\(\int (A+B x) \sqrt {c+d x} (c^2-d^2 x^2)^{5/2} \, dx\) [82]

Optimal result

Integrand size = 31, antiderivative size = 201 \[ \int (A+B x) \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2} \, dx=-\frac {256 c^3 (B c+15 A d) \left (c^2-d^2 x^2\right )^{7/2}}{45045 d^2 (c+d x)^{7/2}}-\frac {64 c^2 (B c+15 A d) \left (c^2-d^2 x^2\right )^{7/2}}{6435 d^2 (c+d x)^{5/2}}-\frac {8 c (B c+15 A d) \left (c^2-d^2 x^2\right )^{7/2}}{715 d^2 (c+d x)^{3/2}}-\frac {2 (B c+15 A d) \left (c^2-d^2 x^2\right )^{7/2}}{195 d^2 \sqrt {c+d x}}-\frac {2 B \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{7/2}}{15 d^2} \] Output:

-256/45045*c^3*(15*A*d+B*c)*(-d^2*x^2+c^2)^(7/2)/d^2/(d*x+c)^(7/2)-64/6435 
*c^2*(15*A*d+B*c)*(-d^2*x^2+c^2)^(7/2)/d^2/(d*x+c)^(5/2)-8/715*c*(15*A*d+B 
*c)*(-d^2*x^2+c^2)^(7/2)/d^2/(d*x+c)^(3/2)-2/195*(15*A*d+B*c)*(-d^2*x^2+c^ 
2)^(7/2)/d^2/(d*x+c)^(1/2)-2/15*B*(d*x+c)^(1/2)*(-d^2*x^2+c^2)^(7/2)/d^2
 

Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.59 \[ \int (A+B x) \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2} \, dx=-\frac {2 (c-d x)^3 \sqrt {c^2-d^2 x^2} \left (15 A d \left (835 c^3+1421 c^2 d x+945 c d^2 x^2+231 d^3 x^3\right )+B \left (3838 c^4+13433 c^3 d x+18963 c^2 d^2 x^2+12243 c d^3 x^3+3003 d^4 x^4\right )\right )}{45045 d^2 \sqrt {c+d x}} \] Input:

Integrate[(A + B*x)*Sqrt[c + d*x]*(c^2 - d^2*x^2)^(5/2),x]
 

Output:

(-2*(c - d*x)^3*Sqrt[c^2 - d^2*x^2]*(15*A*d*(835*c^3 + 1421*c^2*d*x + 945* 
c*d^2*x^2 + 231*d^3*x^3) + B*(3838*c^4 + 13433*c^3*d*x + 18963*c^2*d^2*x^2 
 + 12243*c*d^3*x^3 + 3003*d^4*x^4)))/(45045*d^2*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {672, 459, 459, 459, 458}

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.

\(\displaystyle \int (A+B x) \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2} \, dx\)

\(\Big \downarrow \) 672

\(\displaystyle \frac {(15 A d+B c) \int \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2}dx}{15 d}-\frac {2 B \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{7/2}}{15 d^2}\)

\(\Big \downarrow \) 459

\(\displaystyle \frac {(15 A d+B c) \left (\frac {12}{13} c \int \frac {\left (c^2-d^2 x^2\right )^{5/2}}{\sqrt {c+d x}}dx-\frac {2 \left (c^2-d^2 x^2\right )^{7/2}}{13 d \sqrt {c+d x}}\right )}{15 d}-\frac {2 B \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{7/2}}{15 d^2}\)

\(\Big \downarrow \) 459

\(\displaystyle \frac {(15 A d+B c) \left (\frac {12}{13} c \left (\frac {8}{11} c \int \frac {\left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^{3/2}}dx-\frac {2 \left (c^2-d^2 x^2\right )^{7/2}}{11 d (c+d x)^{3/2}}\right )-\frac {2 \left (c^2-d^2 x^2\right )^{7/2}}{13 d \sqrt {c+d x}}\right )}{15 d}-\frac {2 B \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{7/2}}{15 d^2}\)

