\(\int (d+e x)^2 (f+g x) (a+b x+c x^2)^{5/4} \, dx\) [1107]

Optimal result

Integrand size = 27, antiderivative size = 488 \[ \int (d+e x)^2 (f+g x) \left (a+b x+c x^2\right )^{5/4} \, dx=-\frac {5 \left (b^2-4 a c\right ) \left (88 c^3 d^2 f-17 b^3 e^2 g+2 b c e (12 a e g+13 b (e f+2 d g))-4 c^2 (11 b d (2 e f+d g)+4 a e (e f+2 d g))\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{7392 c^5}+\frac {\left (88 c^3 d^2 f-17 b^3 e^2 g+2 b c e (12 a e g+13 b (e f+2 d g))-4 c^2 (11 b d (2 e f+d g)+4 a e (e f+2 d g))\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{616 c^4}+\frac {2 g (d+e x)^2 \left (a+b x+c x^2\right )^{9/4}}{13 c}+\frac {\left (221 b^2 e^2 g+88 c^2 d (13 e f+2 d g)-2 c e (88 a e g+169 b (e f+2 d g))+18 c e (26 c e f+8 c d g-17 b e g) x\right ) \left (a+b x+c x^2\right )^{9/4}}{2574 c^3}+\frac {5 \left (-b^2+4 a c\right )^{5/2} \left (88 c^3 d^2 f-17 b^3 e^2 g+2 b c e (12 a e g+13 b (e f+2 d g))-4 c^2 (11 b d (2 e f+d g)+4 a e (e f+2 d g))\right ) \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right ),2\right )}{7392 \sqrt {2} c^6 \left (a+b x+c x^2\right )^{3/4}} \] Output:

-5/7392*(-4*a*c+b^2)*(88*c^3*d^2*f-17*b^3*e^2*g+2*b*c*e*(12*a*e*g+13*b*(2* 
d*g+e*f))-4*c^2*(11*b*d*(d*g+2*e*f)+4*a*e*(2*d*g+e*f)))*(2*c*x+b)*(c*x^2+b 
*x+a)^(1/4)/c^5+1/616*(88*c^3*d^2*f-17*b^3*e^2*g+2*b*c*e*(12*a*e*g+13*b*(2 
*d*g+e*f))-4*c^2*(11*b*d*(d*g+2*e*f)+4*a*e*(2*d*g+e*f)))*(2*c*x+b)*(c*x^2+ 
b*x+a)^(5/4)/c^4+2/13*g*(e*x+d)^2*(c*x^2+b*x+a)^(9/4)/c+1/2574*(221*b^2*e^ 
2*g+88*c^2*d*(2*d*g+13*e*f)-2*c*e*(88*a*e*g+169*b*(2*d*g+e*f))+18*c*e*(-17 
*b*e*g+8*c*d*g+26*c*e*f)*x)*(c*x^2+b*x+a)^(9/4)/c^3+5/14784*(4*a*c-b^2)^(5 
/2)*(88*c^3*d^2*f-17*b^3*e^2*g+2*b*c*e*(12*a*e*g+13*b*(2*d*g+e*f))-4*c^2*( 
11*b*d*(d*g+2*e*f)+4*a*e*(2*d*g+e*f)))*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(3/ 
4)*InverseJacobiAM(1/2*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2)),2^(1/2))*2^(1/2 
)/c^6/(c*x^2+b*x+a)^(3/4)
 

Mathematica [A] (warning: unable to verify)

