\(\int \frac {A+B x}{x^3 (d+e x) \sqrt {a+b x+c x^2}} \, dx\) [19]

Optimal result

Integrand size = 30, antiderivative size = 248 \[ \int \frac {A+B x}{x^3 (d+e x) \sqrt {a+b x+c x^2}} \, dx=-\frac {A \sqrt {a+b x+c x^2}}{2 a d x^2}+\frac {(3 A b d-4 a B d+4 a A e) \sqrt {a+b x+c x^2}}{4 a^2 d^2 x}-\frac {((b d+2 a e) (3 A b d-4 a B d+4 a A e)-2 a A d (2 c d+3 b e)) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 a^{5/2} d^3}+\frac {e^2 (B d-A e) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{d^3 \sqrt {c d^2-b d e+a e^2}} \] Output:

-1/2*A*(c*x^2+b*x+a)^(1/2)/a/d/x^2+1/4*(4*A*a*e+3*A*b*d-4*B*a*d)*(c*x^2+b* 
x+a)^(1/2)/a^2/d^2/x-1/8*((2*a*e+b*d)*(4*A*a*e+3*A*b*d-4*B*a*d)-2*a*A*d*(3 
*b*e+2*c*d))*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/a^(5/2)/d^ 
3+e^2*(-A*e+B*d)*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2 
)^(1/2)/(c*x^2+b*x+a)^(1/2))/d^3/(a*e^2-b*d*e+c*d^2)^(1/2)
 

Mathematica [A] (verified)

Time = 1.98 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.07 \[ \int \frac {A+B x}{x^3 (d+e x) \sqrt {a+b x+c x^2}} \, dx=\frac {\frac {d \sqrt {a+x (b+c x)} (3 A b d x-2 a (2 B d x+A (d-2 e x)))}{a^2 x^2}+\frac {8 e^2 (B d-A e) \sqrt {-c d^2+e (b d-a e)} \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )}{c d^2+e (-b d+a e)}+\frac {8 A e^2 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {d \left (4 a B (b d+2 a e)+A \left (-3 b^2 d+4 a c d-4 a b e\right )\right ) \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{a^{5/2}}}{4 d^3} \] Input:

Integrate[(A + B*x)/(x^3*(d + e*x)*Sqrt[a + b*x + c*x^2]),x]
 

Output:

((d*Sqrt[a + x*(b + c*x)]*(3*A*b*d*x - 2*a*(2*B*d*x + A*(d - 2*e*x))))/(a^ 
2*x^2) + (8*e^2*(B*d - A*e)*Sqrt[-(c*d^2) + e*(b*d - a*e)]*ArcTan[(Sqrt[c] 
*(d + e*x) - e*Sqrt[a + x*(b + c*x)])/Sqrt[-(c*d^2) + e*(b*d - a*e)]])/(c* 
d^2 + e*(-(b*d) + a*e)) + (8*A*e^2*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c* 
x)])/Sqrt[a]])/Sqrt[a] + (d*(4*a*B*(b*d + 2*a*e) + A*(-3*b^2*d + 4*a*c*d - 
 4*a*b*e))*ArcTanh[(-(Sqrt[c]*x) + Sqrt[a + x*(b + c*x)])/Sqrt[a]])/a^(5/2 
))/(4*d^3)
 

Rubi [A] (verified)

Time = 1.15 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.37, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {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.

\(\displaystyle \int \frac {A+B x}{x^3 (d+e x) \sqrt {a+b x+c x^2}} \, dx\)

\(\Big \downarrow \) 2153

\(\displaystyle \int \left (\frac {e^2 (B d-A e)}{d^3 (d+e x) \sqrt {a+b x+c x^2}}-\frac {e (B d-A e)}{d^3 x \sqrt {a+b x+c x^2}}+\frac {B d-A e}{d^2 x^2 \sqrt {a+b x+c x^2}}+\frac {A}{d x^3 \sqrt {a+b x+c x^2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {A \left (3 b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 a^{5/2} d}+\frac {b (B d-A e) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 a^{3/2} d^2}+\frac {3 A b \sqrt {a+b x+c x^2}}{4 a^2 d x}+\frac {e (B d-A e) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{\sqrt {a} d^3}+\frac {e^2 (B d-A e) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{d^3 \sqrt {a e^2-b d e+c d^2}}-\frac {\sqrt {a+b x+c x^2} (B d-A e)}{a d^2 x}-\frac {A \sqrt {a+b x+c x^2}}{2 a d x^2}\)

