\(\int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^4} \, dx\) [609]

Optimal result

Integrand size = 28, antiderivative size = 217 \[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^4} \, dx=-\frac {\left (b^2-4 a c\right ) e (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{8 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac {(2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac {\left (b^2-4 a c\right )^2 e \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{16 \left (c d^2-b d e+a e^2\right )^{5/2}} \] Output:

-1/8*(-4*a*c+b^2)*e*(b*d-2*a*e+(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(1/2)/(a*e^2- 
b*d*e+c*d^2)^2/(e*x+d)^2+1/3*(-b*e+2*c*d)*(c*x^2+b*x+a)^(3/2)/(a*e^2-b*d*e 
+c*d^2)/(e*x+d)^3+1/16*(-4*a*c+b^2)^2*e*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))/(a*e^2-b*d*e+c*d^2)^(5 
/2)
 

Mathematica [A] (verified)

Time = 10.48 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.99 \[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^4} \, dx=\frac {\frac {2 (2 c d-b e) (a+x (b+c x))^{3/2}}{(d+e x)^3}-3 \left (b^2-4 a c\right ) e \left (\frac {\sqrt {a+x (b+c x)} (-2 a e+2 c d x+b (d-e x))}{4 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}+\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{8 \left (c d^2+e (-b d+a e)\right )^{3/2}}\right )}{6 \left (c d^2+e (-b d+a e)\right )} \] Input:

Integrate[((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(d + e*x)^4,x]
 

Output:

((2*(2*c*d - b*e)*(a + x*(b + c*x))^(3/2))/(d + e*x)^3 - 3*(b^2 - 4*a*c)*e 
*((Sqrt[a + x*(b + c*x)]*(-2*a*e + 2*c*d*x + b*(d - e*x)))/(4*(c*d^2 + e*( 
-(b*d) + a*e))*(d + e*x)^2) + ((b^2 - 4*a*c)*ArcTanh[(-(b*d) + 2*a*e - 2*c 
*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/( 
8*(c*d^2 + e*(-(b*d) + a*e))^(3/2))))/(6*(c*d^2 + e*(-(b*d) + a*e)))
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1228, 1152, 1154, 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.

\(\displaystyle \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^4} \, dx\)

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1152

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}-\frac {e \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

Int[((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(d + e*x)^4,x]
 

Output:

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

Defintions of rubi rules used

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 1152

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b 
*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a 
*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)))   Int[(d + e*x)^(m + 2)*(a + b*x + 
 c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] 
 && GtQ[p, 0]
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^ 
(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2400\) vs. \(2(199)=398\).

Time = 2.40 (sec) , antiderivative size = 2401, normalized size of antiderivative = 11.06

method	result	size
default	\(\text {Expression too large to display}\)	\(2401\)

Input:

int((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^4,x,method=_RETURNVERBOSE)
 

Output:

(b*e-2*c*d)/e^5*(-1/3/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^3*(c*(x+d/e)^2+(b*e- 
2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)-1/2*(b*e-2*c*d)*e/(a*e^2-b 
*d*e+c*d^2)*(-1/2/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^2*(c*(x+d/e)^2+(b*e-2*c* 
d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)-1/4*(b*e-2*c*d)*e/(a*e^2-b*d*e 
+c*d^2)*(-1/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+ 
d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)+1/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)* 
((c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/2*(b* 
e-2*c*d)/e*ln((1/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+(b*e-2*c* 
d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/c^(1/2)-(a*e^2-b*d*e+c*d^2)/e 
^2/((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)*(c*(x+d/e)^2+(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)))+2*c/(a*e^2-b*d*e+c*d^2)* 
e^2*(1/4*(2*c*(x+d/e)+(b*e-2*c*d)/e)/c*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+ 
(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d 
)^2/e^2)/c^(3/2)*ln((1/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+(b* 
e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))))+1/2*c/(a*e^2-b*d*e+c* 
d^2)*e^2*((c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2 
)+1/2*(b*e-2*c*d)/e*ln((1/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+ 
(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/c^(1/2)-(a*e^2-b*d*e 
+c*d^2)/e^2/((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...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 999 vs. \(2 (199) = 398\).