Time = 11.46 (sec) , antiderivative size = 342, normalized size of antiderivative = 0.70 \[ \int (d+e x)^2 (f+g x) \left (a+b x+c x^2\right )^{5/4} \, dx=\frac {88704 c^5 g (d+e x)^2 (a+x (b+c x))^3+224 c^3 (a+x (b+c x))^3 \left (221 b^2 e^2 g+4 c^2 \left (286 d e f+44 d^2 g+117 e^2 f x+36 d e g x\right )-2 c e (88 a e g+b (169 e f+338 d g+153 e g x))\right )+39 \left (88 c^3 d^2 f-17 b^3 e^2 g+2 b c e (12 a e g+13 b (e f+2 d g))-4 c^2 (11 b d (2 e f+d g)+4 a e (e f+2 d g))\right ) \left (2 c (b+2 c x) \left (32 a^2 c+a \left (-5 b^2+44 b c x+44 c^2 x^2\right )+x \left (-5 b^3+7 b^2 c x+24 b c^2 x^2+12 c^3 x^3\right )\right )+5 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \left (\frac {c (a+x (b+c x))}{-b^2+4 a c}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ),2\right )\right )}{576576 c^6 (a+x (b+c x))^{3/4}} \] Input:

Integrate[(d + e*x)^2*(f + g*x)*(a + b*x + c*x^2)^(5/4),x]
 

Output:

(88704*c^5*g*(d + e*x)^2*(a + x*(b + c*x))^3 + 224*c^3*(a + x*(b + c*x))^3 
*(221*b^2*e^2*g + 4*c^2*(286*d*e*f + 44*d^2*g + 117*e^2*f*x + 36*d*e*g*x) 
- 2*c*e*(88*a*e*g + b*(169*e*f + 338*d*g + 153*e*g*x))) + 39*(88*c^3*d^2*f 
 - 17*b^3*e^2*g + 2*b*c*e*(12*a*e*g + 13*b*(e*f + 2*d*g)) - 4*c^2*(11*b*d* 
(2*e*f + d*g) + 4*a*e*(e*f + 2*d*g)))*(2*c*(b + 2*c*x)*(32*a^2*c + a*(-5*b 
^2 + 44*b*c*x + 44*c^2*x^2) + x*(-5*b^3 + 7*b^2*c*x + 24*b*c^2*x^2 + 12*c^ 
3*x^3)) + 5*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a 
*c))^(3/4)*EllipticF[ArcSin[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]/2, 2]))/(576576 
*c^6*(a + x*(b + c*x))^(3/4))
 

Rubi [A] (warning: unable to verify)

Time = 1.12 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1236, 27, 1225, 1087, 1087, 1094, 761}

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.

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {2 \int \frac {1}{4} (d+e x) (26 c d f-9 b d g-8 a e g+(26 c e f+8 c d g-17 b e g) x) \left (c x^2+b x+a\right )^{5/4}dx}{13 c}+\frac {2 g (d+e x)^2 \left (a+b x+c x^2\right )^{9/4}}{13 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (d+e x) (26 c d f-9 b d g-8 a e g+(26 c e f+8 c d g-17 b e g) x) \left (c x^2+b x+a\right )^{5/4}dx}{26 c}+\frac {2 g (d+e x)^2 \left (a+b x+c x^2\right )^{9/4}}{13 c}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {\frac {13 \left (-4 c^2 (4 a e (2 d g+e f)+11 b d (d g+2 e f))+2 b c e (12 a e g+13 b (2 d g+e f))-17 b^3 e^2 g+88 c^3 d^2 f\right ) \int \left (c x^2+b x+a\right )^{5/4}dx}{44 c^2}+\frac {\left (a+b x+c x^2\right )^{9/4} \left (-2 c e (88 a e g+169 b (2 d g+e f))+221 b^2 e^2 g+18 c e x (-17 b e g+8 c d g+26 c e f)+88 c^2 d (2 d g+13 e f)\right )}{99 c^2}}{26 c}+\frac {2 g (d+e x)^2 \left (a+b x+c x^2\right )^{9/4}}{13 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\frac {13 \left (-4 c^2 (4 a e (2 d g+e f)+11 b d (d g+2 e f))+2 b c e (12 a e g+13 b (2 d g+e f))-17 b^3 e^2 g+88 c^3 d^2 f\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{7 c}-\frac {5 \left (b^2-4 a c\right ) \int \sqrt [4]{c x^2+b x+a}dx}{28 c}\right )}{44 c^2}+\frac {\left (a+b x+c x^2\right )^{9/4} \left (-2 c e (88 a e g+169 b (2 d g+e f))+221 b^2 e^2 g+18 c e x (-17 b e g+8 c d g+26 c e f)+88 c^2 d (2 d g+13 e f)\right )}{99 c^2}}{26 c}+\frac {2 g (d+e x)^2 \left (a+b x+c x^2\right )^{9/4}}{13 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\frac {13 \left (-4 c^2 (4 a e (2 d g+e f)+11 b d (d g+2 e f))+2 b c e (12 a e g+13 b (2 d g+e f))-17 b^3 e^2 g+88 c^3 d^2 f\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{7 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt [4]{a+b x+c x^2}}{3 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\left (c x^2+b x+a\right )^{3/4}}dx}{12 c}\right )}{28 c}\right )}{44 c^2}+\frac {\left (a+b x+c x^2\right )^{9/4} \left (-2 c e (88 a e g+169 b (2 d g+e f))+221 b^2 e^2 g+18 c e x (-17 b e g+8 c d g+26 c e f)+88 c^2 d (2 d g+13 e f)\right )}{99 c^2}}{26 c}+\frac {2 g (d+e x)^2 \left (a+b x+c x^2\right )^{9/4}}{13 c}\)

