\(\int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^6} \, dx\) [160]

Optimal result

Integrand size = 26, antiderivative size = 79 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^6} \, dx=\frac {2 \left (a+b x+c x^2\right )^{3/2}}{5 \left (b^2-4 a c\right ) d^6 (b+2 c x)^5}+\frac {4 \left (a+b x+c x^2\right )^{3/2}}{15 \left (b^2-4 a c\right )^2 d^6 (b+2 c x)^3} \] Output:

2/5*(c*x^2+b*x+a)^(3/2)/(-4*a*c+b^2)/d^6/(2*c*x+b)^5+4/15*(c*x^2+b*x+a)^(3 
/2)/(-4*a*c+b^2)^2/d^6/(2*c*x+b)^3
 

Mathematica [A] (verified)

Time = 10.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^6} \, dx=\frac {2 (a+x (b+c x))^{3/2} \left (5 b^2+8 b c x+4 c \left (-3 a+2 c x^2\right )\right )}{15 \left (b^2-4 a c\right )^2 d^6 (b+2 c x)^5} \] Input:

Integrate[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x)^6,x]
 

Output:

(2*(a + x*(b + c*x))^(3/2)*(5*b^2 + 8*b*c*x + 4*c*(-3*a + 2*c*x^2)))/(15*( 
b^2 - 4*a*c)^2*d^6*(b + 2*c*x)^5)
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1117, 27, 1106}

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.

\(\displaystyle \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^6} \, dx\)

\(\Big \downarrow \) 1117

\(\displaystyle \frac {2 \int \frac {\sqrt {c x^2+b x+a}}{d^4 (b+2 c x)^4}dx}{5 d^2 \left (b^2-4 a c\right )}+\frac {2 \left (a+b x+c x^2\right )^{3/2}}{5 d^6 \left (b^2-4 a c\right ) (b+2 c x)^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \int \frac {\sqrt {c x^2+b x+a}}{(b+2 c x)^4}dx}{5 d^6 \left (b^2-4 a c\right )}+\frac {2 \left (a+b x+c x^2\right )^{3/2}}{5 d^6 \left (b^2-4 a c\right ) (b+2 c x)^5}\)

\(\Big \downarrow \) 1106

\(\displaystyle \frac {4 \left (a+b x+c x^2\right )^{3/2}}{15 d^6 \left (b^2-4 a c\right )^2 (b+2 c x)^3}+\frac {2 \left (a+b x+c x^2\right )^{3/2}}{5 d^6 \left (b^2-4 a c\right ) (b+2 c x)^5}\)

Input:

Int[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x)^6,x]
 

Output:

(2*(a + b*x + c*x^2)^(3/2))/(5*(b^2 - 4*a*c)*d^6*(b + 2*c*x)^5) + (4*(a + 
b*x + c*x^2)^(3/2))/(15*(b^2 - 4*a*c)^2*d^6*(b + 2*c*x)^3)
 

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 1106

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S 
ymbol] :> Simp[2*c*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)* 
(b^2 - 4*a*c))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[2*c*d - b*e, 
0] && EqQ[m + 2*p + 3, 0]
 

rule 1117

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S 
ymbol] :> Simp[-2*b*d*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(d^2*(m 
+ 1)*(b^2 - 4*a*c))), x] + Simp[b^2*((m + 2*p + 3)/(d^2*(m + 1)*(b^2 - 4*a* 
c)))   Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, 
 d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[m, -1] & 
& (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || IntegerQ[(m + 2*p + 3) 
/2])
 

Maple [A] (verified)

Time = 1.47 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89

method	result	size
gosper	\(-\frac {2 \left (-8 c^{2} x^{2}-8 c b x +12 a c -5 b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{15 \left (2 c x +b \right )^{5} d^{6} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}\)	\(70\)
orering	\(-\frac {2 \left (-8 c^{2} x^{2}-8 c b x +12 a c -5 b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (2 c x +b \right )}{15 \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) \left (2 c d x +b d \right )^{6}}\)	\(76\)
trager	\(-\frac {2 \left (-8 c^{3} x^{4}-16 b \,c^{2} x^{3}+4 a \,c^{2} x^{2}-13 b^{2} c \,x^{2}+4 a b c x -5 b^{3} x +12 a^{2} c -5 a \,b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{15 d^{6} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) \left (2 c x +b \right )^{5}}\)	\(107\)
default	\(\frac {-\frac {4 c \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{5 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{5}}+\frac {32 c^{3} \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{15 \left (4 a c -b^{2}\right )^{2} \left (x +\frac {b}{2 c}\right )^{3}}}{64 d^{6} c^{6}}\)	\(122\)

Input:

int((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d)^6,x,method=_RETURNVERBOSE)
 

Output:

-2/15*(-8*c^2*x^2-8*b*c*x+12*a*c-5*b^2)*(c*x^2+b*x+a)^(3/2)/(2*c*x+b)^5/d^ 
6/(16*a^2*c^2-8*a*b^2*c+b^4)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (71) = 142\).

