Integrand size = 17, antiderivative size = 100 \[ \int (a+b x)^3 (c+d x)^{5/2} \, dx=-\frac {2 (b c-a d)^3 (c+d x)^{7/2}}{7 d^4}+\frac {2 b (b c-a d)^2 (c+d x)^{9/2}}{3 d^4}-\frac {6 b^2 (b c-a d) (c+d x)^{11/2}}{11 d^4}+\frac {2 b^3 (c+d x)^{13/2}}{13 d^4} \] Output:
-2/7*(-a*d+b*c)^3*(d*x+c)^(7/2)/d^4+2/3*b*(-a*d+b*c)^2*(d*x+c)^(9/2)/d^4-6 /11*b^2*(-a*d+b*c)*(d*x+c)^(11/2)/d^4+2/13*b^3*(d*x+c)^(13/2)/d^4
Time = 0.07 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.02 \[ \int (a+b x)^3 (c+d x)^{5/2} \, dx=\frac {2 (c+d x)^{7/2} \left (429 a^3 d^3+143 a^2 b d^2 (-2 c+7 d x)+13 a b^2 d \left (8 c^2-28 c d x+63 d^2 x^2\right )+b^3 \left (-16 c^3+56 c^2 d x-126 c d^2 x^2+231 d^3 x^3\right )\right )}{3003 d^4} \] Input:
Integrate[(a + b*x)^3*(c + d*x)^(5/2),x]
Output:
(2*(c + d*x)^(7/2)*(429*a^3*d^3 + 143*a^2*b*d^2*(-2*c + 7*d*x) + 13*a*b^2* d*(8*c^2 - 28*c*d*x + 63*d^2*x^2) + b^3*(-16*c^3 + 56*c^2*d*x - 126*c*d^2* x^2 + 231*d^3*x^3)))/(3003*d^4)
Time = 0.21 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {53, 2009}
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.
| \(\displaystyle \int (a+b x)^3 (c+d x)^{5/2} \, dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {3 b^2 (c+d x)^{9/2} (b c-a d)}{d^3}+\frac {3 b (c+d x)^{7/2} (b c-a d)^2}{d^3}+\frac {(c+d x)^{5/2} (a d-b c)^3}{d^3}+\frac {b^3 (c+d x)^{11/2}}{d^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {6 b^2 (c+d x)^{11/2} (b c-a d)}{11 d^4}+\frac {2 b (c+d x)^{9/2} (b c-a d)^2}{3 d^4}-\frac {2 (c+d x)^{7/2} (b c-a d)^3}{7 d^4}+\frac {2 b^3 (c+d x)^{13/2}}{13 d^4}\) |
Input:
Int[(a + b*x)^3*(c + d*x)^(5/2),x]
Output:
(-2*(b*c - a*d)^3*(c + d*x)^(7/2))/(7*d^4) + (2*b*(b*c - a*d)^2*(c + d*x)^ (9/2))/(3*d^4) - (6*b^2*(b*c - a*d)*(c + d*x)^(11/2))/(11*d^4) + (2*b^3*(c + d*x)^(13/2))/(13*d^4)
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Time = 0.32 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.78
| method | result | size |
| derivativedivides | \(\frac {\frac {2 b^{3} \left (x d +c \right )^{\frac {13}{2}}}{13}+\frac {6 \left (a d -b c \right ) b^{2} \left (x d +c \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a d -b c \right )^{2} b \left (x d +c \right )^{\frac {9}{2}}}{3}+\frac {2 \left (a d -b c \right )^{3} \left (x d +c \right )^{\frac {7}{2}}}{7}}{d^{4}}\) | \(78\) |
| default | \(\frac {\frac {2 b^{3} \left (x d +c \right )^{\frac {13}{2}}}{13}+\frac {6 \left (a d -b c \right ) b^{2} \left (x d +c \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a d -b c \right )^{2} b \left (x d +c \right )^{\frac {9}{2}}}{3}+\frac {2 \left (a d -b c \right )^{3} \left (x d +c \right )^{\frac {7}{2}}}{7}}{d^{4}}\) | \(78\) |