Time = 1.89 (sec) , antiderivative size = 2040, normalized size of antiderivative = 9.40 \[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^4} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/96*(3*((b^4 - 8*a*b^2*c + 16*a^2*c^2)*e^4*x^3 + 3*(b^4 - 8*a*b^2*c + 16 
*a^2*c^2)*d*e^3*x^2 + 3*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*d^2*e^2*x + (b^4 - 
8*a*b^2*c + 16*a^2*c^2)*d^3*e)*sqrt(c*d^2 - b*d*e + a*e^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)) + 4*(16*a*c^3*d^5 - 8*a^3*b*e^5 - (3*b^3*c + 28* 
a*b*c^2)*d^4*e + (3*b^4 + 26*a*b^2*c + 8*a^2*c^2)*d^3*e^2 - (17*a*b^3 + 12 
*a^2*b*c)*d^2*e^3 + 2*(11*a^2*b^2 - 4*a^3*c)*d*e^4 + (16*c^4*d^5 - 40*b*c^ 
3*d^4*e + 2*(13*b^2*c^2 + 28*a*c^3)*d^3*e^2 + (b^3*c - 84*a*b*c^2)*d^2*e^3 
 - (3*b^4 - 22*a*b^2*c - 40*a^2*c^2)*d*e^4 + (3*a*b^3 - 20*a^2*b*c)*e^5)*x 
^2 + 2*(8*b*c^3*d^5 - (23*b^2*c^2 - 12*a*c^3)*d^4*e + (19*b^3*c + 4*a*b*c^ 
2)*d^3*e^2 - 4*(b^4 + 6*a*b^2*c)*d^2*e^3 + 5*(a*b^3 + 4*a^2*b*c)*d*e^4 - ( 
a^2*b^2 + 12*a^3*c)*e^5)*x)*sqrt(c*x^2 + b*x + a))/(c^3*d^9 - 3*b*c^2*d^8* 
e - 3*a^2*b*d^4*e^5 + a^3*d^3*e^6 + 3*(b^2*c + a*c^2)*d^7*e^2 - (b^3 + 6*a 
*b*c)*d^6*e^3 + 3*(a*b^2 + a^2*c)*d^5*e^4 + (c^3*d^6*e^3 - 3*b*c^2*d^5*e^4 
 - 3*a^2*b*d*e^8 + a^3*e^9 + 3*(b^2*c + a*c^2)*d^4*e^5 - (b^3 + 6*a*b*c)*d 
^3*e^6 + 3*(a*b^2 + a^2*c)*d^2*e^7)*x^3 + 3*(c^3*d^7*e^2 - 3*b*c^2*d^6*e^3 
 - 3*a^2*b*d^2*e^7 + a^3*d*e^8 + 3*(b^2*c + a*c^2)*d^5*e^4 - (b^3 + 6*a*b* 
c)*d^4*e^5 + 3*(a*b^2 + a^2*c)*d^3*e^6)*x^2 + 3*(c^3*d^8*e - 3*b*c^2*d^...
 

Sympy [F]

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

integrate((2*c*x+b)*(c*x**2+b*x+a)**(1/2)/(e*x+d)**4,x)
 

Output:

Integral((b + 2*c*x)*sqrt(a + b*x + c*x**2)/(d + e*x)**4, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^4} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^4,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(a*e^2-b*d*e>0)', see `assume?` f 
or more de
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2677 vs. \(2 (199) = 398\).

Time = 0.48 (sec) , antiderivative size = 2677, normalized size of antiderivative = 12.34 \[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^4} \, dx=\text {Too large to display} \] Input:

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

Output:

1/8*(b^4*e - 8*a*b^2*c*e + 16*a^2*c^2*e)*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))/((c^2*d^4 - 2*b*c 
*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*sqrt(-c*d^2 
+ b*d*e - a*e^2)) + 1/24*(96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*c^4*d^4 
*e^2 - 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b*c^3*d^3*e^3 + 96*(sqrt( 
c)*x - sqrt(c*x^2 + b*x + a))^5*b^2*c^2*d^2*e^4 + 192*(sqrt(c)*x - sqrt(c* 
x^2 + b*x + a))^5*a*c^3*d^2*e^4 - 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^ 
5*a*b*c^2*d*e^5 - 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^4*e^6 + 24*(sq 
rt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^2*c*e^6 + 48*(sqrt(c)*x - sqrt(c*x^ 
2 + b*x + a))^5*a^2*c^2*e^6 + 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*c^ 
(9/2)*d^5*e - 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b*c^(7/2)*d^4*e^2 
- 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^2*c^(5/2)*d^3*e^3 + 384*(sqrt 
(c)*x - sqrt(c*x^2 + b*x + a))^4*a*c^(7/2)*d^3*e^3 + 144*(sqrt(c)*x - sqrt 
(c*x^2 + b*x + a))^4*b^3*c^(3/2)*d^2*e^4 - 96*(sqrt(c)*x - sqrt(c*x^2 + b* 
x + a))^4*a*b*c^(5/2)*d^2*e^4 - 15*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b 
^4*sqrt(c)*d*e^5 - 168*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^2*c^(3/2) 
*d*e^5 - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*c^(5/2)*d*e^5 + 144* 
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b*c^(3/2)*e^6 + 128*(sqrt(c)*x - 
 sqrt(c*x^2 + b*x + a))^3*c^5*d^6 - 272*(sqrt(c)*x - sqrt(c*x^2 + b*x + a) 
)^3*b^2*c^3*d^4*e^2 + 128*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c^4*d...
 

Mupad [F(-1)]

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

int(((b + 2*c*x)*(a + b*x + c*x^2)^(1/2))/(d + e*x)^4,x)
 

Output:

int(((b + 2*c*x)*(a + b*x + c*x^2)^(1/2))/(d + e*x)^4, x)
 

Reduce [B] (verification not implemented)

Time = 1.38 (sec) , antiderivative size = 2669, normalized size of antiderivative = 12.30 \[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^4} \, dx =\text {Too large to display} \] Input:

int((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^4,x)
 

Output:

(48*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**2*c**2*d**3*e + 144 
*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**2*c**2*d**2*e**2*x + 1 
44*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**2*c**2*d*e**3*x**2 + 
 48*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**2*c**2*e**4*x**3 - 
24*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*b**2*c*d**3*e - 72*sq 
rt(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*b**2*c*d**2*e**2*x - 72*sq 
rt(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*b**2*c*d*e**3*x**2 - 24*sq 
rt(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*b**2*c*e**4*x**3 + 3*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)*b**4*d**3*e + 9*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)*b**4*d**2*e**2*x + 9*sqrt(a*e**2 - b*d...