\(\int \frac {(f+g x) \sqrt {a+b x+c x^2}}{(d+e x)^5} \, dx\) [878]

Optimal result
Mathematica [A] (verified)
Rubi [A] (verified)
Maple [B] (verified)
Fricas [B] (verification not implemented)
Sympy [F]
Maxima [F(-2)]
Giac [F(-1)]
Mupad [F(-1)]
Reduce [B] (verification not implemented)

Optimal result

Integrand size = 27, antiderivative size = 392 \[ \int \frac {(f+g x) \sqrt {a+b x+c x^2}}{(d+e x)^5} \, dx=\frac {\left (16 c^2 d^2 f+b e (5 b e f+3 b d g-8 a e g)-4 c (a e (e f-5 d g)+2 b d (2 e f+d g))\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{64 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}-\frac {(e f-d g) \left (a+b x+c x^2\right )^{3/2}}{4 \left (c d^2-b d e+a e^2\right ) (d+e x)^4}-\frac {(2 c d (5 e f-d g)-e (5 b e f+3 b d g-8 a e g)) \left (a+b x+c x^2\right )^{3/2}}{24 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}-\frac {\left (b^2-4 a c\right ) \left (16 c^2 d^2 f+b e (5 b e f+3 b d g-8 a e g)-4 c (a e (e f-5 d g)+2 b d (2 e f+d g))\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{128 \left (c d^2-b d e+a e^2\right )^{7/2}} \] Output:

1/64*(16*c^2*d^2*f+b*e*(-8*a*e*g+3*b*d*g+5*b*e*f)-4*c*(a*e*(-5*d*g+e*f)+2* 
b*d*(d*g+2*e*f)))*(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)^3/(e*x+d)^2-1/4*(-d*g+e*f)*(c*x^2+b*x+a)^(3/2)/(a*e^2-b*d*e+c*d 
^2)/(e*x+d)^4-1/24*(2*c*d*(-d*g+5*e*f)-e*(-8*a*e*g+3*b*d*g+5*b*e*f))*(c*x^ 
2+b*x+a)^(3/2)/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^3-1/128*(-4*a*c+b^2)*(16*c^2* 
d^2*f+b*e*(-8*a*e*g+3*b*d*g+5*b*e*f)-4*c*(a*e*(-5*d*g+e*f)+2*b*d*(d*g+2*e* 
f)))*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)^(7/2)
 

Mathematica [A] (verified)

Time = 11.35 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.84 \[ \int \frac {(f+g x) \sqrt {a+b x+c x^2}}{(d+e x)^5} \, dx=\frac {\frac {6 \left (c d^2+e (-b d+a e)\right ) (-e f+d g) (a+x (b+c x))^{3/2}}{(d+e x)^4}+\frac {(2 c d (-5 e f+d g)+e (5 b e f+3 b d g-8 a e g)) (a+x (b+c x))^{3/2}}{(d+e x)^3}+3 \left (8 c^2 d^2 f+\frac {1}{2} b e (5 b e f+3 b d g-8 a e g)-2 c (a e (e f-5 d g)+2 b d (2 e f+d g))\right ) \left (\frac {\sqrt {a+x (b+c x)} (-2 a e+2 c d x+b (d-e x))}{4 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}+\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{8 \left (c d^2+e (-b d+a e)\right )^{3/2}}\right )}{24 \left (c d^2+e (-b d+a e)\right )^2} \] Input:

Integrate[((f + g*x)*Sqrt[a + b*x + c*x^2])/(d + e*x)^5,x]
 

Output:

((6*(c*d^2 + e*(-(b*d) + a*e))*(-(e*f) + d*g)*(a + x*(b + c*x))^(3/2))/(d 
+ e*x)^4 + ((2*c*d*(-5*e*f + d*g) + e*(5*b*e*f + 3*b*d*g - 8*a*e*g))*(a + 
x*(b + c*x))^(3/2))/(d + e*x)^3 + 3*(8*c^2*d^2*f + (b*e*(5*b*e*f + 3*b*d*g 
 - 8*a*e*g))/2 - 2*c*(a*e*(e*f - 5*d*g) + 2*b*d*(2*e*f + d*g)))*((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))))/(24*(c*d^2 + e*(-(b*d) + a*e))^2)
 

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1237, 27, 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.