Input:

Int[(A + B*x)/(x^3*(d + e*x)*Sqrt[a + b*x + c*x^2]),x]
 

Output:

-1/2*(A*Sqrt[a + b*x + c*x^2])/(a*d*x^2) + (3*A*b*Sqrt[a + b*x + c*x^2])/( 
4*a^2*d*x) - ((B*d - A*e)*Sqrt[a + b*x + c*x^2])/(a*d^2*x) - (A*(3*b^2 - 4 
*a*c)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(8*a^(5/2)*d 
) + (b*(B*d - A*e)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])]) 
/(2*a^(3/2)*d^2) + (e*(B*d - A*e)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + 
b*x + c*x^2])])/(Sqrt[a]*d^3) + (e^2*(B*d - A*e)*ArcTanh[(b*d - 2*a*e + (2 
*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(d^ 
3*Sqrt[c*d^2 - b*d*e + a*e^2])
 

Defintions of rubi rules used

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])
 

Maple [A] (verified)

Time = 0.73 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.26

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (-4 A a e x -3 x A b d +4 B a d x +2 A a d \right )}{4 a^{2} d^{2} x^{2}}+\frac {-\frac {\left (8 A \,a^{2} e^{2}+4 A a b d e -4 A c \,d^{2} a +3 A \,b^{2} d^{2}-8 B \,a^{2} d e -4 B b \,d^{2} a \right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{d \sqrt {a}}+\frac {8 a^{2} e \left (A e -B d \right ) \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}}{8 d^{2} a^{2}}\)	\(313\)
default	\(\frac {A \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{2 a \,x^{2}}-\frac {3 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}+\frac {c \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{d}-\frac {\left (A e -B d \right ) \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{d^{2}}-\frac {\left (A e -B d \right ) e \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{d^{3} \sqrt {a}}+\frac {\left (A e -B d \right ) e \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d^{3} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\)	\(404\)

Input:

int((B*x+A)/x^3/(e*x+d)/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/4*(c*x^2+b*x+a)^(1/2)*(-4*A*a*e*x-3*A*b*d*x+4*B*a*d*x+2*A*a*d)/a^2/d^2/ 
x^2+1/8/d^2/a^2*(-(8*A*a^2*e^2+4*A*a*b*d*e-4*A*a*c*d^2+3*A*b^2*d^2-8*B*a^2 
*d*e-4*B*a*b*d^2)/d/a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+ 
8*a^2*e*(A*e-B*d)/d/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d 
^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2 
*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 499 vs. \(2 (222) = 444\).

Time = 38.60 (sec) , antiderivative size = 2096, normalized size of antiderivative = 8.45 \[ \int \frac {A+B x}{x^3 (d+e x) \sqrt {a+b x+c x^2}} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)/x^3/(e*x+d)/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
 

Output:

[-1/16*(8*(B*a^3*d*e^2 - A*a^3*e^3)*sqrt(c*d^2 - b*d*e + a*e^2)*x^2*log((8 
*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 
 4*a*c)*e^2)*x^2 + 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b* 
d - 2*a*e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)* 
d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2)) - (8*A*a^3*e^4 - (4*A*a*c^2 + (4*B*a*b 
- 3*A*b^2)*c)*d^4 + (4*B*a*b^2 - 3*A*b^3 - 8*(B*a^2 - A*a*b)*c)*d^3*e + (4 
*B*a^2*b - A*a*b^2 + 4*A*a^2*c)*d^2*e^2 - 4*(2*B*a^3 + A*a^2*b)*d*e^3)*sqr 
t(a)*x^2*log(-(8*a*b*x + (b^2 + 4*a*c)*x^2 - 4*sqrt(c*x^2 + b*x + a)*(b*x 
+ 2*a)*sqrt(a) + 8*a^2)/x^2) + 4*(2*A*a^2*c*d^4 - 2*A*a^2*b*d^3*e + 2*A*a^ 
3*d^2*e^2 - (4*A*a^3*d*e^3 - (4*B*a^2 - 3*A*a*b)*c*d^4 + (4*B*a^2*b - 3*A* 
a*b^2 + 4*A*a^2*c)*d^3*e - (4*B*a^3 + A*a^2*b)*d^2*e^2)*x)*sqrt(c*x^2 + b* 
x + a))/((a^3*c*d^5 - a^3*b*d^4*e + a^4*d^3*e^2)*x^2), 1/16*(16*(B*a^3*d*e 
^2 - A*a^3*e^3)*sqrt(-c*d^2 + b*d*e - a*e^2)*x^2*arctan(-1/2*sqrt(-c*d^2 + 
 b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x)/(a*c 
*d^2 - a*b*d*e + a^2*e^2 + (c^2*d^2 - b*c*d*e + a*c*e^2)*x^2 + (b*c*d^2 - 
b^2*d*e + a*b*e^2)*x)) + (8*A*a^3*e^4 - (4*A*a*c^2 + (4*B*a*b - 3*A*b^2)*c 
)*d^4 + (4*B*a*b^2 - 3*A*b^3 - 8*(B*a^2 - A*a*b)*c)*d^3*e + (4*B*a^2*b - A 
*a*b^2 + 4*A*a^2*c)*d^2*e^2 - 4*(2*B*a^3 + A*a^2*b)*d*e^3)*sqrt(a)*x^2*log 
(-(8*a*b*x + (b^2 + 4*a*c)*x^2 - 4*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt( 
a) + 8*a^2)/x^2) - 4*(2*A*a^2*c*d^4 - 2*A*a^2*b*d^3*e + 2*A*a^3*d^2*e^2...
 

Sympy [F]

\[ \int \frac {A+B x}{x^3 (d+e x) \sqrt {a+b x+c x^2}} \, dx=\int \frac {A + B x}{x^{3} \left (d + e x\right ) \sqrt {a + b x + c x^{2}}}\, dx \] Input:

integrate((B*x+A)/x**3/(e*x+d)/(c*x**2+b*x+a)**(1/2),x)
 

Output:

Integral((A + B*x)/(x**3*(d + e*x)*sqrt(a + b*x + c*x**2)), x)
 

Maxima [F]

\[ \int \frac {A+B x}{x^3 (d+e x) \sqrt {a+b x+c x^2}} \, dx=\int { \frac {B x + A}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )} x^{3}} \,d x } \] Input:

integrate((B*x+A)/x^3/(e*x+d)/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")
 

Output:

integrate((B*x + A)/(sqrt(c*x^2 + b*x + a)*(e*x + d)*x^3), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 536 vs. \(2 (222) = 444\).

Time = 0.15 (sec) , antiderivative size = 536, normalized size of antiderivative = 2.16 \[ \int \frac {A+B x}{x^3 (d+e x) \sqrt {a+b x+c x^2}} \, dx=\frac {2 \, {\left (B d e^{2} - A e^{3}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e - a e^{2}}}\right )}{\sqrt {-c d^{2} + b d e - a e^{2}} d^{3}} - \frac {{\left (4 \, B a b d^{2} - 3 \, A b^{2} d^{2} + 4 \, A a c d^{2} + 8 \, B a^{2} d e - 4 \, A a b d e - 8 \, A a^{2} e^{2}\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a^{2} d^{3}} + \frac {4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a b d - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A b^{2} d + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a c d - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a b e + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{2} \sqrt {c} d - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a^{2} \sqrt {c} e - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{2} b d + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a b^{2} d + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{2} c d + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{2} b e - 8 \, B a^{3} \sqrt {c} d + 8 \, A a^{2} b \sqrt {c} d + 8 \, A a^{3} \sqrt {c} e}{4 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{2} a^{2} d^{2}} \] Input:

integrate((B*x+A)/x^3/(e*x+d)/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")
 

