Integrand size = 20, antiderivative size = 156 \[ \int x^3 (c+d x)^{3/2} \left (a+b x^2\right ) \, dx=-\frac {2 c^3 \left (b c^2+a d^2\right ) (c+d x)^{5/2}}{5 d^6}+\frac {2 c^2 \left (5 b c^2+3 a d^2\right ) (c+d x)^{7/2}}{7 d^6}-\frac {2 c \left (10 b c^2+3 a d^2\right ) (c+d x)^{9/2}}{9 d^6}+\frac {2 \left (10 b c^2+a d^2\right ) (c+d x)^{11/2}}{11 d^6}-\frac {10 b c (c+d x)^{13/2}}{13 d^6}+\frac {2 b (c+d x)^{15/2}}{15 d^6} \] Output:
-2/5*c^3*(a*d^2+b*c^2)*(d*x+c)^(5/2)/d^6+2/7*c^2*(3*a*d^2+5*b*c^2)*(d*x+c) ^(7/2)/d^6-2/9*c*(3*a*d^2+10*b*c^2)*(d*x+c)^(9/2)/d^6+2/11*(a*d^2+10*b*c^2 )*(d*x+c)^(11/2)/d^6-10/13*b*c*(d*x+c)^(13/2)/d^6+2/15*b*(d*x+c)^(15/2)/d^ 6
Time = 0.08 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.69 \[ \int x^3 (c+d x)^{3/2} \left (a+b x^2\right ) \, dx=\frac {2 (c+d x)^{5/2} \left (39 a d^2 \left (-16 c^3+40 c^2 d x-70 c d^2 x^2+105 d^3 x^3\right )+b \left (-256 c^5+640 c^4 d x-1120 c^3 d^2 x^2+1680 c^2 d^3 x^3-2310 c d^4 x^4+3003 d^5 x^5\right )\right )}{45045 d^6} \] Input:
Integrate[x^3*(c + d*x)^(3/2)*(a + b*x^2),x]
Output:
(2*(c + d*x)^(5/2)*(39*a*d^2*(-16*c^3 + 40*c^2*d*x - 70*c*d^2*x^2 + 105*d^ 3*x^3) + b*(-256*c^5 + 640*c^4*d*x - 1120*c^3*d^2*x^2 + 1680*c^2*d^3*x^3 - 2310*c*d^4*x^4 + 3003*d^5*x^5)))/(45045*d^6)
Time = 0.48 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {522, 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.
| \(\displaystyle \int x^3 \left (a+b x^2\right ) (c+d x)^{3/2} \, dx\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle \int \left (\frac {(c+d x)^{7/2} \left (-3 a c d^2-10 b c^3\right )}{d^5}+\frac {(c+d x)^{9/2} \left (a d^2+10 b c^2\right )}{d^5}+\frac {(c+d x)^{5/2} \left (3 a c^2 d^2+5 b c^4\right )}{d^5}+\frac {c^3 (c+d x)^{3/2} \left (-a d^2-b c^2\right )}{d^5}+\frac {b (c+d x)^{13/2}}{d^5}-\frac {5 b c (c+d x)^{11/2}}{d^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 (c+d x)^{11/2} \left (a d^2+10 b c^2\right )}{11 d^6}-\frac {2 c (c+d x)^{9/2} \left (3 a d^2+10 b c^2\right )}{9 d^6}+\frac {2 c^2 (c+d x)^{7/2} \left (3 a d^2+5 b c^2\right )}{7 d^6}-\frac {2 c^3 (c+d x)^{5/2} \left (a d^2+b c^2\right )}{5 d^6}+\frac {2 b (c+d x)^{15/2}}{15 d^6}-\frac {10 b c (c+d x)^{13/2}}{13 d^6}\) |
Input:
Int[x^3*(c + d*x)^(3/2)*(a + b*x^2),x]
Output:
(-2*c^3*(b*c^2 + a*d^2)*(c + d*x)^(5/2))/(5*d^6) + (2*c^2*(5*b*c^2 + 3*a*d ^2)*(c + d*x)^(7/2))/(7*d^6) - (2*c*(10*b*c^2 + 3*a*d^2)*(c + d*x)^(9/2))/ (9*d^6) + (2*(10*b*c^2 + a*d^2)*(c + d*x)^(11/2))/(11*d^6) - (10*b*c*(c + d*x)^(13/2))/(13*d^6) + (2*b*(c + d*x)^(15/2))/(15*d^6)
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.60
| method | result | size |