\(\Big \downarrow \) 459

\(\displaystyle \frac {(15 A d+B c) \left (\frac {12}{13} c \left (\frac {8}{11} c \left (\frac {4}{9} c \int \frac {\left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^{5/2}}dx-\frac {2 \left (c^2-d^2 x^2\right )^{7/2}}{9 d (c+d x)^{5/2}}\right )-\frac {2 \left (c^2-d^2 x^2\right )^{7/2}}{11 d (c+d x)^{3/2}}\right )-\frac {2 \left (c^2-d^2 x^2\right )^{7/2}}{13 d \sqrt {c+d x}}\right )}{15 d}-\frac {2 B \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{7/2}}{15 d^2}\)

\(\Big \downarrow \) 458

\(\displaystyle \frac {\left (\frac {12}{13} c \left (\frac {8}{11} c \left (-\frac {2 \left (c^2-d^2 x^2\right )^{7/2}}{9 d (c+d x)^{5/2}}-\frac {8 c \left (c^2-d^2 x^2\right )^{7/2}}{63 d (c+d x)^{7/2}}\right )-\frac {2 \left (c^2-d^2 x^2\right )^{7/2}}{11 d (c+d x)^{3/2}}\right )-\frac {2 \left (c^2-d^2 x^2\right )^{7/2}}{13 d \sqrt {c+d x}}\right ) (15 A d+B c)}{15 d}-\frac {2 B \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{7/2}}{15 d^2}\)

Input:

Int[(A + B*x)*Sqrt[c + d*x]*(c^2 - d^2*x^2)^(5/2),x]
 

Output:

(-2*B*Sqrt[c + d*x]*(c^2 - d^2*x^2)^(7/2))/(15*d^2) + ((B*c + 15*A*d)*((-2 
*(c^2 - d^2*x^2)^(7/2))/(13*d*Sqrt[c + d*x]) + (12*c*((-2*(c^2 - d^2*x^2)^ 
(7/2))/(11*d*(c + d*x)^(3/2)) + (8*c*((-8*c*(c^2 - d^2*x^2)^(7/2))/(63*d*( 
c + d*x)^(7/2)) - (2*(c^2 - d^2*x^2)^(7/2))/(9*d*(c + d*x)^(5/2))))/11))/1 
3))/(15*d)
 

Defintions of rubi rules used

rule 458

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(p + 1))), x] /; FreeQ[{a, b, c 
, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + p, 0]
 

rule 459

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[2*c* 
(Simplify[n + p]/(n + 2*p + 1))   Int[(c + d*x)^(n - 1)*(a + b*x^2)^p, x], 
x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && IGtQ[Simplif 
y[n + p], 0]
 

rule 672

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

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.57

method	result	size
gosper	\(-\frac {2 \left (-d x +c \right ) \left (3003 B \,d^{4} x^{4}+3465 A \,d^{4} x^{3}+12243 B c \,d^{3} x^{3}+14175 A c \,d^{3} x^{2}+18963 x^{2} c^{2} B \,d^{2}+21315 A \,c^{2} d^{2} x +13433 B \,c^{3} d x +12525 A \,c^{3} d +3838 B \,c^{4}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{45045 d^{2} \left (d x +c \right )^{\frac {5}{2}}}\)	\(115\)
orering	\(-\frac {2 \left (-d x +c \right ) \left (3003 B \,d^{4} x^{4}+3465 A \,d^{4} x^{3}+12243 B c \,d^{3} x^{3}+14175 A c \,d^{3} x^{2}+18963 x^{2} c^{2} B \,d^{2}+21315 A \,c^{2} d^{2} x +13433 B \,c^{3} d x +12525 A \,c^{3} d +3838 B \,c^{4}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{45045 d^{2} \left (d x +c \right )^{\frac {5}{2}}}\)	\(115\)
default	\(-\frac {2 \sqrt {-d^{2} x^{2}+c^{2}}\, \left (-d x +c \right )^{3} \left (3003 B \,d^{4} x^{4}+3465 A \,d^{4} x^{3}+12243 B c \,d^{3} x^{3}+14175 A c \,d^{3} x^{2}+18963 x^{2} c^{2} B \,d^{2}+21315 A \,c^{2} d^{2} x +13433 B \,c^{3} d x +12525 A \,c^{3} d +3838 B \,c^{4}\right )}{45045 \sqrt {d x +c}\, d^{2}}\)	\(117\)
risch	\(-\frac {2 \sqrt {\frac {-d^{2} x^{2}+c^{2}}{d x +c}}\, \sqrt {d x +c}\, \left (-3003 B \,d^{7} x^{7}-3465 A \,d^{7} x^{6}-3234 B c \,d^{6} x^{6}-3780 A c \,d^{6} x^{5}+8757 B \,c^{2} d^{5} x^{5}+10815 A \,c^{2} d^{5} x^{4}+9730 B \,c^{3} d^{4} x^{4}+12360 A \,c^{3} d^{4} x^{3}-8185 B \,c^{4} d^{3} x^{3}-12195 c^{4} x^{2} A \,d^{3}-9822 c^{5} x^{2} B \,d^{2}-16260 A \,c^{5} d^{2} x +1919 B \,c^{6} d x +12525 A \,c^{6} d +3838 B \,c^{7}\right ) \sqrt {-d x +c}}{45045 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{2}}\)	\(211\)