\(\Big \downarrow \) 1094

\(\displaystyle \frac {\frac {13 \left (-4 c^2 (4 a e (2 d g+e f)+11 b d (d g+2 e f))+2 b c e (12 a e g+13 b (2 d g+e f))-17 b^3 e^2 g+88 c^3 d^2 f\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{7 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt [4]{a+b x+c x^2}}{3 c}-\frac {\left (b^2-4 a c\right ) \sqrt {(b+2 c x)^2} \int \frac {1}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{3 c (b+2 c x)}\right )}{28 c}\right )}{44 c^2}+\frac {\left (a+b x+c x^2\right )^{9/4} \left (-2 c e (88 a e g+169 b (2 d g+e f))+221 b^2 e^2 g+18 c e x (-17 b e g+8 c d g+26 c e f)+88 c^2 d (2 d g+13 e f)\right )}{99 c^2}}{26 c}+\frac {2 g (d+e x)^2 \left (a+b x+c x^2\right )^{9/4}}{13 c}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {\frac {13 \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{7 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt [4]{a+b x+c x^2}}{3 c}-\frac {\left (b^2-4 a c\right )^{5/4} \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{6 \sqrt {2} c^{5/4} (b+2 c x) \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}\right )}{28 c}\right ) \left (-4 c^2 (4 a e (2 d g+e f)+11 b d (d g+2 e f))+2 b c e (12 a e g+13 b (2 d g+e f))-17 b^3 e^2 g+88 c^3 d^2 f\right )}{44 c^2}+\frac {\left (a+b x+c x^2\right )^{9/4} \left (-2 c e (88 a e g+169 b (2 d g+e f))+221 b^2 e^2 g+18 c e x (-17 b e g+8 c d g+26 c e f)+88 c^2 d (2 d g+13 e f)\right )}{99 c^2}}{26 c}+\frac {2 g (d+e x)^2 \left (a+b x+c x^2\right )^{9/4}}{13 c}\)

Input:

Int[(d + e*x)^2*(f + g*x)*(a + b*x + c*x^2)^(5/4),x]
 

Output:

(2*g*(d + e*x)^2*(a + b*x + c*x^2)^(9/4))/(13*c) + (((221*b^2*e^2*g + 88*c 
^2*d*(13*e*f + 2*d*g) - 2*c*e*(88*a*e*g + 169*b*(e*f + 2*d*g)) + 18*c*e*(2 
6*c*e*f + 8*c*d*g - 17*b*e*g)*x)*(a + b*x + c*x^2)^(9/4))/(99*c^2) + (13*( 
88*c^3*d^2*f - 17*b^3*e^2*g + 2*b*c*e*(12*a*e*g + 13*b*(e*f + 2*d*g)) - 4* 
c^2*(11*b*d*(2*e*f + d*g) + 4*a*e*(e*f + 2*d*g)))*(((b + 2*c*x)*(a + b*x + 
 c*x^2)^(5/4))/(7*c) - (5*(b^2 - 4*a*c)*(((b + 2*c*x)*(a + b*x + c*x^2)^(1 
/4))/(3*c) - ((b^2 - 4*a*c)^(5/4)*Sqrt[(b + 2*c*x)^2]*(1 + (2*Sqrt[c]*Sqrt 
[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])*Sqrt[(b^2 - 4*a*c + 4*c*(a + b*x + c 
*x^2))/((b^2 - 4*a*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4* 
a*c])^2)]*EllipticF[2*ArcTan[(Sqrt[2]*c^(1/4)*(a + b*x + c*x^2)^(1/4))/(b^ 
2 - 4*a*c)^(1/4)], 1/2])/(6*Sqrt[2]*c^(5/4)*(b + 2*c*x)*Sqrt[b^2 - 4*a*c + 
 4*c*(a + b*x + c*x^2)])))/(28*c)))/(44*c^2))/(26*c)
 

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 761

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 
1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* 
EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 1094

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[4*(Sqrt[(b 
+ 2*c*x)^2]/(b + 2*c*x))   Subst[Int[x^(4*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4 
*c*x^4], x], x, (a + b*x + c*x^2)^(1/4)], x] /; FreeQ[{a, b, c}, x] && Inte 
gerQ[4*p]
 

rule 1225

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

rule 1236

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

Maple [F]

\[\int \left (e x +d \right )^{2} \left (g x +f \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{4}}d x\]

Input:

int((e*x+d)^2*(g*x+f)*(c*x^2+b*x+a)^(5/4),x)
 

Output:

int((e*x+d)^2*(g*x+f)*(c*x^2+b*x+a)^(5/4),x)
 

Fricas [F]

\[ \int (d+e x)^2 (f+g x) \left (a+b x+c x^2\right )^{5/4} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {5}{4}} {\left (e x + d\right )}^{2} {\left (g x + f\right )} \,d x } \] Input:

integrate((e*x+d)^2*(g*x+f)*(c*x^2+b*x+a)^(5/4),x, algorithm="fricas")
 

Output:

integral((c*e^2*g*x^5 + (c*e^2*f + (2*c*d*e + b*e^2)*g)*x^4 + a*d^2*f + (( 
2*c*d*e + b*e^2)*f + (c*d^2 + 2*b*d*e + a*e^2)*g)*x^3 + ((c*d^2 + 2*b*d*e 
+ a*e^2)*f + (b*d^2 + 2*a*d*e)*g)*x^2 + (a*d^2*g + (b*d^2 + 2*a*d*e)*f)*x) 
*(c*x^2 + b*x + a)^(1/4), x)
 

Sympy [F]

\[ \int (d+e x)^2 (f+g x) \left (a+b x+c x^2\right )^{5/4} \, dx=\int \left (d + e x\right )^{2} \left (f + g x\right ) \left (a + b x + c x^{2}\right )^{\frac {5}{4}}\, dx \] Input:

integrate((e*x+d)**2*(g*x+f)*(c*x**2+b*x+a)**(5/4),x)
 

Output:

Integral((d + e*x)**2*(f + g*x)*(a + b*x + c*x**2)**(5/4), x)
 

Maxima [F]

\[ \int (d+e x)^2 (f+g x) \left (a+b x+c x^2\right )^{5/4} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {5}{4}} {\left (e x + d\right )}^{2} {\left (g x + f\right )} \,d x } \] Input:

integrate((e*x+d)^2*(g*x+f)*(c*x^2+b*x+a)^(5/4),x, algorithm="maxima")
 

Output:

integrate((c*x^2 + b*x + a)^(5/4)*(e*x + d)^2*(g*x + f), x)
 