| pseudoelliptic | \(-\frac {32 \left (d x +c \right )^{\frac {5}{2}} \left (-\frac {105 \left (\frac {11 b \,x^{2}}{15}+a \right ) x^{3} d^{5}}{16}+\frac {35 x^{2} \left (\frac {11 b \,x^{2}}{13}+a \right ) c \,d^{4}}{8}-\frac {5 x \left (\frac {14 b \,x^{2}}{13}+a \right ) c^{2} d^{3}}{2}+c^{3} \left (\frac {70 b \,x^{2}}{39}+a \right ) d^{2}-\frac {40 b \,c^{4} d x}{39}+\frac {16 b \,c^{5}}{39}\right )}{1155 d^{6}}\) | \(93\) |
| gosper | \(-\frac {2 \left (d x +c \right )^{\frac {5}{2}} \left (-3003 b \,d^{5} x^{5}+2310 b c \,d^{4} x^{4}-4095 x^{3} a \,d^{5}-1680 b \,c^{2} d^{3} x^{3}+2730 x^{2} a c \,d^{4}+1120 b \,c^{3} d^{2} x^{2}-1560 a \,c^{2} d^{3} x -640 b \,c^{4} d x +624 a \,c^{3} d^{2}+256 b \,c^{5}\right )}{45045 d^{6}}\) | \(109\) |
| orering | \(-\frac {2 \left (d x +c \right )^{\frac {5}{2}} \left (-3003 b \,d^{5} x^{5}+2310 b c \,d^{4} x^{4}-4095 x^{3} a \,d^{5}-1680 b \,c^{2} d^{3} x^{3}+2730 x^{2} a c \,d^{4}+1120 b \,c^{3} d^{2} x^{2}-1560 a \,c^{2} d^{3} x -640 b \,c^{4} d x +624 a \,c^{3} d^{2}+256 b \,c^{5}\right )}{45045 d^{6}}\) | \(109\) |
| derivativedivides | \(\frac {\frac {2 b \left (d x +c \right )^{\frac {15}{2}}}{15}-\frac {10 b c \left (d x +c \right )^{\frac {13}{2}}}{13}+\frac {2 \left (a \,d^{2}+10 b \,c^{2}\right ) \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {2 \left (-7 b \,c^{3}-3 c \left (a \,d^{2}+b \,c^{2}\right )\right ) \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {2 \left (2 b \,c^{4}+3 c^{2} \left (a \,d^{2}+b \,c^{2}\right )\right ) \left (d x +c \right )^{\frac {7}{2}}}{7}-\frac {2 c^{3} \left (a \,d^{2}+b \,c^{2}\right ) \left (d x +c \right )^{\frac {5}{2}}}{5}}{d^{6}}\) | \(134\) |
| default | \(\frac {\frac {2 b \left (d x +c \right )^{\frac {15}{2}}}{15}-\frac {10 b c \left (d x +c \right )^{\frac {13}{2}}}{13}-\frac {2 \left (-a \,d^{2}-10 b \,c^{2}\right ) \left (d x +c \right )^{\frac {11}{2}}}{11}-\frac {2 \left (7 b \,c^{3}+3 c \left (a \,d^{2}+b \,c^{2}\right )\right ) \left (d x +c \right )^{\frac {9}{2}}}{9}-\frac {2 \left (-2 b \,c^{4}-3 c^{2} \left (a \,d^{2}+b \,c^{2}\right )\right ) \left (d x +c \right )^{\frac {7}{2}}}{7}-\frac {2 c^{3} \left (a \,d^{2}+b \,c^{2}\right ) \left (d x +c \right )^{\frac {5}{2}}}{5}}{d^{6}}\) | \(135\) |
| trager | \(-\frac {2 \left (-3003 b \,d^{7} x^{7}-3696 b c \,d^{6} x^{6}-4095 a \,d^{7} x^{5}-63 b \,c^{2} d^{5} x^{5}-5460 a c \,d^{6} x^{4}+70 b \,c^{3} d^{4} x^{4}-195 d^{5} a \,c^{2} x^{3}-80 b \,c^{4} d^{3} x^{3}+234 a \,c^{3} d^{4} x^{2}+96 b \,c^{5} d^{2} x^{2}-312 a \,c^{4} d^{3} x -128 b \,c^{6} d x +624 a \,c^{5} d^{2}+256 b \,c^{7}\right ) \sqrt {d x +c}}{45045 d^{6}}\) | \(157\) |
| risch | \(-\frac {2 \left (-3003 b \,d^{7} x^{7}-3696 b c \,d^{6} x^{6}-4095 a \,d^{7} x^{5}-63 b \,c^{2} d^{5} x^{5}-5460 a c \,d^{6} x^{4}+70 b \,c^{3} d^{4} x^{4}-195 d^{5} a \,c^{2} x^{3}-80 b \,c^{4} d^{3} x^{3}+234 a \,c^{3} d^{4} x^{2}+96 b \,c^{5} d^{2} x^{2}-312 a \,c^{4} d^{3} x -128 b \,c^{6} d x +624 a \,c^{5} d^{2}+256 b \,c^{7}\right ) \sqrt {d x +c}}{45045 d^{6}}\) | \(157\) |