\(\displaystyle \int \frac {(f+g x) \sqrt {a+b x+c x^2}}{(d+e x)^5} \, dx\)

\(\Big \downarrow \) 1237

\(\displaystyle -\frac {\int -\frac {(8 c d f-5 b e f-3 b d g+8 a e g-2 c (e f-d g) x) \sqrt {c x^2+b x+a}}{2 (d+e x)^4}dx}{4 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} (e f-d g)}{4 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(8 c d f-5 b e f-3 b d g+8 a e g-2 c (e f-d g) x) \sqrt {c x^2+b x+a}}{(d+e x)^4}dx}{8 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} (e f-d g)}{4 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {\frac {\left (-4 c (a e (e f-5 d g)+2 b d (d g+2 e f))+b e (-8 a e g+3 b d g+5 b e f)+16 c^2 d^2 f\right ) \int \frac {\sqrt {c x^2+b x+a}}{(d+e x)^3}dx}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d (5 e f-d g)-e (-8 a e g+3 b d g+5 b e f))}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}}{8 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} (e f-d g)}{4 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1152

\(\displaystyle \frac {\frac {\left (-4 c (a e (e f-5 d g)+2 b d (d g+2 e f))+b e (-8 a e g+3 b d g+5 b e f)+16 c^2 d^2 f\right ) \left (\frac {\sqrt {a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d (5 e f-d g)-e (-8 a e g+3 b d g+5 b e f))}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}}{8 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} (e f-d g)}{4 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {\left (-4 c (a e (e f-5 d g)+2 b d (d g+2 e f))+b e (-8 a e g+3 b d g+5 b e f)+16 c^2 d^2 f\right ) \left (\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d (5 e f-d g)-e (-8 a e g+3 b d g+5 b e f))}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}}{8 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} (e f-d g)}{4 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\left (\frac {\sqrt {a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{3/2}}\right ) \left (-4 c (a e (e f-5 d g)+2 b d (d g+2 e f))+b e (-8 a e g+3 b d g+5 b e f)+16 c^2 d^2 f\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d (5 e f-d g)-e (-8 a e g+3 b d g+5 b e f))}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}}{8 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} (e f-d g)}{4 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}\)

Input:

Int[((f + g*x)*Sqrt[a + b*x + c*x^2])/(d + e*x)^5,x]
 

Output:

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

Defintions of rubi rules used

rule 27
Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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]
 

rule 1237
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)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) 
*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ 
(c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m 
+ 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 
] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 
Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3787\) vs. \(2(370)=740\).

Time = 3.29 (sec) , antiderivative size = 3788, normalized size of antiderivative = 9.66

method result size
default \(\text {Expression too large to display}\) \(3788\)

Input:

int((g*x+f)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^5,x,method=_RETURNVERBOSE)
 

Output:

g/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^2+(b*e-2*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2824 vs. \(2 (370) = 740\).

Time = 156.21 (sec) , antiderivative size = 5690, normalized size of antiderivative = 14.52 \[ \int \frac {(f+g x) \sqrt {a+b x+c x^2}}{(d+e x)^5} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^5,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {(f+g x) \sqrt {a+b x+c x^2}}{(d+e x)^5} \, dx=\int \frac {\left (f + g x\right ) \sqrt {a + b x + c x^{2}}}{\left (d + e x\right )^{5}}\, dx \] Input:

integrate((g*x+f)*(c*x**2+b*x+a)**(1/2)/(e*x+d)**5,x)
 

Output:

Integral((f + g*x)*sqrt(a + b*x + c*x**2)/(d + e*x)**5, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(f+g x) \sqrt {a+b x+c x^2}}{(d+e x)^5} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((g*x+f)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^5,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 [F(-1)]

Timed out. \[ \int \frac {(f+g x) \sqrt {a+b x+c x^2}}{(d+e x)^5} \, dx=\text {Timed out} \] Input:

integrate((g*x+f)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^5,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) \sqrt {a+b x+c x^2}}{(d+e x)^5} \, dx=\int \frac {\left (f+g\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (d+e\,x\right )}^5} \,d x \] Input:

int(((f + g*x)*(a + b*x + c*x^2)^(1/2))/(d + e*x)^5,x)
 

Output:

int(((f + g*x)*(a + b*x + c*x^2)^(1/2))/(d + e*x)^5, x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 12.29 (sec) , antiderivative size = 12044, normalized size of antiderivative = 30.72 \[ \int \frac {(f+g x) \sqrt {a+b x+c x^2}}{(d+e x)^5} \, dx =\text {Too large to display} \] Input:

int((g*x+f)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^5,x)
 

Output:

( - 96*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*b*c*d**4*e**2* 
g - 384*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*b*c*d**3*e**3 
*g*x - 576*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*b*c*d**2*e 
**4*g*x**2 - 384*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*b*c* 
d*e**5*g*x**3 - 96*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*b* 
c*e**6*g*x**4 + 240*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**5*e*g - 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**4*e**2*f + 960*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**4*e**2*g*x - 192*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**3*f*x + 1440*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...