Time = 1.15 (sec) , antiderivative size = 274, normalized size of antiderivative = 3.47 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^6} \, dx=\frac {2 \, {\left (8 \, c^{3} x^{4} + 16 \, b c^{2} x^{3} + 5 \, a b^{2} - 12 \, a^{2} c + {\left (13 \, b^{2} c - 4 \, a c^{2}\right )} x^{2} + {\left (5 \, b^{3} - 4 \, a b c\right )} x\right )} \sqrt {c x^{2} + b x + a}}{15 \, {\left (32 \, {\left (b^{4} c^{5} - 8 \, a b^{2} c^{6} + 16 \, a^{2} c^{7}\right )} d^{6} x^{5} + 80 \, {\left (b^{5} c^{4} - 8 \, a b^{3} c^{5} + 16 \, a^{2} b c^{6}\right )} d^{6} x^{4} + 80 \, {\left (b^{6} c^{3} - 8 \, a b^{4} c^{4} + 16 \, a^{2} b^{2} c^{5}\right )} d^{6} x^{3} + 40 \, {\left (b^{7} c^{2} - 8 \, a b^{5} c^{3} + 16 \, a^{2} b^{3} c^{4}\right )} d^{6} x^{2} + 10 \, {\left (b^{8} c - 8 \, a b^{6} c^{2} + 16 \, a^{2} b^{4} c^{3}\right )} d^{6} x + {\left (b^{9} - 8 \, a b^{7} c + 16 \, a^{2} b^{5} c^{2}\right )} d^{6}\right )}} \] Input:

integrate((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d)^6,x, algorithm="fricas")
 

Output:

2/15*(8*c^3*x^4 + 16*b*c^2*x^3 + 5*a*b^2 - 12*a^2*c + (13*b^2*c - 4*a*c^2) 
*x^2 + (5*b^3 - 4*a*b*c)*x)*sqrt(c*x^2 + b*x + a)/(32*(b^4*c^5 - 8*a*b^2*c 
^6 + 16*a^2*c^7)*d^6*x^5 + 80*(b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*d^6*x 
^4 + 80*(b^6*c^3 - 8*a*b^4*c^4 + 16*a^2*b^2*c^5)*d^6*x^3 + 40*(b^7*c^2 - 8 
*a*b^5*c^3 + 16*a^2*b^3*c^4)*d^6*x^2 + 10*(b^8*c - 8*a*b^6*c^2 + 16*a^2*b^ 
4*c^3)*d^6*x + (b^9 - 8*a*b^7*c + 16*a^2*b^5*c^2)*d^6)
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^6} \, dx=\frac {\int \frac {\sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx}{d^{6}} \] Input:

integrate((c*x**2+b*x+a)**(1/2)/(2*c*d*x+b*d)**6,x)
 

Output:

Integral(sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x + 60*b**4*c**2*x**2 + 
160*b**3*c**3*x**3 + 240*b**2*c**4*x**4 + 192*b*c**5*x**5 + 64*c**6*x**6), 
 x)/d**6
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^6} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d)^6,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(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 417 vs. \(2 (71) = 142\).

Time = 0.45 (sec) , antiderivative size = 417, normalized size of antiderivative = 5.28 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^6} \, dx=\frac {60 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} c^{3} + 180 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} b c^{\frac {5}{2}} + 220 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} b^{2} c^{2} + 20 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} a c^{3} + 140 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b^{3} c^{\frac {3}{2}} + 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a b c^{\frac {5}{2}} + 50 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{4} c + 20 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a b^{2} c^{2} + 20 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{2} c^{3} + 10 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{5} \sqrt {c} + 20 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{2} b c^{\frac {5}{2}} + b^{6} - 2 \, a b^{4} c + 8 \, a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}}{30 \, {\left (2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b \sqrt {c} + b^{2} - 2 \, a c\right )}^{5} c^{\frac {3}{2}} d^{6}} \] Input:

integrate((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d)^6,x, algorithm="giac")
 

Output:

