Integrand size = 24, antiderivative size = 287 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^2} \, dx=\frac {d (c d-b e)^2 (A e (5 c d-2 b e)-3 B d (2 c d-b e)) x}{e^7}+\frac {(c d-b e)^2 (B d (5 c d-2 b e)-A e (4 c d-b e)) x^2}{2 e^6}-\frac {(c d-b e)^2 (4 B c d-b B e-3 A c e) x^3}{3 e^5}-\frac {c \left (A c e (2 c d-3 b e)-3 B (c d-b e)^2\right ) x^4}{4 e^4}-\frac {c^2 (2 B c d-3 b B e-A c e) x^5}{5 e^3}+\frac {B c^3 x^6}{6 e^2}+\frac {d^3 (B d-A e) (c d-b e)^3}{e^8 (d+e x)}+\frac {d^2 (c d-b e)^2 (B d (7 c d-4 b e)-3 A e (2 c d-b e)) \log (d+e x)}{e^8} \] Output:
d*(-b*e+c*d)^2*(A*e*(-2*b*e+5*c*d)-3*B*d*(-b*e+2*c*d))*x/e^7+1/2*(-b*e+c*d )^2*(B*d*(-2*b*e+5*c*d)-A*e*(-b*e+4*c*d))*x^2/e^6-1/3*(-b*e+c*d)^2*(-3*A*c *e-B*b*e+4*B*c*d)*x^3/e^5-1/4*c*(A*c*e*(-3*b*e+2*c*d)-3*B*(-b*e+c*d)^2)*x^ 4/e^4-1/5*c^2*(-A*c*e-3*B*b*e+2*B*c*d)*x^5/e^3+1/6*B*c^3*x^6/e^2+d^3*(-A*e +B*d)*(-b*e+c*d)^3/e^8/(e*x+d)+d^2*(-b*e+c*d)^2*(B*d*(-4*b*e+7*c*d)-3*A*e* (-b*e+2*c*d))*ln(e*x+d)/e^8
Time = 0.15 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.95 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^2} \, dx=\frac {-60 d e (c d-b e)^2 (3 B d (2 c d-b e)+A e (-5 c d+2 b e)) x+30 e^2 (c d-b e)^2 (B d (5 c d-2 b e)+A e (-4 c d+b e)) x^2+20 e^3 (c d-b e)^2 (-4 B c d+b B e+3 A c e) x^3-15 c e^4 \left (A c e (2 c d-3 b e)-3 B (c d-b e)^2\right ) x^4+12 c^2 e^5 (-2 B c d+3 b B e+A c e) x^5+10 B c^3 e^6 x^6+\frac {60 d^3 (B d-A e) (c d-b e)^3}{d+e x}+60 d^2 (c d-b e)^2 (B d (7 c d-4 b e)+3 A e (-2 c d+b e)) \log (d+e x)}{60 e^8} \] Input:
Integrate[((A + B*x)*(b*x + c*x^2)^3)/(d + e*x)^2,x]
Output:
(-60*d*e*(c*d - b*e)^2*(3*B*d*(2*c*d - b*e) + A*e*(-5*c*d + 2*b*e))*x + 30 *e^2*(c*d - b*e)^2*(B*d*(5*c*d - 2*b*e) + A*e*(-4*c*d + b*e))*x^2 + 20*e^3 *(c*d - b*e)^2*(-4*B*c*d + b*B*e + 3*A*c*e)*x^3 - 15*c*e^4*(A*c*e*(2*c*d - 3*b*e) - 3*B*(c*d - b*e)^2)*x^4 + 12*c^2*e^5*(-2*B*c*d + 3*b*B*e + A*c*e) *x^5 + 10*B*c^3*e^6*x^6 + (60*d^3*(B*d - A*e)*(c*d - b*e)^3)/(d + e*x) + 6 0*d^2*(c*d - b*e)^2*(B*d*(7*c*d - 4*b*e) + 3*A*e*(-2*c*d + b*e))*Log[d + e *x])/(60*e^8)
Time = 1.23 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1195, 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.
