQ 
[m] && ILtQ[n, 0])) &&  !(IGtQ[m, 0] && IGtQ[n, 0])
 

rule 7239

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

rule 7270

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p 
]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p])))   Int[u*v 
^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !Free 
Q[v, x] &&  !FreeQ[w, x]
 

Maple [F]

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

Input:

int((-(2*a-b)*b^2+(4*a-b)*b*x-(2*a+b)*x^2+x^3)/(-a+x)/((-a+x)*(-b+x)^2)^(1 
/4)/(b^2+a*d-(2*b+d)*x+x^2),x)
 

Output:

int((-(2*a-b)*b^2+(4*a-b)*b*x-(2*a+b)*x^2+x^3)/(-a+x)/((-a+x)*(-b+x)^2)^(1 
/4)/(b^2+a*d-(2*b+d)*x+x^2),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {-\left ((2 a-b) b^2\right )+(4 a-b) b x-(2 a+b) x^2+x^3}{(-a+x) \sqrt [4]{(-a+x) (-b+x)^2} \left (b^2+a d-(2 b+d) x+x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((-(2*a-b)*b^2+(4*a-b)*b*x-(2*a+b)*x^2+x^3)/(-a+x)/((-a+x)*(-b+x) 
^2)^(1/4)/(b^2+a*d-(2*b+d)*x+x^2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {-\left ((2 a-b) b^2\right )+(4 a-b) b x-(2 a+b) x^2+x^3}{(-a+x) \sqrt [4]{(-a+x) (-b+x)^2} \left (b^2+a d-(2 b+d) x+x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((-(2*a-b)*b**2+(4*a-b)*b*x-(2*a+b)*x**2+x**3)/(-a+x)/((-a+x)*(-b 
+x)**2)**(1/4)/(b**2+a*d-(2*b+d)*x+x**2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {-\left ((2 a-b) b^2\right )+(4 a-b) b x-(2 a+b) x^2+x^3}{(-a+x) \sqrt [4]{(-a+x) (-b+x)^2} \left (b^2+a d-(2 b+d) x+x^2\right )} \, dx=\int { \frac {{\left (2 \, a - b\right )} b^{2} - {\left (4 \, a - b\right )} b x + {\left (2 \, a + b\right )} x^{2} - x^{3}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{4}} {\left (b^{2} + a d - {\left (2 \, b + d\right )} x + x^{2}\right )} {\left (a - x\right )}} \,d x } \] Input:

integrate((-(2*a-b)*b^2+(4*a-b)*b*x-(2*a+b)*x^2+x^3)/(-a+x)/((-a+x)*(-b+x) 
^2)^(1/4)/(b^2+a*d-(2*b+d)*x+x^2),x, algorithm="maxima")
 

Output:

integrate(((2*a - b)*b^2 - (4*a - b)*b*x + (2*a + b)*x^2 - x^3)/((-(a - x) 
*(b - x)^2)^(1/4)*(b^2 + a*d - (2*b + d)*x + x^2)*(a - x)), x)
 

Giac [F]

\[ \int \frac {-\left ((2 a-b) b^2\right )+(4 a-b) b x-(2 a+b) x^2+x^3}{(-a+x) \sqrt [4]{(-a+x) (-b+x)^2} \left (b^2+a d-(2 b+d) x+x^2\right )} \, dx=\int { \frac {{\left (2 \, a - b\right )} b^{2} - {\left (4 \, a - b\right )} b x + {\left (2 \, a + b\right )} x^{2} - x^{3}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{4}} {\left (b^{2} + a d - {\left (2 \, b + d\right )} x + x^{2}\right )} {\left (a - x\right )}} \,d x } \] Input:

integrate((-(2*a-b)*b^2+(4*a-b)*b*x-(2*a+b)*x^2+x^3)/(-a+x)/((-a+x)*(-b+x) 
^2)^(1/4)/(b^2+a*d-(2*b+d)*x+x^2),x, algorithm="giac")
 

Output:

integrate(((2*a - b)*b^2 - (4*a - b)*b*x + (2*a + b)*x^2 - x^3)/((-(a - x) 
*(b - x)^2)^(1/4)*(b^2 + a*d - (2*b + d)*x + x^2)*(a - x)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-\left ((2 a-b) b^2\right )+(4 a-b) b x-(2 a+b) x^2+x^3}{(-a+x) \sqrt [4]{(-a+x) (-b+x)^2} \left (b^2+a d-(2 b+d) x+x^2\right )} \, dx=-\int -\frac {b^2\,\left (2\,a-b\right )+x^2\,\left (2\,a+b\right )-x^3-b\,x\,\left (4\,a-b\right )}{\left (a-x\right )\,{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/4}\,\left (a\,d-x\,\left (2\,b+d\right )+b^2+x^2\right )} \,d x \] Input:

int((b^2*(2*a - b) + x^2*(2*a + b) - x^3 - b*x*(4*a - b))/((a - x)*(-(a - 
x)*(b - x)^2)^(1/4)*(a*d - x*(2*b + d) + b^2 + x^2)),x)
 

Output:

-int(-(b^2*(2*a - b) + x^2*(2*a + b) - x^3 - b*x*(4*a - b))/((a - x)*(-(a 
- x)*(b - x)^2)^(1/4)*(a*d - x*(2*b + d) + b^2 + x^2)), x)
 

Reduce [F]

\[ \int \frac {-\left ((2 a-b) b^2\right )+(4 a-b) b x-(2 a+b) x^2+x^3}{(-a+x) \sqrt [4]{(-a+x) (-b+x)^2} \left (b^2+a d-(2 b+d) x+x^2\right )} \, dx=\text {too large to display} \] Input:

int((-(2*a-b)*b^2+(4*a-b)*b*x-(2*a+b)*x^2+x^3)/(-a+x)/((-a+x)*(-b+x)^2)^(1 
/4)/(b^2+a*d-(2*b+d)*x+x^2),x)
 

Output:

 - int(x**3/(( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3)**(1 
/4)*a**2*d + ( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3)**(1 
/4)*a*b**2 - 2*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3)** 
(1/4)*a*b*x - 2*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3)* 
*(1/4)*a*d*x + ( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3)** 
(1/4)*a*x**2 - ( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3)** 
(1/4)*b**2*x + 2*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3) 
**(1/4)*b*x**2 + ( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3) 
**(1/4)*d*x**2 - ( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2*b*x**2 + x**3) 
**(1/4)*x**3),x) + 2*int(x**2/(( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2* 
b*x**2 + x**3)**(1/4)*a**2*d + ( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 2* 
b*x**2 + x**3)**(1/4)*a*b**2 - 2*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 
2*b*x**2 + x**3)**(1/4)*a*b*x - 2*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 
 2*b*x**2 + x**3)**(1/4)*a*d*x + ( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 
2*b*x**2 + x**3)**(1/4)*a*x**2 - ( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x - 
2*b*x**2 + x**3)**(1/4)*b**2*x + 2*( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x 
- 2*b*x**2 + x**3)**(1/4)*b*x**2 + ( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x 
- 2*b*x**2 + x**3)**(1/4)*d*x**2 - ( - a*b**2 + 2*a*b*x - a*x**2 + b**2*x 
- 2*b*x**2 + x**3)**(1/4)*x**3),x)*a + int(x**2/(( - a*b**2 + 2*a*b*x - a* 
x**2 + b**2*x - 2*b*x**2 + x**3)**(1/4)*a**2*d + ( - a*b**2 + 2*a*b*x -...