| pseudoelliptic | \(\frac {2 \left (x d +c \right )^{\frac {7}{2}} \left (\left (\frac {7}{13} x^{3} d^{3}-\frac {42}{143} c \,d^{2} x^{2}+\frac {56}{429} c^{2} d x -\frac {16}{429} c^{3}\right ) b^{3}+\frac {8 a d \left (\frac {63}{8} d^{2} x^{2}-\frac {7}{2} c d x +c^{2}\right ) b^{2}}{33}-\frac {2 a^{2} \left (-\frac {7 x d}{2}+c \right ) d^{2} b}{3}+a^{3} d^{3}\right )}{7 d^{4}}\) | \(94\) |
| gosper | \(\frac {2 \left (x d +c \right )^{\frac {7}{2}} \left (231 d^{3} x^{3} b^{3}+819 x^{2} a \,b^{2} d^{3}-126 x^{2} b^{3} c \,d^{2}+1001 x \,a^{2} b \,d^{3}-364 x a \,b^{2} c \,d^{2}+56 x \,b^{3} c^{2} d +429 a^{3} d^{3}-286 a^{2} b c \,d^{2}+104 a \,b^{2} c^{2} d -16 b^{3} c^{3}\right )}{3003 d^{4}}\) | \(116\) |
| orering | \(\frac {2 \left (x d +c \right )^{\frac {7}{2}} \left (231 d^{3} x^{3} b^{3}+819 x^{2} a \,b^{2} d^{3}-126 x^{2} b^{3} c \,d^{2}+1001 x \,a^{2} b \,d^{3}-364 x a \,b^{2} c \,d^{2}+56 x \,b^{3} c^{2} d +429 a^{3} d^{3}-286 a^{2} b c \,d^{2}+104 a \,b^{2} c^{2} d -16 b^{3} c^{3}\right )}{3003 d^{4}}\) | \(116\) |
| trager | \(\frac {2 \left (231 b^{3} d^{6} x^{6}+819 a \,b^{2} d^{6} x^{5}+567 b^{3} c \,d^{5} x^{5}+1001 a^{2} b \,d^{6} x^{4}+2093 a \,b^{2} c \,d^{5} x^{4}+371 b^{3} c^{2} d^{4} x^{4}+429 a^{3} d^{6} x^{3}+2717 a^{2} b c \,d^{5} x^{3}+1469 a \,b^{2} c^{2} d^{4} x^{3}+5 b^{3} c^{3} d^{3} x^{3}+1287 a^{3} c \,d^{5} x^{2}+2145 a^{2} b \,c^{2} d^{4} x^{2}+39 a \,b^{2} c^{3} d^{3} x^{2}-6 b^{3} c^{4} d^{2} x^{2}+1287 a^{3} c^{2} d^{4} x +143 a^{2} b \,c^{3} d^{3} x -52 a \,b^{2} c^{4} d^{2} x +8 b^{3} c^{5} d x +429 a^{3} c^{3} d^{3}-286 a^{2} b \,c^{4} d^{2}+104 a \,b^{2} c^{5} d -16 b^{3} c^{6}\right ) \sqrt {x d +c}}{3003 d^{4}}\) | \(286\) |
| risch | \(\frac {2 \left (231 b^{3} d^{6} x^{6}+819 a \,b^{2} d^{6} x^{5}+567 b^{3} c \,d^{5} x^{5}+1001 a^{2} b \,d^{6} x^{4}+2093 a \,b^{2} c \,d^{5} x^{4}+371 b^{3} c^{2} d^{4} x^{4}+429 a^{3} d^{6} x^{3}+2717 a^{2} b c \,d^{5} x^{3}+1469 a \,b^{2} c^{2} d^{4} x^{3}+5 b^{3} c^{3} d^{3} x^{3}+1287 a^{3} c \,d^{5} x^{2}+2145 a^{2} b \,c^{2} d^{4} x^{2}+39 a \,b^{2} c^{3} d^{3} x^{2}-6 b^{3} c^{4} d^{2} x^{2}+1287 a^{3} c^{2} d^{4} x +143 a^{2} b \,c^{3} d^{3} x -52 a \,b^{2} c^{4} d^{2} x +8 b^{3} c^{5} d x +429 a^{3} c^{3} d^{3}-286 a^{2} b \,c^{4} d^{2}+104 a \,b^{2} c^{5} d -16 b^{3} c^{6}\right ) \sqrt {x d +c}}{3003 d^{4}}\) | \(286\) |
Input:
int((b*x+a)^3*(d*x+c)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/d^4*(1/13*b^3*(d*x+c)^(13/2)+3/11*(a*d-b*c)*b^2*(d*x+c)^(11/2)+1/3*(a*d- b*c)^2*b*(d*x+c)^(9/2)+1/7*(a*d-b*c)^3*(d*x+c)^(7/2))
Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (84) = 168\).