| \(\displaystyle \int \frac {(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {c^2 x^4 (A c e+3 b B e-2 B c d)}{e^3}-\frac {d^3 (B d-A e) (c d-b e)^3}{e^7 (d+e x)^2}+\frac {d^2 (c d-b e)^2 (B d (7 c d-4 b e)-3 A e (2 c d-b e))}{e^7 (d+e x)}+\frac {d (c d-b e)^2 (A e (5 c d-2 b e)-3 B d (2 c d-b e))}{e^7}+\frac {x (c d-b e)^2 (B d (5 c d-2 b e)-A e (4 c d-b e))}{e^6}+\frac {x^2 (b e-c d)^2 (3 A c e+b B e-4 B c d)}{e^5}+\frac {c x^3 \left (3 B (c d-b e)^2-A c e (2 c d-3 b e)\right )}{e^4}+\frac {B c^3 x^5}{e^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {c^2 x^5 (-A c e-3 b B e+2 B c d)}{5 e^3}+\frac {d^3 (B d-A e) (c d-b e)^3}{e^8 (d+e x)}+\frac {d^2 (c d-b e)^2 \log (d+e x) (B d (7 c d-4 b e)-3 A e (2 c d-b e))}{e^8}+\frac {d x (c d-b e)^2 (A e (5 c d-2 b e)-3 B d (2 c d-b e))}{e^7}+\frac {x^2 (c d-b e)^2 (B d (5 c d-2 b e)-A e (4 c d-b e))}{2 e^6}-\frac {x^3 (c d-b e)^2 (-3 A c e-b B e+4 B c d)}{3 e^5}-\frac {c x^4 \left (A c e (2 c d-3 b e)-3 B (c d-b e)^2\right )}{4 e^4}+\frac {B c^3 x^6}{6 e^2}\) |
Input:
Int[((A + B*x)*(b*x + c*x^2)^3)/(d + e*x)^2,x]
Output:
(d*(c*d - b*e)^2*(A*e*(5*c*d - 2*b*e) - 3*B*d*(2*c*d - b*e))*x)/e^7 + ((c* d - b*e)^2*(B*d*(5*c*d - 2*b*e) - A*e*(4*c*d - b*e))*x^2)/(2*e^6) - ((c*d - b*e)^2*(4*B*c*d - b*B*e - 3*A*c*e)*x^3)/(3*e^5) - (c*(A*c*e*(2*c*d - 3*b *e) - 3*B*(c*d - b*e)^2)*x^4)/(4*e^4) - (c^2*(2*B*c*d - 3*b*B*e - A*c*e)*x ^5)/(5*e^3) + (B*c^3*x^6)/(6*e^2) + (d^3*(B*d - A*e)*(c*d - b*e)^3)/(e^8*( d + e*x)) + (d^2*(c*d - b*e)^2*(B*d*(7*c*d - 4*b*e) - 3*A*e*(2*c*d - b*e)) *Log[d + e*x])/e^8
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(570\) vs. \(2(277)=554\).