Input:

int((B*x+A)*(d*x+c)^(1/2)*(-d^2*x^2+c^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-2/45045*(-d*x+c)*(3003*B*d^4*x^4+3465*A*d^4*x^3+12243*B*c*d^3*x^3+14175*A 
*c*d^3*x^2+18963*B*c^2*d^2*x^2+21315*A*c^2*d^2*x+13433*B*c^3*d*x+12525*A*c 
^3*d+3838*B*c^4)*(-d^2*x^2+c^2)^(5/2)/d^2/(d*x+c)^(5/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.96 \[ \int (A+B x) \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2} \, dx=\frac {2 \, {\left (3003 \, B d^{7} x^{7} - 3838 \, B c^{7} - 12525 \, A c^{6} d + 231 \, {\left (14 \, B c d^{6} + 15 \, A d^{7}\right )} x^{6} - 63 \, {\left (139 \, B c^{2} d^{5} - 60 \, A c d^{6}\right )} x^{5} - 35 \, {\left (278 \, B c^{3} d^{4} + 309 \, A c^{2} d^{5}\right )} x^{4} + 5 \, {\left (1637 \, B c^{4} d^{3} - 2472 \, A c^{3} d^{4}\right )} x^{3} + 3 \, {\left (3274 \, B c^{5} d^{2} + 4065 \, A c^{4} d^{3}\right )} x^{2} - {\left (1919 \, B c^{6} d - 16260 \, A c^{5} d^{2}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c}}{45045 \, {\left (d^{3} x + c d^{2}\right )}} \] Input:

integrate((B*x+A)*(d*x+c)^(1/2)*(-d^2*x^2+c^2)^(5/2),x, algorithm="fricas" 
)
 

Output:

2/45045*(3003*B*d^7*x^7 - 3838*B*c^7 - 12525*A*c^6*d + 231*(14*B*c*d^6 + 1 
5*A*d^7)*x^6 - 63*(139*B*c^2*d^5 - 60*A*c*d^6)*x^5 - 35*(278*B*c^3*d^4 + 3 
09*A*c^2*d^5)*x^4 + 5*(1637*B*c^4*d^3 - 2472*A*c^3*d^4)*x^3 + 3*(3274*B*c^ 
5*d^2 + 4065*A*c^4*d^3)*x^2 - (1919*B*c^6*d - 16260*A*c^5*d^2)*x)*sqrt(-d^ 
2*x^2 + c^2)*sqrt(d*x + c)/(d^3*x + c*d^2)
 

Sympy [F]

\[ \int (A+B x) \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2} \, dx=\int \left (- \left (- c + d x\right ) \left (c + d x\right )\right )^{\frac {5}{2}} \left (A + B x\right ) \sqrt {c + d x}\, dx \] Input:

integrate((B*x+A)*(d*x+c)**(1/2)*(-d**2*x**2+c**2)**(5/2),x)
 

Output:

Integral((-(-c + d*x)*(c + d*x))**(5/2)*(A + B*x)*sqrt(c + d*x), x)
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.97 \[ \int (A+B x) \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2} \, dx=\frac {2 \, {\left (231 \, d^{6} x^{6} + 252 \, c d^{5} x^{5} - 721 \, c^{2} d^{4} x^{4} - 824 \, c^{3} d^{3} x^{3} + 813 \, c^{4} d^{2} x^{2} + 1084 \, c^{5} d x - 835 \, c^{6}\right )} {\left (d x + c\right )} \sqrt {-d x + c} A}{3003 \, {\left (d^{2} x + c d\right )}} + \frac {2 \, {\left (3003 \, d^{7} x^{7} + 3234 \, c d^{6} x^{6} - 8757 \, c^{2} d^{5} x^{5} - 9730 \, c^{3} d^{4} x^{4} + 8185 \, c^{4} d^{3} x^{3} + 9822 \, c^{5} d^{2} x^{2} - 1919 \, c^{6} d x - 3838 \, c^{7}\right )} {\left (d x + c\right )} \sqrt {-d x + c} B}{45045 \, {\left (d^{3} x + c d^{2}\right )}} \] Input:

integrate((B*x+A)*(d*x+c)^(1/2)*(-d^2*x^2+c^2)^(5/2),x, algorithm="maxima" 
)
 

Output:

2/3003*(231*d^6*x^6 + 252*c*d^5*x^5 - 721*c^2*d^4*x^4 - 824*c^3*d^3*x^3 + 
813*c^4*d^2*x^2 + 1084*c^5*d*x - 835*c^6)*(d*x + c)*sqrt(-d*x + c)*A/(d^2* 
x + c*d) + 2/45045*(3003*d^7*x^7 + 3234*c*d^6*x^6 - 8757*c^2*d^5*x^5 - 973 
0*c^3*d^4*x^4 + 8185*c^4*d^3*x^3 + 9822*c^5*d^2*x^2 - 1919*c^6*d*x - 3838* 
c^7)*(d*x + c)*sqrt(-d*x + c)*B/(d^3*x + c*d^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 674 vs. \(2 (171) = 342\).

Time = 0.13 (sec) , antiderivative size = 674, normalized size of antiderivative = 3.35 \[ \int (A+B x) \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(d*x+c)^(1/2)*(-d^2*x^2+c^2)^(5/2),x, algorithm="giac")
 

Output:

-2/45045*(45045*sqrt(-d*x + c)*A*c^6*d - 15015*((-d*x + c)^(3/2) - 3*sqrt( 
-d*x + c)*c)*B*c^6 - 9009*(3*(d*x - c)^2*sqrt(-d*x + c) - 10*(-d*x + c)^(3 
/2)*c + 15*sqrt(-d*x + c)*c^2)*A*c^4*d - 3861*(5*(d*x - c)^3*sqrt(-d*x + c 
) + 21*(d*x - c)^2*sqrt(-d*x + c)*c - 35*(-d*x + c)^(3/2)*c^2 + 35*sqrt(-d 
*x + c)*c^3)*B*c^4 + 429*(35*(d*x - c)^4*sqrt(-d*x + c) + 180*(d*x - c)^3* 
sqrt(-d*x + c)*c + 378*(d*x - c)^2*sqrt(-d*x + c)*c^2 - 420*(-d*x + c)^(3/ 
2)*c^3 + 315*sqrt(-d*x + c)*c^4)*A*c^2*d + 195*(63*(d*x - c)^5*sqrt(-d*x + 
 c) + 385*(d*x - c)^4*sqrt(-d*x + c)*c + 990*(d*x - c)^3*sqrt(-d*x + c)*c^ 
2 + 1386*(d*x - c)^2*sqrt(-d*x + c)*c^3 - 1155*(-d*x + c)^(3/2)*c^4 + 693* 
sqrt(-d*x + c)*c^5)*B*c^2 - 15*(231*(d*x - c)^6*sqrt(-d*x + c) + 1638*(d*x 
 - c)^5*sqrt(-d*x + c)*c + 5005*(d*x - c)^4*sqrt(-d*x + c)*c^2 + 8580*(d*x 
 - c)^3*sqrt(-d*x + c)*c^3 + 9009*(d*x - c)^2*sqrt(-d*x + c)*c^4 - 6006*(- 
d*x + c)^(3/2)*c^5 + 3003*sqrt(-d*x + c)*c^6)*A*d - 7*(429*(d*x - c)^7*sqr 
t(-d*x + c) + 3465*(d*x - c)^6*sqrt(-d*x + c)*c + 12285*(d*x - c)^5*sqrt(- 
d*x + c)*c^2 + 25025*(d*x - c)^4*sqrt(-d*x + c)*c^3 + 32175*(d*x - c)^3*sq 
rt(-d*x + c)*c^4 + 27027*(d*x - c)^2*sqrt(-d*x + c)*c^5 - 15015*(-d*x + c) 
^(3/2)*c^6 + 6435*sqrt(-d*x + c)*c^7)*B)/d^2
 