Giac [F]

\[ \int (d+e x)^2 (f+g x) \left (a+b x+c x^2\right )^{5/4} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {5}{4}} {\left (e x + d\right )}^{2} {\left (g x + f\right )} \,d x } \] Input:

integrate((e*x+d)^2*(g*x+f)*(c*x^2+b*x+a)^(5/4),x, algorithm="giac")
 

Output:

integrate((c*x^2 + b*x + a)^(5/4)*(e*x + d)^2*(g*x + f), x)
 

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^2 (f+g x) \left (a+b x+c x^2\right )^{5/4} \, dx=\int \left (f+g\,x\right )\,{\left (d+e\,x\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{5/4} \,d x \] Input:

int((f + g*x)*(d + e*x)^2*(a + b*x + c*x^2)^(5/4),x)
 

Output:

int((f + g*x)*(d + e*x)^2*(a + b*x + c*x^2)^(5/4), x)
 

Reduce [F]

\[ \int (d+e x)^2 (f+g x) \left (a+b x+c x^2\right )^{5/4} \, dx=\text {too large to display} \] Input:

int((e*x+d)^2*(g*x+f)*(c*x^2+b*x+a)^(5/4),x)
                                                                                    
                                                                                    
 

Output:

(110336*(a + b*x + c*x**2)**(1/4)*a**3*b*c**2*e**2*g - 199680*(a + b*x + c 
*x**2)**(1/4)*a**3*c**3*d*e*g - 99840*(a + b*x + c*x**2)**(1/4)*a**3*c**3* 
e**2*f - 71552*(a + b*x + c*x**2)**(1/4)*a**2*b**3*c*e**2*g + 193024*(a + 
b*x + c*x**2)**(1/4)*a**2*b**2*c**2*d*e*g + 96512*(a + b*x + c*x**2)**(1/4 
)*a**2*b**2*c**2*e**2*f - 27584*(a + b*x + c*x**2)**(1/4)*a**2*b**2*c**2*e 
**2*g*x - 146432*(a + b*x + c*x**2)**(1/4)*a**2*b*c**3*d**2*g - 292864*(a 
+ b*x + c*x**2)**(1/4)*a**2*b*c**3*d*e*f + 49920*(a + b*x + c*x**2)**(1/4) 
*a**2*b*c**3*d*e*g*x + 24960*(a + b*x + c*x**2)**(1/4)*a**2*b*c**3*e**2*f* 
x + 9856*(a + b*x + c*x**2)**(1/4)*a**2*b*c**3*e**2*g*x**2 + 549120*(a + b 
*x + c*x**2)**(1/4)*a**2*c**4*d**2*f + 10608*(a + b*x + c*x**2)**(1/4)*a*b 
**5*e**2*g - 32448*(a + b*x + c*x**2)**(1/4)*a*b**4*c*d*e*g - 16224*(a + b 
*x + c*x**2)**(1/4)*a*b**4*c*e**2*f + 17888*(a + b*x + c*x**2)**(1/4)*a*b* 
*4*c*e**2*g*x + 27456*(a + b*x + c*x**2)**(1/4)*a*b**3*c**2*d**2*g + 54912 
*(a + b*x + c*x**2)**(1/4)*a*b**3*c**2*d*e*f - 48256*(a + b*x + c*x**2)**( 
1/4)*a*b**3*c**2*d*e*g*x - 24128*(a + b*x + c*x**2)**(1/4)*a*b**3*c**2*e** 
2*f*x - 10112*(a + b*x + c*x**2)**(1/4)*a*b**3*c**2*e**2*g*x**2 - 54912*(a 
 + b*x + c*x**2)**(1/4)*a*b**2*c**3*d**2*f + 36608*(a + b*x + c*x**2)**(1/ 
4)*a*b**2*c**3*d**2*g*x + 73216*(a + b*x + c*x**2)**(1/4)*a*b**2*c**3*d*e* 
f*x + 26624*(a + b*x + c*x**2)**(1/4)*a*b**2*c**3*d*e*g*x**2 + 13312*(a + 
b*x + c*x**2)**(1/4)*a*b**2*c**3*e**2*f*x**2 + 6400*(a + b*x + c*x**2)*...