Time = 0.86 (sec) , antiderivative size = 571, normalized size of antiderivative = 1.99
| method | result | size |
| norman | \(\frac {\frac {d \left (3 A \,b^{3} d^{2} e^{4}-12 A \,b^{2} c \,d^{3} e^{3}+15 A b \,c^{2} d^{4} e^{2}-6 A \,c^{3} d^{5} e -4 B \,b^{3} d^{3} e^{3}+15 B \,b^{2} c \,d^{4} e^{2}-18 B b \,c^{2} d^{5} e +7 B \,c^{3} d^{6}\right )}{e^{8}}+\frac {\left (12 A \,b^{2} c \,e^{3}-15 A b \,c^{2} d \,e^{2}+6 A \,c^{3} d^{2} e +4 B \,e^{3} b^{3}-15 B \,b^{2} c d \,e^{2}+18 B b \,c^{2} d^{2} e -7 B \,c^{3} d^{3}\right ) x^{4}}{12 e^{4}}+\frac {\left (3 A \,b^{3} e^{4}-12 A \,b^{2} c d \,e^{3}+15 A b \,c^{2} d^{2} e^{2}-6 A \,c^{3} d^{3} e -4 B \,b^{3} d \,e^{3}+15 B \,b^{2} c \,d^{2} e^{2}-18 B b \,c^{2} d^{3} e +7 B \,c^{3} d^{4}\right ) x^{3}}{6 e^{5}}+\frac {B \,c^{3} x^{7}}{6 e}+\frac {c \left (15 A b c \,e^{2}-6 A \,c^{2} d e +15 B \,e^{2} b^{2}-18 B b c d e +7 B \,c^{2} d^{2}\right ) x^{5}}{20 e^{3}}+\frac {c^{2} \left (6 A c e +18 B b e -7 B c d \right ) x^{6}}{30 e^{2}}-\frac {d \left (3 A \,b^{3} e^{4}-12 A \,b^{2} c d \,e^{3}+15 A b \,c^{2} d^{2} e^{2}-6 A \,c^{3} d^{3} e -4 B \,b^{3} d \,e^{3}+15 B \,b^{2} c \,d^{2} e^{2}-18 B b \,c^{2} d^{3} e +7 B \,c^{3} d^{4}\right ) x^{2}}{2 e^{6}}}{e x +d}+\frac {d^{2} \left (3 A \,b^{3} e^{4}-12 A \,b^{2} c d \,e^{3}+15 A b \,c^{2} d^{2} e^{2}-6 A \,c^{3} d^{3} e -4 B \,b^{3} d \,e^{3}+15 B \,b^{2} c \,d^{2} e^{2}-18 B b \,c^{2} d^{3} e +7 B \,c^{3} d^{4}\right ) \ln \left (e x +d \right )}{e^{8}}\) | \(571\) |
| default | \(-\frac {\frac {4}{3} B \,c^{3} d^{3} e^{2} x^{3}+2 A \,c^{3} d^{3} e^{2} x^{2}-\frac {5}{2} B \,c^{3} d^{4} e \,x^{2}-3 B \,b^{3} d^{2} e^{3} x -\frac {3}{5} B b \,c^{2} e^{5} x^{5}+\frac {1}{2} A \,c^{3} d \,e^{4} x^{4}-\frac {3}{4} B \,b^{2} c \,e^{5} x^{4}-\frac {3}{4} B \,c^{3} d^{2} e^{3} x^{4}+2 A b \,c^{2} d \,e^{4} x^{3}-\frac {1}{2} A \,b^{3} e^{5} x^{2}+2 A d \,b^{3} e^{4} x -5 A \,c^{3} d^{4} e x -A \,b^{2} c \,e^{5} x^{3}-A \,c^{3} d^{2} e^{3} x^{3}+\frac {2}{5} B \,c^{3} d \,e^{4} x^{5}-\frac {3}{4} A b \,c^{2} e^{5} x^{4}-\frac {9}{2} B \,b^{2} c \,d^{2} e^{3} x^{2}+6 B b \,c^{2} d^{3} e^{2} x^{2}-\frac {9}{2} A b \,c^{2} d^{2} e^{3} x^{2}+\frac {3}{2} B b \,c^{2} d \,e^{4} x^{4}+12 A b \,c^{2} d^{3} e^{2} x +12 B \,b^{2} c \,d^{3} e^{2} x -15 B b \,c^{2} d^{4} e x +6 B \,c^{3} d^{5} x +B \,b^{3} d \,e^{4} x^{2}+2 