Mupad [B] (verification not implemented)

Time = 10.11 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.08 \[ \int (A+B x) \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2} \, dx=\frac {\sqrt {c^2-d^2\,x^2}\,\left (\frac {2\,B\,d^4\,x^7\,\sqrt {c+d\,x}}{15}-\frac {\left (7676\,B\,c^7+25050\,A\,d\,c^6\right )\,\sqrt {c+d\,x}}{45045\,d^3}-\frac {2\,c^3\,x^3\,\left (2472\,A\,d-1637\,B\,c\right )\,\sqrt {c+d\,x}}{9009}+\frac {2\,d^3\,x^6\,\left (15\,A\,d+14\,B\,c\right )\,\sqrt {c+d\,x}}{195}+\frac {2\,c\,d^2\,x^5\,\left (60\,A\,d-139\,B\,c\right )\,\sqrt {c+d\,x}}{715}-\frac {2\,c^2\,d\,x^4\,\left (309\,A\,d+278\,B\,c\right )\,\sqrt {c+d\,x}}{1287}+\frac {2\,c^5\,x\,\left (16260\,A\,d-1919\,B\,c\right )\,\sqrt {c+d\,x}}{45045\,d^2}+\frac {2\,c^4\,x^2\,\left (4065\,A\,d+3274\,B\,c\right )\,\sqrt {c+d\,x}}{15015\,d}\right )}{x+\frac {c}{d}} \] Input:

int((c^2 - d^2*x^2)^(5/2)*(A + B*x)*(c + d*x)^(1/2),x)
 

Output:

((c^2 - d^2*x^2)^(1/2)*((2*B*d^4*x^7*(c + d*x)^(1/2))/15 - ((7676*B*c^7 + 
25050*A*c^6*d)*(c + d*x)^(1/2))/(45045*d^3) - (2*c^3*x^3*(2472*A*d - 1637* 
B*c)*(c + d*x)^(1/2))/9009 + (2*d^3*x^6*(15*A*d + 14*B*c)*(c + d*x)^(1/2)) 
/195 + (2*c*d^2*x^5*(60*A*d - 139*B*c)*(c + d*x)^(1/2))/715 - (2*c^2*d*x^4 
*(309*A*d + 278*B*c)*(c + d*x)^(1/2))/1287 + (2*c^5*x*(16260*A*d - 1919*B* 
c)*(c + d*x)^(1/2))/(45045*d^2) + (2*c^4*x^2*(4065*A*d + 3274*B*c)*(c + d* 
x)^(1/2))/(15015*d)))/(x + c/d)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.83 \[ \int (A+B x) \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2} \, dx=\frac {2 \sqrt {-d x +c}\, \left (3003 b \,d^{7} x^{7}+3465 a \,d^{7} x^{6}+3234 b c \,d^{6} x^{6}+3780 a c \,d^{6} x^{5}-8757 b \,c^{2} d^{5} x^{5}-10815 a \,c^{2} d^{5} x^{4}-9730 b \,c^{3} d^{4} x^{4}-12360 a \,c^{3} d^{4} x^{3}+8185 b \,c^{4} d^{3} x^{3}+12195 a \,c^{4} d^{3} x^{2}+9822 b \,c^{5} d^{2} x^{2}+16260 a \,c^{5} d^{2} x -1919 b \,c^{6} d x -12525 a \,c^{6} d -3838 b \,c^{7}\right )}{45045 d^{2}} \] Input:

int((B*x+A)*(d*x+c)^(1/2)*(-d^2*x^2+c^2)^(5/2),x)
 

Output:

(2*sqrt(c - d*x)*( - 12525*a*c**6*d + 16260*a*c**5*d**2*x + 12195*a*c**4*d 
**3*x**2 - 12360*a*c**3*d**4*x**3 - 10815*a*c**2*d**5*x**4 + 3780*a*c*d**6 
*x**5 + 3465*a*d**7*x**6 - 3838*b*c**7 - 1919*b*c**6*d*x + 9822*b*c**5*d** 
2*x**2 + 8185*b*c**4*d**3*x**3 - 9730*b*c**3*d**4*x**4 - 8757*b*c**2*d**5* 
x**5 + 3234*b*c*d**6*x**6 + 3003*b*d**7*x**7))/(45045*d**2)