Time = 0.07 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.68 \[ \int (a+b x)^3 (c+d x)^{5/2} \, dx=\frac {2 \, {\left (231 \, b^{3} d^{6} x^{6} - 16 \, b^{3} c^{6} + 104 \, a b^{2} c^{5} d - 286 \, a^{2} b c^{4} d^{2} + 429 \, a^{3} c^{3} d^{3} + 63 \, {\left (9 \, b^{3} c d^{5} + 13 \, a b^{2} d^{6}\right )} x^{5} + 7 \, {\left (53 \, b^{3} c^{2} d^{4} + 299 \, a b^{2} c d^{5} + 143 \, a^{2} b d^{6}\right )} x^{4} + {\left (5 \, b^{3} c^{3} d^{3} + 1469 \, a b^{2} c^{2} d^{4} + 2717 \, a^{2} b c d^{5} + 429 \, a^{3} d^{6}\right )} x^{3} - 3 \, {\left (2 \, b^{3} c^{4} d^{2} - 13 \, a b^{2} c^{3} d^{3} - 715 \, a^{2} b c^{2} d^{4} - 429 \, a^{3} c d^{5}\right )} x^{2} + {\left (8 \, b^{3} c^{5} d - 52 \, a b^{2} c^{4} d^{2} + 143 \, a^{2} b c^{3} d^{3} + 1287 \, a^{3} c^{2} d^{4}\right )} x\right )} \sqrt {d x + c}}{3003 \, d^{4}} \] Input:
integrate((b*x+a)^3*(d*x+c)^(5/2),x, algorithm="fricas")
Output:
2/3003*(231*b^3*d^6*x^6 - 16*b^3*c^6 + 104*a*b^2*c^5*d - 286*a^2*b*c^4*d^2 + 429*a^3*c^3*d^3 + 63*(9*b^3*c*d^5 + 13*a*b^2*d^6)*x^5 + 7*(53*b^3*c^2*d ^4 + 299*a*b^2*c*d^5 + 143*a^2*b*d^6)*x^4 + (5*b^3*c^3*d^3 + 1469*a*b^2*c^ 2*d^4 + 2717*a^2*b*c*d^5 + 429*a^3*d^6)*x^3 - 3*(2*b^3*c^4*d^2 - 13*a*b^2* c^3*d^3 - 715*a^2*b*c^2*d^4 - 429*a^3*c*d^5)*x^2 + (8*b^3*c^5*d - 52*a*b^2 *c^4*d^2 + 143*a^2*b*c^3*d^3 + 1287*a^3*c^2*d^4)*x)*sqrt(d*x + c)/d^4
Leaf count of result is larger than twice the leaf count of optimal. 549 vs. \(2 (92) = 184\).
Time = 0.42 (sec) , antiderivative size = 549, normalized size of antiderivative = 5.49 \[ \int (a+b x)^3 (c+d x)^{5/2} \, dx=\begin {cases} \frac {2 a^{3} c^{3} \sqrt {c + d x}}{7 d} + \frac {6 a^{3} c^{2} x \sqrt {c + d x}}{7} + \frac {6 a^{3} c d x^{2} \sqrt {c + d x}}{7} + \frac {2 a^{3} d^{2} x^{3} \sqrt {c + d x}}{7} - \frac {4 a^{2} b c^{4} \sqrt {c + d x}}{21 d^{2}} + \frac {2 a^{2} b c^{3} x \sqrt {c + d x}}{21 d} + \frac {10 a^{2} b c^{2} x^{2} \sqrt {c + d x}}{7} + \frac {38 a^{2} b c d x^{3} \sqrt {c + d x}}{21} + \frac {2 a^{2} b d^{2} x^{4} \sqrt {c + d x}}{3} + \frac {16 a b^{2} c^{5} \sqrt {c + d x}}{231 d^{3}} - \frac {8 a b^{2} c^{4} x \sqrt {c + d x}}{231 d^{2}} + \frac {2 a b^{2} c^{3} x^{2} \sqrt {c + d x}}{77 d} + \frac {226 a b^{2} c^{2} x^{3} \sqrt {c + d x}}{231} + \frac {46 a b^{2} c d x^{4} \sqrt {c + d x}}{33} + \frac {6 a b^{2} d^{2} x^{5} \sqrt {c + d x}}{11} - \frac {32 b^{3} c^{6} \sqrt {c + d x}}{3003 d^{4}} + \frac {16 b^{3} c^{5} x \sqrt {c + d x}}{3003 d^{3}} - \frac {4 b^{3} c^{4} x^{2} \sqrt {c + d x}}{1001 d^{2}} + \frac {10 b^{3} c^{3} x^{3} \sqrt {c + d x}}{3003 d} + \frac {106 b^{3} c^{2} x^{4} \sqrt {c + d x}}{429} + \frac {54 b^{3} c d x^{5} \sqrt {c + d x}}{143} + \frac {2 b^{3} d^{2} x^{6} \sqrt {c + d x}}{13} & \text {for}\: d \neq 0 \\c^{\frac {5}{2}} \left (a^{3} x + \frac {3 a^{2} b x^{2}}{2} + a b^{2} x^{3} + \frac {b^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a)**3*(d*x+c)**(5/2),x)
Output:
Piecewise((2*a**3*c**3*sqrt(c + d*x)/(7*d) + 6*a**3*c**2*x*sqrt(c + d*x)/7 + 6*a**3*c*d*x**2*sqrt(c + d*x)/7 + 2*a**3*d**2*x**3*sqrt(c + d*x)/7 - 4* a**2*b*c**4*sqrt(c + d*x)/(21*d**2) + 2*a**2*b*c**3*x*sqrt(c + d*x)/(21*d) + 10*a**2*b*c**2*x**2*sqrt(c + d*x)/7 + 38*a**2*b*c*d*x**3*sqrt(c + d*x)/ 21 + 2*a**2*b*d**2*x**4*sqrt(c + d*x)/3 + 16*a*b**2*c**5*sqrt(c + d*x)/(23 1*d**3) - 8*a*b**2*c**4*x*sqrt(c + d*x)/(231*d**2) + 2*a*b**2*c**3*x**2*sq rt(c + d*x)/(77*d) + 226*a*b**2*c**2*x**3*sqrt(c + d*x)/231 + 46*a*b**2*c* d*x**4*sqrt(c + d*x)/33 + 6*a*b**2*d**2*x**5*sqrt(c + d*x)/11 - 32*b**3*c* *6*sqrt(c + d*x)/(3003*d**4) + 16*b**3*c**5*x*sqrt(c + d*x)/(3003*d**3) - 4*b**3*c**4*x**2*sqrt(c + d*x)/(1001*d**2) + 10*b**3*c**3*x**3*sqrt(c + d* x)/(3003*d) + 106*b**3*c**2*x**4*sqrt(c + d*x)/429 + 54*b**3*c*d*x**5*sqrt (c + d*x)/143 + 2*b**3*d**2*x**6*sqrt(c + d*x)/13, Ne(d, 0)), (c**(5/2)*(a **3*x + 3*a**2*b*x**2/2 + a*b**2*x**3 + b**3*x**4/4), True))
Time = 0.03 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.18 \[ \int (a+b x)^3 (c+d x)^{5/2} \, dx=\frac {2 \, {\left (231 \, {\left (d x + c\right )}^{\frac {13}{2}} b^{3} - 819 \, {\left (b^{3} c - a b^{2} d\right )} {\left (d x + c\right )}^{\frac {11}{2}} + 1001 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} {\left (d x + c\right )}^{\frac {9}{2}} - 429 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left (d x + c\right )}^{\frac {7}{2}}\right )}}{3003 \, d^{4}} \] Input:
integrate((b*x+a)^3*(d*x+c)^(5/2),x, algorithm="maxima")
Output:
2/3003*(231*(d*x + c)^(13/2)*b^3 - 819*(b^3*c - a*b^2*d)*(d*x + c)^(11/2) + 1001*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*(d*x + c)^(9/2) - 429*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(d*x + c)^(7/2))/d^4
Leaf count of result is larger than twice the leaf count of optimal. 857 vs. \(2 (84) = 168\).
Time = 0.14 (sec) , antiderivative size = 857, normalized size of antiderivative = 8.57 \[ \int (a+b x)^3 (c+d x)^{5/2} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^3*(d*x+c)^(5/2),x, algorithm="giac")
Output:
2/15015*(15015*sqrt(d*x + c)*a^3*c^3 + 15015*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^3*c^2 + 15015*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^2*b*c^3/d + 3003*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)* a^3*c + 3003*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)* c^2)*a*b^2*c^3/d^2 + 9009*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*s qrt(d*x + c)*c^2)*a^2*b*c^2/d + 429*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/2 )*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*a^3 + 429*(5*(d*x + c )^(7/2) - 21*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c) *c^3)*b^3*c^3/d^3 + 3861*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/2)*c + 35*(d *x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*a*b^2*c^2/d^2 + 3861*(5*(d*x + c )^(7/2) - 21*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c) *c^3)*a^2*b*c/d + 143*(35*(d*x + c)^(9/2) - 180*(d*x + c)^(7/2)*c + 378*(d *x + c)^(5/2)*c^2 - 420*(d*x + c)^(3/2)*c^3 + 315*sqrt(d*x + c)*c^4)*b^3*c ^2/d^3 + 429*(35*(d*x + c)^(9/2) - 180*(d*x + c)^(7/2)*c + 378*(d*x + c)^( 5/2)*c^2 - 420*(d*x + c)^(3/2)*c^3 + 315*sqrt(d*x + c)*c^4)*a*b^2*c/d^2 + 143*(35*(d*x + c)^(9/2) - 180*(d*x + c)^(7/2)*c + 378*(d*x + c)^(5/2)*c^2 - 420*(d*x + c)^(3/2)*c^3 + 315*sqrt(d*x + c)*c^4)*a^2*b/d + 65*(63*(d*x + c)^(11/2) - 385*(d*x + c)^(9/2)*c + 990*(d*x + c)^(7/2)*c^2 - 1386*(d*x + c)^(5/2)*c^3 + 1155*(d*x + c)^(3/2)*c^4 - 693*sqrt(d*x + c)*c^5)*b^3*c/d^ 3 + 65*(63*(d*x + c)^(11/2) - 385*(d*x + c)^(9/2)*c + 990*(d*x + c)^(7/...