B \,b^{2} c d \,e^{4} x^{3}-3 B b \,c^{2} d^{2} e^{3} x^{3}+3 A \,b^{2} c d \,e^{4} x^{2}-\frac {1}{3} B \,b^{3} e^{5} x^{3}-9 A \,b^{2} c \,d^{2} e^{3} x -\frac {1}{6} B \,c^{3} x^{6} e^{5}-\frac {1}{5} A \,c^{3} e^{5} x^{5}}{e^{7}}+\frac {d^{2} \left (3 A \,b^{3} e^{4}-12 A \,b^{2} c d \,e^{3}+15 A b \,c^{2} d^{2} e^{2}-6 A \,c^{3} d^{3} e -4 B \,b^{3} d \,e^{3}+15 B \,b^{2} c \,d^{2} e^{2}-18 B b \,c^{2} d^{3} e +7 B \,c^{3} d^{4}\right ) \ln \left (e x +d \right )}{e^{8}}+\frac {d^{3} \left (A \,b^{3} e^{4}-3 A \,b^{2} c d \,e^{3}+3 A b \,c^{2} d^{2} e^{2}-A \,c^{3} d^{3} e -B \,b^{3} d \,e^{3}+3 B \,b^{2} c \,d^{2} e^{2}-3 B b \,c^{2} d^{3} e +B \,c^{3} d^{4}\right )}{e^{8} \left (e x +d \right )}\) | \(637\) |
| risch | \(-\frac {4 d^{3} \ln \left (e x +d \right ) B \,b^{3}}{e^{5}}+\frac {7 d^{6} \ln \left (e x +d \right ) B \,c^{3}}{e^{8}}-\frac {4 B \,c^{3} d^{3} x^{3}}{3 e^{5}}-\frac {2 A \,c^{3} d^{3} x^{2}}{e^{5}}-\frac {2 A d \,b^{3} x}{e^{3}}+\frac {5 A \,c^{3} d^{4} x}{e^{6}}+\frac {3 d^{2} \ln \left (e x +d \right ) A \,b^{3}}{e^{4}}-\frac {6 d^{5} \ln \left (e x +d \right ) A \,c^{3}}{e^{7}}-\frac {d^{4} B \,b^{3}}{e^{5} \left (e x +d \right )}+\frac {d^{7} B \,c^{3}}{e^{8} \left (e x +d \right )}+\frac {5 B \,c^{3} d^{4} x^{2}}{2 e^{6}}+\frac {3 B \,b^{3} d^{2} x}{e^{4}}+\frac {3 B b \,c^{2} x^{5}}{5 e^{2}}-\frac {A \,c^{3} d \,x^{4}}{2 e^{3}}+\frac {3 B \,b^{2} c \,x^{4}}{4 e^{2}}+\frac {3 B \,c^{3} d^{2} x^{4}}{4 e^{4}}+\frac {A \,b^{2} c \,x^{3}}{e^{2}}+\frac {A \,b^{3} x^{2}}{2 e^{2}}+\frac {B \,b^{3} x^{3}}{3 e^{2}}+\frac {A \,c^{3} x^{5}}{5 e^{2}}+\frac {A \,c^{3} d^{2} x^{3}}{e^{4}}-\frac {2 B \,c^{3} d \,x^{5}}{5 e^{3}}+\frac {3 A b \,c^{2} x^{4}}{4 e^{2}}-\frac {6 B \,c^{3} d^{5} x}{e^{7}}-\frac {B \,b^{3} d \,x^{2}}{e^{3}}+\frac {d^{3} A \,b^{3}}{e^{4} \left (e x +d \right )}-\frac {d^{6} A \,c^{3}}{e^{7} \left (e x +d \right )}-\frac {2 A b \,c^{2} d \,x^{3}}{e^{3}}-\frac {3 d^{6} B b \,c^{2}}{e^{7} \left (e x +d \right )}-\frac {12 d^{3} \ln \left (e x +d \right ) A \,b^{2} c}{e^{5}}+\frac {15 d^{4} \ln \left (e x +d \right ) A b \,c^{2}}{e^{6}}+\frac {15 d^{4} \ln \left (e x +d \right ) B \,b^{2} c}{e^{6}}-\frac {18 d^{5} \ln \left (e x +d \right ) B b \,c^{2}}{e^{7}}-\frac {3 d^{4} A \,b^{2} c}{e^{5} \left (e x +d \right )}+\frac {3 d^{5} A b \,c^{2}}{e^{6} \left (e