Input:
int(x^3*(d*x+c)^(3/2)*(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
-32/1155*(d*x+c)^(5/2)*(-105/16*(11/15*b*x^2+a)*x^3*d^5+35/8*x^2*(11/13*b* x^2+a)*c*d^4-5/2*x*(14/13*b*x^2+a)*c^2*d^3+c^3*(70/39*b*x^2+a)*d^2-40/39*b *c^4*d*x+16/39*b*c^5)/d^6
Time = 0.09 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00 \[ \int x^3 (c+d x)^{3/2} \left (a+b x^2\right ) \, dx=\frac {2 \, {\left (3003 \, b d^{7} x^{7} + 3696 \, b c d^{6} x^{6} - 256 \, b c^{7} - 624 \, a c^{5} d^{2} + 63 \, {\left (b c^{2} d^{5} + 65 \, a d^{7}\right )} x^{5} - 70 \, {\left (b c^{3} d^{4} - 78 \, a c d^{6}\right )} x^{4} + 5 \, {\left (16 \, b c^{4} d^{3} + 39 \, a c^{2} d^{5}\right )} x^{3} - 6 \, {\left (16 \, b c^{5} d^{2} + 39 \, a c^{3} d^{4}\right )} x^{2} + 8 \, {\left (16 \, b c^{6} d + 39 \, a c^{4} d^{3}\right )} x\right )} \sqrt {d x + c}}{45045 \, d^{6}} \] Input:
integrate(x^3*(d*x+c)^(3/2)*(b*x^2+a),x, algorithm="fricas")
Output:
2/45045*(3003*b*d^7*x^7 + 3696*b*c*d^6*x^6 - 256*b*c^7 - 624*a*c^5*d^2 + 6 3*(b*c^2*d^5 + 65*a*d^7)*x^5 - 70*(b*c^3*d^4 - 78*a*c*d^6)*x^4 + 5*(16*b*c ^4*d^3 + 39*a*c^2*d^5)*x^3 - 6*(16*b*c^5*d^2 + 39*a*c^3*d^4)*x^2 + 8*(16*b *c^6*d + 39*a*c^4*d^3)*x)*sqrt(d*x + c)/d^6
Time = 0.78 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.14 \[ \int x^3 (c+d x)^{3/2} \left (a+b x^2\right ) \, dx=\begin {cases} \frac {2 \left (- \frac {5 b c \left (c + d x\right )^{\frac {13}{2}}}{13 d^{2}} + \frac {b \left (c + d x\right )^{\frac {15}{2}}}{15 d^{2}} + \frac {\left (c + d x\right )^{\frac {11}{2}} \left (a d^{2} + 10 b c^{2}\right )}{11 d^{2}} + \frac {\left (c + d x\right )^{\frac {9}{2}} \left (- 3 a c d^{2} - 10 b c^{3}\right )}{9 d^{2}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \cdot \left (3 a c^{2} d^{2} + 5 b c^{4}\right )}{7 d^{2}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \left (- a c^{3} d^{2} - b c^{5}\right )}{5 d^{2}}\right )}{d^{4}} & \text {for}\: d \neq 0 \\c^{\frac {3}{2}} \left (\frac {a x^{4}}{4} + \frac {b x^{6}}{6}\right ) & \text {otherwise} \end {cases} \] Input:
integrate(x**3*(d*x+c)**(3/2)*(b*x**2+a),x)
Output:
Piecewise((2*(-5*b*c*(c + d*x)**(13/2)/(13*d**2) + b*(c + d*x)**(15/2)/(15 *d**2) + (c + d*x)**(11/2)*(a*d**2 + 10*b*c**2)/(11*d**2) + (c + d*x)**(9/ 2)*(-3*a*c*d**2 - 10*b*c**3)/(9*d**2) + (c + d*x)**(7/2)*(3*a*c**2*d**2 + 5*b*c**4)/(7*d**2) + (c + d*x)**(5/2)*(-a*c**3*d**2 - b*c**5)/(5*d**2))/d* *4, Ne(d, 0)), (c**(3/2)*(a*x**4/4 + b*x**6/6), True))