1/30*(60*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*c^3 + 180*(sqrt(c)*x - sqrt 
(c*x^2 + b*x + a))^5*b*c^(5/2) + 220*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4 
*b^2*c^2 + 20*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*c^3 + 140*(sqrt(c)*x 
 - sqrt(c*x^2 + b*x + a))^3*b^3*c^(3/2) + 40*(sqrt(c)*x - sqrt(c*x^2 + b*x 
 + a))^3*a*b*c^(5/2) + 50*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^4*c + 20 
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^2*c^2 + 20*(sqrt(c)*x - sqrt(c* 
x^2 + b*x + a))^2*a^2*c^3 + 10*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^5*sqr 
t(c) + 20*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b*c^(5/2) + b^6 - 2*a*b^ 
4*c + 8*a^2*b^2*c^2 - 4*a^3*c^3)/((2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2 
*c + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*sqrt(c) + b^2 - 2*a*c)^5*c^(3 
/2)*d^6)
 

Mupad [B] (verification not implemented)

Time = 5.74 (sec) , antiderivative size = 589, normalized size of antiderivative = 7.46 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^6} \, dx=\frac {\left (\frac {b\,\left (\frac {4\,c^2\,\left (2\,b^2+4\,a\,c\right )}{15\,d^6\,{\left (4\,a\,c-b^2\right )}^2\,\left (32\,a\,c^3-8\,b^2\,c^2\right )}-\frac {8\,b^2\,c^2}{15\,d^6\,{\left (4\,a\,c-b^2\right )}^2\,\left (32\,a\,c^3-8\,b^2\,c^2\right )}\right )}{2\,c}-\frac {8\,a\,b\,c^2}{15\,d^6\,{\left (4\,a\,c-b^2\right )}^2\,\left (32\,a\,c^3-8\,b^2\,c^2\right )}\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (b+2\,c\,x\right )}^2}-\frac {\left (\frac {2\,a}{15\,d^6\,{\left (4\,a\,c-b^2\right )}^3}-\frac {b^2}{30\,c\,d^6\,{\left (4\,a\,c-b^2\right )}^3}\right )\,\sqrt {c\,x^2+b\,x+a}}{b+2\,c\,x}+\frac {\left (\frac {b\,\left (\frac {4\,c^2\,\left (2\,b^2+4\,a\,c\right )}{5\,d^6\,\left (4\,a\,c-b^2\right )\,\left (64\,a\,c^3-16\,b^2\,c^2\right )}-\frac {8\,b^2\,c^2}{5\,d^6\,\left (4\,a\,c-b^2\right )\,\left (64\,a\,c^3-16\,b^2\,c^2\right )}\right )}{2\,c}-\frac {8\,a\,b\,c^2}{5\,d^6\,\left (4\,a\,c-b^2\right )\,\left (64\,a\,c^3-16\,b^2\,c^2\right )}\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (b+2\,c\,x\right )}^4}-\frac {\left (\frac {8\,a\,c^2}{d^6\,\left (80\,a\,c^3-20\,b^2\,c^2\right )}-\frac {2\,b^2\,c}{d^6\,\left (80\,a\,c^3-20\,b^2\,c^2\right )}\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (b+2\,c\,x\right )}^5}-\frac {\left (\frac {8\,a\,c^2}{5\,d^6\,\left (4\,a\,c-b^2\right )\,\left (48\,a\,c^3-12\,b^2\,c^2\right )}-\frac {2\,b^2\,c}{5\,d^6\,\left (4\,a\,c-b^2\right )\,\left (48\,a\,c^3-12\,b^2\,c^2\right )}\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (b+2\,c\,x\right )}^3}+\frac {\sqrt {c\,x^2+b\,x+a}}{10\,c\,d^6\,{\left (4\,a\,c-b^2\right )}^2\,\left (b+2\,c\,x\right )} \] Input:

int((a + b*x + c*x^2)^(1/2)/(b*d + 2*c*d*x)^6,x)
 

Output:

(((b*((4*c^2*(4*a*c + 2*b^2))/(15*d^6*(4*a*c - b^2)^2*(32*a*c^3 - 8*b^2*c^ 
2)) - (8*b^2*c^2)/(15*d^6*(4*a*c - b^2)^2*(32*a*c^3 - 8*b^2*c^2))))/(2*c) 
- (8*a*b*c^2)/(15*d^6*(4*a*c - b^2)^2*(32*a*c^3 - 8*b^2*c^2)))*(a + b*x + 
c*x^2)^(1/2))/(b + 2*c*x)^2 - (((2*a)/(15*d^6*(4*a*c - b^2)^3) - b^2/(30*c 
*d^6*(4*a*c - b^2)^3))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x) + (((b*((4*c^2 
*(4*a*c + 2*b^2))/(5*d^6*(4*a*c - b^2)*(64*a*c^3 - 16*b^2*c^2)) - (8*b^2*c 
^2)/(5*d^6*(4*a*c - b^2)*(64*a*c^3 - 16*b^2*c^2))))/(2*c) - (8*a*b*c^2)/(5 
*d^6*(4*a*c - b^2)*(64*a*c^3 - 16*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 
 2*c*x)^4 - (((8*a*c^2)/(d^6*(80*a*c^3 - 20*b^2*c^2)) - (2*b^2*c)/(d^6*(80 
*a*c^3 - 20*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^5 - (((8*a*c^2 
)/(5*d^6*(4*a*c - b^2)*(48*a*c^3 - 12*b^2*c^2)) - (2*b^2*c)/(5*d^6*(4*a*c 
- b^2)*(48*a*c^3 - 12*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^3 + 
(a + b*x + c*x^2)^(1/2)/(10*c*d^6*(4*a*c - b^2)^2*(b + 2*c*x))
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 427, normalized size of antiderivative = 5.41 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^6} \, dx=\frac {-48 \sqrt {c \,x^{2}+b x +a}\, a^{2} c^{3}+20 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} c^{2}-16 \sqrt {c \,x^{2}+b x +a}\, a b \,c^{3} x -16 \sqrt {c \,x^{2}+b x +a}\, a \,c^{4} x^{2}+20 \sqrt {c \,x^{2}+b x +a}\, b^{3} c^{2} x +52 \sqrt {c \,x^{2}+b x +a}\, b^{2} c^{3} x^{2}+64 \sqrt {c \,x^{2}+b x +a}\, b \,c^{4} x^{3}+32 \sqrt {c \,x^{2}+b x +a}\, c^{5} x^{4}-\sqrt {c}\, b^{5}-10 \sqrt {c}\, b^{4} c x -40 \sqrt {c}\, b^{3} c^{2} x^{2}-80 \sqrt {c}\, b^{2} c^{3} x^{3}-80 \sqrt {c}\, b \,c^{4} x^{4}-32 \sqrt {c}\, c^{5} x^{5}}{30 c^{2} d^{6} \left (512 a^{2} c^{7} x^{5}-256 a \,b^{2} c^{6} x^{5}+32 b^{4} c^{5} x^{5}+1280 a^{2} b \,c^{6} x^{4}-640 a \,b^{3} c^{5} x^{4}+80 b^{5} c^{4} x^{4}+1280 a^{2} b^{2} c^{5} x^{3}-640 a \,b^{4} c^{4} x^{3}+80 b^{6} c^{3} x^{3}+640 a^{2} b^{3} c^{4} x^{2}-320 a \,b^{5} c^{3} x^{2}+40 b^{7} c^{2} x^{2}+160 a^{2} b^{4} c^{3} x -80 a \,b^{6} c^{2} x +10 b^{8} c x +16 a^{2} b^{5} c^{2}-8 a \,b^{7} c +b^{9}\right )} \] Input:

int((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d)^6,x)
 

Output:

( - 48*sqrt(a + b*x + c*x**2)*a**2*c**3 + 20*sqrt(a + b*x + c*x**2)*a*b**2 
*c**2 - 16*sqrt(a + b*x + c*x**2)*a*b*c**3*x - 16*sqrt(a + b*x + c*x**2)*a 
*c**4*x**2 + 20*sqrt(a + b*x + c*x**2)*b**3*c**2*x + 52*sqrt(a + b*x + c*x 
**2)*b**2*c**3*x**2 + 64*sqrt(a + b*x + c*x**2)*b*c**4*x**3 + 32*sqrt(a + 
b*x + c*x**2)*c**5*x**4 - sqrt(c)*b**5 - 10*sqrt(c)*b**4*c*x - 40*sqrt(c)* 
b**3*c**2*x**2 - 80*sqrt(c)*b**2*c**3*x**3 - 80*sqrt(c)*b*c**4*x**4 - 32*s 
qrt(c)*c**5*x**5)/(30*c**2*d**6*(16*a**2*b**5*c**2 + 160*a**2*b**4*c**3*x 
+ 640*a**2*b**3*c**4*x**2 + 1280*a**2*b**2*c**5*x**3 + 1280*a**2*b*c**6*x* 
*4 + 512*a**2*c**7*x**5 - 8*a*b**7*c - 80*a*b**6*c**2*x - 320*a*b**5*c**3* 
x**2 - 640*a*b**4*c**4*x**3 - 640*a*b**3*c**5*x**4 - 256*a*b**2*c**6*x**5 
+ b**9 + 10*b**8*c*x + 40*b**7*c**2*x**2 + 80*b**6*c**3*x**3 + 80*b**5*c** 
4*x**4 + 32*b**4*c**5*x**5))