Output:

2*(B*d*e^2 - A*e^3)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt( 
c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/(sqrt(-c*d^2 + b*d*e - a*e^2)*d^3) - 1 
/4*(4*B*a*b*d^2 - 3*A*b^2*d^2 + 4*A*a*c*d^2 + 8*B*a^2*d*e - 4*A*a*b*d*e - 
8*A*a^2*e^2)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(- 
a)*a^2*d^3) + 1/4*(4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a*b*d - 3*(sq 
rt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*b^2*d + 4*(sqrt(c)*x - sqrt(c*x^2 + b 
*x + a))^3*A*a*c*d - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a*b*e + 8*( 
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^2*sqrt(c)*d - 8*(sqrt(c)*x - sqrt 
(c*x^2 + b*x + a))^2*A*a^2*sqrt(c)*e - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a 
))*B*a^2*b*d + 5*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a*b^2*d + 4*(sqrt(c 
)*x - sqrt(c*x^2 + b*x + a))*A*a^2*c*d + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + 
 a))*A*a^2*b*e - 8*B*a^3*sqrt(c)*d + 8*A*a^2*b*sqrt(c)*d + 8*A*a^3*sqrt(c) 
*e)/(((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2 - a)^2*a^2*d^2)
 

Mupad [F(-1)]

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

int((A + B*x)/(x^3*(d + e*x)*(a + b*x + c*x^2)^(1/2)),x)
 

Output:

int((A + B*x)/(x^3*(d + e*x)*(a + b*x + c*x^2)^(1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 816, normalized size of antiderivative = 3.29 \[ \int \frac {A+B x}{x^3 (d+e x) \sqrt {a+b x+c x^2}} \, dx =\text {Too large to display} \] Input:

int((B*x+A)/x^3/(e*x+d)/(c*x^2+b*x+a)^(1/2),x)
 

Output:

(8*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e* 
*2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**3*e**3*x**2 - 8*s 
qrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - 
 b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*b*d*e**2*x**2 - 8*s 
qrt(a*e**2 - b*d*e + c*d**2)*log(d + e*x)*a**3*e**3*x**2 + 8*sqrt(a*e**2 - 
 b*d*e + c*d**2)*log(d + e*x)*a**2*b*d*e**2*x**2 - 4*sqrt(a + b*x + c*x**2 
)*a**3*d**2*e**2 + 8*sqrt(a + b*x + c*x**2)*a**3*d*e**3*x + 4*sqrt(a + b*x 
 + c*x**2)*a**2*b*d**3*e - 10*sqrt(a + b*x + c*x**2)*a**2*b*d**2*e**2*x - 
4*sqrt(a + b*x + c*x**2)*a**2*c*d**4 + 8*sqrt(a + b*x + c*x**2)*a**2*c*d** 
3*e*x + 2*sqrt(a + b*x + c*x**2)*a*b**2*d**3*e*x - 2*sqrt(a + b*x + c*x**2 
)*a*b*c*d**4*x + 8*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b* 
x)*a**3*e**4*x**2 - 12*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a 
- b*x)*a**2*b*d*e**3*x**2 + 4*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) 
 - 2*a - b*x)*a**2*c*d**2*e**2*x**2 + 3*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x 
 + c*x**2) - 2*a - b*x)*a*b**2*d**2*e**2*x**2 - 4*sqrt(a)*log(2*sqrt(a)*sq 
rt(a + b*x + c*x**2) - 2*a - b*x)*a*c**2*d**4*x**2 + sqrt(a)*log(2*sqrt(a) 
*sqrt(a + b*x + c*x**2) - 2*a - b*x)*b**3*d**3*e*x**2 - sqrt(a)*log(2*sqrt 
(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*b**2*c*d**4*x**2 - 8*sqrt(a)*log(x 
)*a**3*e**4*x**2 + 12*sqrt(a)*log(x)*a**2*b*d*e**3*x**2 - 4*sqrt(a)*log(x) 
*a**2*c*d**2*e**2*x**2 - 3*sqrt(a)*log(x)*a*b**2*d**2*e**2*x**2 + 4*sqr...