Time = 0.03 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.76 \[ \int x^3 (c+d x)^{3/2} \left (a+b x^2\right ) \, dx=\frac {2 \, {\left (3003 \, {\left (d x + c\right )}^{\frac {15}{2}} b - 17325 \, {\left (d x + c\right )}^{\frac {13}{2}} b c + 4095 \, {\left (10 \, b c^{2} + a d^{2}\right )} {\left (d x + c\right )}^{\frac {11}{2}} - 5005 \, {\left (10 \, b c^{3} + 3 \, a c d^{2}\right )} {\left (d x + c\right )}^{\frac {9}{2}} + 6435 \, {\left (5 \, b c^{4} + 3 \, a c^{2} d^{2}\right )} {\left (d x + c\right )}^{\frac {7}{2}} - 9009 \, {\left (b c^{5} + a c^{3} d^{2}\right )} {\left (d x + c\right )}^{\frac {5}{2}}\right )}}{45045 \, d^{6}} \] Input:
integrate(x^3*(d*x+c)^(3/2)*(b*x^2+a),x, algorithm="maxima")
Output:
2/45045*(3003*(d*x + c)^(15/2)*b - 17325*(d*x + c)^(13/2)*b*c + 4095*(10*b *c^2 + a*d^2)*(d*x + c)^(11/2) - 5005*(10*b*c^3 + 3*a*c*d^2)*(d*x + c)^(9/ 2) + 6435*(5*b*c^4 + 3*a*c^2*d^2)*(d*x + c)^(7/2) - 9009*(b*c^5 + a*c^3*d^ 2)*(d*x + c)^(5/2))/d^6
Leaf count of result is larger than twice the leaf count of optimal. 458 vs. \(2 (132) = 264\).
Time = 0.13 (sec) , antiderivative size = 458, normalized size of antiderivative = 2.94 \[ \int x^3 (c+d x)^{3/2} \left (a+b x^2\right ) \, dx=\frac {2 \, {\left (\frac {1287 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} a c^{2}}{d^{3}} + \frac {286 \, {\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} - 180 \, {\left (d x + c\right )}^{\frac {7}{2}} c + 378 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{2} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {d x + c} c^{4}\right )} a c}{d^{3}} + \frac {65 \, {\left (63 \, {\left (d x + c\right )}^{\frac {11}{2}} - 385 \, {\left (d x + c\right )}^{\frac {9}{2}} c + 990 \, {\left (d x + c\right )}^{\frac {7}{2}} c^{2} - 1386 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{3} + 1155 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{4} - 693 \, \sqrt {d x + c} c^{5}\right )} b c^{2}}{d^{5}} + \frac {65 \, {\left (63 \, {\left (d x + c\right )}^{\frac {11}{2}} - 385 \, {\left (d x + c\right )}^{\frac {9}{2}} c + 990 \, {\left (d x + c\right )}^{\frac {7}{2}} c^{2} - 1386 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{3} + 1155 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{4} - 693 \, \sqrt {d x + c} c^{5}\right )} a}{d^{3}} + \frac {30 \, {\left (231 \, {\left (d x + c\right )}^{\frac {13}{2}} - 1638 \, {\left (d x + c\right )}^{\frac {11}{2}} c + 5005 \, {\left (d x + c\right )}^{\frac {9}{2}} c^{2} - 8580 \, {\left (d x + c\right )}^{\frac {7}{2}} c^{3} + 9009 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{4} - 6006 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{5} + 3003 \, \sqrt {d x + c} c^{6}\right )} b c}{d^{5}} + \frac {7 \, {\left (429 \, {\left (d x + c\right )}^{\frac {15}{2}} - 3465 \, {\left (d x + c\right )}^{\frac {13}{2}} c + 12285 \, {\left (d x + c\right )}^{\frac {11}{2}} c^{2} - 25025 \, {\left (d x + c\right )}^{\frac {9}{2}} c^{3} + 32175 \, {\left (d x + c\right )}^{\frac {7}{2}} c^{4} - 27027 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{5} + 15015 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{6} - 6435 \, \sqrt {d x + c} c^{7}\right )} b}{d^{5}}\right )}}{45045 \, d} \] Input:
integrate(x^3*(d*x+c)^(3/2)*(b*x^2+a),x, algorithm="giac")
Output:
2/45045*(1287*(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*c^2/d^3 + 286*(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*c/d^3 + 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*c^2/d^5 + 65*(63*(d*x + c)^(11/2) - 3 85*(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)*a/d^3 + 30*(231*(d*x + c)^(13/2) - 1638*(d*x + c)^(11/2)*c + 5005*(d*x + c)^(9/2)*c^2 - 8580*(d* x + c)^(7/2)*c^3 + 9009*(d*x + c)^(5/2)*c^4 - 6006*(d*x + c)^(3/2)*c^5 + 3 003*sqrt(d*x + c)*c^6)*b*c/d^5 + 7*(429*(d*x + c)^(15/2) - 3465*(d*x + c)^ (13/2)*c + 12285*(d*x + c)^(11/2)*c^2 - 25025*(d*x + c)^(9/2)*c^3 + 32175* (d*x + c)^(7/2)*c^4 - 27027*(d*x + c)^(5/2)*c^5 + 15015*(d*x + c)^(3/2)*c^ 6 - 6435*sqrt(d*x + c)*c^7)*b/d^5)/d
Time = 7.72 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.87 \[ \int x^3 (c+d x)^{3/2} \left (a+b x^2\right ) \, dx=\frac {\left (10\,b\,c^4+6\,a\,c^2\,d^2\right )\,{\left (c+d\,x\right )}^{7/2}}{7\,d^6}-\frac {\left (2\,b\,c^5+2\,a\,c^3\,d^2\right )\,{\left (c+d\,x\right )}^{5/2}}{5\,d^6}-\frac {\left (20\,b\,c^3+6\,a\,c\,d^2\right )\,{\left (c+d\,x\right )}^{9/2}}{9\,d^6}+\frac {2\,b\,{\left (c+d\,x\right )}^{15/2}}{15\,d^6}+\frac {\left (20\,b\,c^2+2\,a\,d^2\right )\,{\left (c+d\,x\right )}^{11/2}}{11\,d^6}-\frac {10\,b\,c\,{\left (c+d\,x\right )}^{13/2}}{13\,d^6} \] Input:
int(x^3*(a + b*x^2)*(c + d*x)^(3/2),x)
Output:
((10*b*c^4 + 6*a*c^2*d^2)*(c + d*x)^(7/2))/(7*d^6) - ((2*b*c^5 + 2*a*c^3*d ^2)*(c + d*x)^(5/2))/(5*d^6) - ((20*b*c^3 + 6*a*c*d^2)*(c + d*x)^(9/2))/(9 *d^6) + (2*b*(c + d*x)^(15/2))/(15*d^6) + ((2*a*d^2 + 20*b*c^2)*(c + d*x)^ (11/2))/(11*d^6) - (10*b*c*(c + d*x)^(13/2))/(13*d^6)
Time = 0.20 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.99 \[ \int x^3 (c+d x)^{3/2} \left (a+b x^2\right ) \, dx=\frac {2 \sqrt {d x +c}\, \left (3003 b \,d^{7} x^{7}+3696 b c \,d^{6} x^{6}+4095 a \,d^{7} x^{5}+63 b \,c^{2} d^{5} x^{5}+5460 a c \,d^{6} x^{4}-70 b \,c^{3} d^{4} x^{4}+195 a \,c^{2} d^{5} x^{3}+80 b \,c^{4} d^{3} x^{3}-234 a \,c^{3} d^{4} x^{2}-96 b \,c^{5} d^{2} x^{2}+312 a \,c^{4} d^{3} x +128 b \,c^{6} d x -624 a \,c^{5} d^{2}-256 b \,c^{7}\right )}{45045 d^{6}} \] Input:
int(x^3*(d*x+c)^(3/2)*(b*x^2+a),x)
Output:
(2*sqrt(c + d*x)*( - 624*a*c**5*d**2 + 312*a*c**4*d**3*x - 234*a*c**3*d**4 *x**2 + 195*a*c**2*d**5*x**3 + 5460*a*c*d**6*x**4 + 4095*a*d**7*x**5 - 256 *b*c**7 + 128*b*c**6*d*x - 96*b*c**5*d**2*x**2 + 80*b*c**4*d**3*x**3 - 70* b*c**3*d**4*x**4 + 63*b*c**2*d**5*x**5 + 3696*b*c*d**6*x**6 + 3003*b*d**7* x**7))/(45045*d**6)