Time = 0.15 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.87 \[ \int (a+b x)^3 (c+d x)^{5/2} \, dx=\frac {2\,b^3\,{\left (c+d\,x\right )}^{13/2}}{13\,d^4}-\frac {\left (6\,b^3\,c-6\,a\,b^2\,d\right )\,{\left (c+d\,x\right )}^{11/2}}{11\,d^4}+\frac {2\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{7/2}}{7\,d^4}+\frac {2\,b\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{9/2}}{3\,d^4} \] Input:
int((a + b*x)^3*(c + d*x)^(5/2),x)
Output:
(2*b^3*(c + d*x)^(13/2))/(13*d^4) - ((6*b^3*c - 6*a*b^2*d)*(c + d*x)^(11/2 ))/(11*d^4) + (2*(a*d - b*c)^3*(c + d*x)^(7/2))/(7*d^4) + (2*b*(a*d - b*c) ^2*(c + d*x)^(9/2))/(3*d^4)
Time = 0.16 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.84 \[ \int (a+b x)^3 (c+d x)^{5/2} \, dx=\frac {2 \sqrt {d x +c}\, \left (231 b^{3} d^{6} x^{6}+819 a \,b^{2} d^{6} x^{5}+567 b^{3} c \,d^{5} x^{5}+1001 a^{2} b \,d^{6} x^{4}+2093 a \,b^{2} c \,d^{5} x^{4}+371 b^{3} c^{2} d^{4} x^{4}+429 a^{3} d^{6} x^{3}+2717 a^{2} b c \,d^{5} x^{3}+1469 a \,b^{2} c^{2} d^{4} x^{3}+5 b^{3} c^{3} d^{3} x^{3}+1287 a^{3} c \,d^{5} x^{2}+2145 a^{2} b \,c^{2} d^{4} x^{2}+39 a \,b^{2} c^{3} d^{3} x^{2}-6 b^{3} c^{4} d^{2} x^{2}+1287 a^{3} c^{2} d^{4} x +143 a^{2} b \,c^{3} d^{3} x -52 a \,b^{2} c^{4} d^{2} x +8 b^{3} c^{5} d x +429 a^{3} c^{3} d^{3}-286 a^{2} b \,c^{4} d^{2}+104 a \,b^{2} c^{5} d -16 b^{3} c^{6}\right )}{3003 d^{4}} \] Input:
int((b*x+a)^3*(d*x+c)^(5/2),x)
Output:
(2*sqrt(c + d*x)*(429*a**3*c**3*d**3 + 1287*a**3*c**2*d**4*x + 1287*a**3*c *d**5*x**2 + 429*a**3*d**6*x**3 - 286*a**2*b*c**4*d**2 + 143*a**2*b*c**3*d **3*x + 2145*a**2*b*c**2*d**4*x**2 + 2717*a**2*b*c*d**5*x**3 + 1001*a**2*b *d**6*x**4 + 104*a*b**2*c**5*d - 52*a*b**2*c**4*d**2*x + 39*a*b**2*c**3*d* *3*x**2 + 1469*a*b**2*c**2*d**4*x**3 + 2093*a*b**2*c*d**5*x**4 + 819*a*b** 2*d**6*x**5 - 16*b**3*c**6 + 8*b**3*c**5*d*x - 6*b**3*c**4*d**2*x**2 + 5*b **3*c**3*d**3*x**3 + 371*b**3*c**2*d**4*x**4 + 567*b**3*c*d**5*x**5 + 231* b**3*d**6*x**6))/(3003*d**4)