x +d \right )}+\frac {3 d^{5} B \,b^{2} c}{e^{6} \left (e x +d \right )}-\frac {2 B \,b^{2} c d \,x^{3}}{e^{3}}+\frac {3 B b \,c^{2} d^{2} x^{3}}{e^{4}}-\frac {3 A \,b^{2} c d \,x^{2}}{e^{3}}+\frac {9 A \,b^{2} c \,d^{2} x}{e^{4}}+\frac {9 B \,b^{2} c \,d^{2} x^{2}}{2 e^{4}}-\frac {6 B b \,c^{2} d^{3} x^{2}}{e^{5}}+\frac {9 A b \,c^{2} d^{2} x^{2}}{2 e^{4}}-\frac {3 B b \,c^{2} d \,x^{4}}{2 e^{3}}-\frac {12 A b \,c^{2} d^{3} x}{e^{5}}-\frac {12 B \,b^{2} c \,d^{3} x}{e^{5}}+\frac {15 B b \,c^{2} d^{4} x}{e^{6}}+\frac {B \,c^{3} x^{6}}{6 e^{2}}\) | \(742\) |
| parallelrisch | \(\frac {-720 A \,b^{2} c \,d^{4} e^{3}+70 B \,x^{3} c^{3} d^{4} e^{3}-90 A \,x^{2} b^{3} d \,e^{6}+36 B \,x^{6} b \,c^{2} e^{7}+180 A \ln \left (e x +d \right ) b^{3} d^{3} e^{4}-360 A \ln \left (e x +d \right ) c^{3} d^{6} e +900 A b \,c^{2} d^{5} e^{2}+900 B \,b^{2} c \,d^{5} e^{2}-1080 B b \,c^{2} d^{6} e +21 B \,x^{5} c^{3} d^{2} e^{5}+60 A \,x^{4} b^{2} c \,e^{7}+45 A \,x^{5} b \,c^{2} e^{7}-18 A \,x^{5} c^{3} d \,e^{6}-14 B \,x^{6} c^{3} d \,e^{6}+420 B \ln \left (e x +d \right ) c^{3} d^{7}+900 B \ln \left (e x +d \right ) x \,b^{2} c \,d^{4} e^{3}-1080 B \ln \left (e x +d \right ) x b \,c^{2} d^{5} e^{2}-720 A \ln \left (e x +d \right ) x \,b^{2} c \,d^{3} e^{4}+900 A \ln \left (e x +d \right ) x b \,c^{2} d^{4} e^{3}-54 B \,x^{5} b \,c^{2} d \,e^{6}+180 A \,b^{3} d^{3} e^{4}-360 A \,c^{3} d^{6} e -240 B \,b^{3} d^{4} e^{3}-60 A \,x^{3} c^{3} d^{3} e^{4}-40 B \,x^{3} b^{3} d \,e^{6}-210 B \,x^{2} c^{3} d^{5} e^{2}+45 B \,x^{5} b^{2} c \,e^{7}+180 A \,x^{2} c^{3} d^{4} e^{3}-240 B \ln \left (e x +d \right ) b^{3} d^{4} e^{3}+120 B \,x^{2} b^{3} d^{2} e^{5}+30 A \,x^{4} c^{3} d^{2} e^{5}-35 B \,x^{4} c^{3} d^{3} e^{4}+20 B \,x^{4} b^{3} e^{7}+30 A \,x^{3} b^{3} e^{7}+10 B \,x^{7} c^{3} e^{7}+12 A \,x^{6} c^{3} e^{7}-720 A \ln \left (e x +d \right ) b^{2} c \,d^{4} e^{3}+900 A \ln \left (e x +d \right ) b \,c^{2} d^{5} e^{2}+420 B \ln \left (e x +d \right ) x \,c^{3} d^{6} e +180 A \ln \left (e x +d \right ) x \,b^{3} d^{2} e^{5}-360 A \ln \left (e x +d \right ) x \,c^{3} d^{5} e^{2}-240 B \ln \left (e x +d \right ) x \,b^{3} d^{3} e^{4}-120 A \,x^{3} b^{2} c d \,e^{6}+150 A \,x^{3} b \,c^{2} d^{2} e^{5}+150 B \,x^{3} b^{2} c \,d^{2} e^{5}-180 B \,x^{3} b \,c^{2} d^{3} e^{4}+360 A \,x^{2} b^{2} c \,d^{2} e^{5}-450 A \,x^{2} b \,c^{2} d^{3} e^{4}-450 B \,x^{2} b^{2} c \,d^{3} e^{4}+420 B \,c^{3} d^{7}-75 A \,x^{4} b \,c^{2} d \,e^{6}+900 B \ln \left (e x +d \right ) b^{2} c \,d^{5} e^{2}-1080 B \ln \left (e x +d \right ) b \,c^{2} d^{6} e -75 B \,x^{4} b^{2} c d \,e^{6}+90 B \,x^{4} b \,c^{2} d^{2} e^{5}+540 B \,x^{2} b \,c^{2} d^{4} e^{3}}{60 e^{8} \left (e x +d \right )}\) | \(858\) |
Input:
int((B*x+A)*(c*x^2+b*x)^3/(e*x+d)^2,x,method=_RETURNVERBOSE)
Output:
(d*(3*A*b^3*d^2*e^4-12*A*b^2*c*d^3*e^3+15*A*b*c^2*d^4*e^2-6*A*c^3*d^5*e-4* B*b^3*d^3*e^3+15*B*b^2*c*d^4*e^2-18*B*b*c^2*d^5*e+7*B*c^3*d^6)/e^8+1/12*(1 2*A*b^2*c*e^3-15*A*b*c^2*d*e^2+6*A*c^3*d^2*e+4*B*b^3*e^3-15*B*b^2*c*d*e^2+ 18*B*b*c^2*d^2*e-7*B*c^3*d^3)/e^4*x^4+1/6*(3*A*b^3*e^4-12*A*b^2*c*d*e^3+15 *A*b*c^2*d^2*e^2-6*A*c^3*d^3*e-4*B*b^3*d*e^3+15*B*b^2*c*d^2*e^2-18*B*b*c^2 *d^3*e+7*B*c^3*d^4)/e^5*x^3+1/6*B*c^3*x^7/e+1/20*c*(15*A*b*c*e^2-6*A*c^2*d *e+15*B*b^2*e^2-18*B*b*c*d*e+7*B*c^2*d^2)/e^3*x^5+1/30*c^2*(6*A*c*e+18*B*b *e-7*B*c*d)/e^2*x^6-1/2*d*(3*A*b^3*e^4-12*A*b^2*c*d*e^3+15*A*b*c^2*d^2*e^2 -6*A*c^3*d^3*e-4*B*b^3*d*e^3+15*B*b^2*c*d^2*e^2-18*B*b*c^2*d^3*e+7*B*c^3*d ^4)/e^6*x^2)/(e*x+d)+d^2/e^8*(3*A*b^3*e^4-12*A*b^2*c*d*e^3+15*A*b*c^2*d^2* e^2-6*A*c^3*d^3*e-4*B*b^3*d*e^3+15*B*b^2*c*d^2*e^2-18*B*b*c^2*d^3*e+7*B*c^ 3*d^4)*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 722 vs. \(2 (279) = 558\).
Time = 0.09 (sec) , antiderivative size = 722, normalized size of antiderivative = 2.52 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^2} \, dx =\text {Too large to display} \] Input:
integrate((B*x+A)*(c*x^2+b*x)^3/(e*x+d)^2,x, algorithm="fricas")
Output:
1/60*(10*B*c^3*e^7*x^7 + 60*B*c^3*d^7 + 60*A*b^3*d^3*e^4 - 60*(3*B*b*c^2 + A*c^3)*d^6*e + 180*(B*b^2*c + A*b*c^2)*d^5*e^2 - 60*(B*b^3 + 3*A*b^2*c)*d ^4*e^3 - 2*(7*B*c^3*d*e^6 - 6*(3*B*b*c^2 + A*c^3)*e^7)*x^6 + 3*(7*B*c^3*d^ 2*e^5 - 6*(3*B*b*c^2 + A*c^3)*d*e^6 + 15*(B*b^2*c + A*b*c^2)*e^7)*x^5 - 5* (7*B*c^3*d^3*e^4 - 6*(3*B*b*c^2 + A*c^3)*d^2*e^5 + 15*(B*b^2*c + A*b*c^2)* d*e^6 - 4*(B*b^3 + 3*A*b^2*c)*e^7)*x^4 + 10*(7*B*c^3*d^4*e^3 + 3*A*b^3*e^7 - 6*(3*B*b*c^2 + A*c^3)*d^3*e^4 + 15*(B*b^2*c + A*b*c^2)*d^2*e^5 - 4*(B*b ^3 + 3*A*b^2*c)*d*e^6)*x^3 - 30*(7*B*c^3*d^5*e^2 + 3*A*b^3*d*e^6 - 6*(3*B* b*c^2 + A*c^3)*d^4*e^3 + 15*(B*b^2*c + A*b*c^2)*d^3*e^4 - 4*(B*b^3 + 3*A*b ^2*c)*d^2*e^5)*x^2 - 60*(6*B*c^3*d^6*e + 2*A*b^3*d^2*e^5 - 5*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 12*(B*b^2*c + A*b*c^2)*d^4*e^3 - 3*(B*b^3 + 3*A*b^2*c)*d^ 3*e^4)*x + 60*(7*B*c^3*d^7 + 3*A*b^3*d^3*e^4 - 6*(3*B*b*c^2 + A*c^3)*d^6*e + 15*(B*b^2*c + A*b*c^2)*d^5*e^2 - 4*(B*b^3 + 3*A*b^2*c)*d^4*e^3 + (7*B*c ^3*d^6*e + 3*A*b^3*d^2*e^5 - 6*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 15*(B*b^2*c + A*b*c^2)*d^4*e^3 - 4*(B*b^3 + 3*A*b^2*c)*d^3*e^4)*x)*log(e*x + d))/(e^9*x + d*e^8)
Leaf count of result is larger than twice the leaf count of optimal. 619 vs. \(2 (277) = 554\).
Time = 1.38 (sec) , antiderivative size = 619, normalized size of antiderivative = 2.16 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^2} \, dx=\frac {B c^{3} x^{6}}{6 e^{2}} - \frac {d^{2} \left (b e - c d\right )^{2} \left (- 3 A b e^{2} + 6 A c d e + 4 B b d e - 7 B c d^{2}\right ) \log {\left (d + e x \right )}}{e^{8}} + x^{5} \left (\frac {A c^{3}}{5 e^{2}} + \frac {3 B b c^{2}}{5 e^{2}} - \frac {2 B c^{3} d}{5 e^{3}}\right ) + x^{4} \cdot \left (\frac {3 A b c^{2}}{4 e^{2}} - \frac {A c^{3} d}{2 e^{3}} + \frac {3 B b^{2} c}{4 e^{2}} - \frac {3 B b c^{2} d}{2 e^{3}} + \frac {3 B c^{3} d^{2}}{4 e^{4}}\right ) + x^{3} \left (\frac {A b^{2} c}{e^{2}} - \frac {2 A b c^{2} d}{e^{3}} + \frac {A c^{3} d^{2}}{e^{4}} + \frac {B b^{3}}{3 e^{2}} - \frac {2 B b^{2} c d}{e^{3}} + \frac {3 B b c^{2} d^{2}}{e^{4}} - \frac {4 B c^{3} d^{3}}{3 e^{5}}\right ) + x^{2} \left (\frac {A b^{3}}{2 e^{2}} - \frac {3 A b^{2} c d}{e^{3}} + \frac {9 A b c^{2} d^{2}}{2 e^{4}} - \frac {2 A c^{3} d^{3}}{e^{5}} - \frac {B b^{3} d}{e^{3}} + \frac {9 B b^{2} c d^{2}}{2 e^{4}} - \frac {6 B b c^{2} d^{3}}{e^{5}} + \frac {5 B c^{3} d^{4}}{2 e^{6}}\right ) + x \left (- \frac {2 A b^{3} d}{e^{3}} + \frac {9 A b^{2} c d^{2}}{e^{4}} - \frac {12 A b c^{2} d^{3}}{e^{5}} + \frac {5 A c^{3} d^{4}}{e^{6}} + \frac {3 B b^{3} d^{2}}{e^{4}} - \frac {12 B b^{2} c d^{3}}{e^{5}} + \frac {15 B b c^{2} d^{4}}{e^{6}} - \frac {6 B c^{3} d^{5}}{e^{7}}\right ) + \frac {A b^{3} d^{3} e^{4} - 3 A b^{2} c d^{4} e^{3} + 3 A b c^{2} d^{5} e^{2} - A c^{3} d^{6} e - B b^{3} d^{4} e^{3} + 3 B b^{2} c d^{5} e^{2} - 3 B b c^{2} d^{6} e + B c^{3} d^{7}}{d e^{8} + e^{9} x} \] Input:
integrate((B*x+A)*(c*x**2+b*x)**3/(e*x+d)**2,x)
Output:
B*c**3*x**6/(6*e**2) - d**2*(b*e - c*d)**2*(-3*A*b*e**2 + 6*A*c*d*e + 4*B* b*d*e - 7*B*c*d**2)*log(d + e*x)/e**8 + x**5*(A*c**3/(5*e**2) + 3*B*b*c**2 /(5*e**2) - 2*B*c**3*d/(5*e**3)) + x**4*(3*A*b*c**2/(4*e**2) - A*c**3*d/(2 *e**3) + 3*B*b**2*c/(4*e**2) - 3*B*b*c**2*d/(2*e**3) + 3*B*c**3*d**2/(4*e* *4)) + x**3*(A*b**2*c/e**2 - 2*A*b*c**2*d/e**3 + A*c**3*d**2/e**4 + B*b**3 /(3*e**2) - 2*B*b**2*c*d/e**3 + 3*B*b*c**2*d**2/e**4 - 4*B*c**3*d**3/(3*e* *5)) + x**2*(A*b**3/(2*e**2) - 3*A*b**2*c*d/e**3 + 9*A*b*c**2*d**2/(2*e**4 ) - 2*A*c**3*d**3/e**5 - B*b**3*d/e**3 + 9*B*b**2*c*d**2/(2*e**4) - 6*B*b* c**2*d**3/e**5 + 5*B*c**3*d**4/(2*e**6)) + x*(-2*A*b**3*d/e**3 + 9*A*b**2* c*d**2/e**4 - 12*A*b*c**2*d**3/e**5 + 5*A*c**3*d**4/e**6 + 3*B*b**3*d**2/e **4 - 12*B*b**2*c*d**3/e**5 + 15*B*b*c**2*d**4/e**6 - 6*B*c**3*d**5/e**7) + (A*b**3*d**3*e**4 - 3*A*b**2*c*d**4*e**3 + 3*A*b*c**2*d**5*e**2 - A*c**3 *d**6*e - B*b**3*d**4*e**3 + 3*B*b**2*c*d**5*e**2 - 3*B*b*c**2*d**6*e + B* c**3*d**7)/(d*e**8 + e**9*x)
Time = 0.04 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.89 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^2} \, dx=\frac {B c^{3} d^{7} + A b^{3} d^{3} e^{4} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 3 \, {\left (B b^{2} c + A b c^{2}\right )} d^{5} e^{2} - {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{4} e^{3}}{e^{9} x + d e^{8}} + \frac {10 \, B c^{3} e^{5} x^{6} - 12 \, {\left (2 \, B c^{3} d e^{4} - {\left (3 \, B b c^{2} + A c^{3}\right )} e^{5}\right )} x^{5} + 15 \, {\left (3 \, B c^{3} d^{2} e^{3} - 2 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{4} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} e^{5}\right )} x^{4} - 20 \, {\left (4 \, B c^{3} d^{3} e^{2} - 3 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{3} + 6 \, {\left (B b^{2} c + A b c^{2}\right )} d e^{4} - {\left (B b^{3} + 3 \, A b^{2} c\right )} e^{5}\right )} x^{3} + 30 \, {\left (5 \, B c^{3} d^{4} e + A b^{3} e^{5} - 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{2} + 9 \, {\left (B b^{2} c + A b c^{2}\right )} d^{2} e^{3} - 2 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d e^{4}\right )} x^{2} - 60 \, {\left (6 \, B c^{3} d^{5} + 2 \, A b^{3} d e^{4} - 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e + 12 \, {\left (B b^{2} c + A b c^{2}\right )} d^{3} e^{2} - 3 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2} e^{3}\right )} x}{60 \, e^{7}} + \frac {{\left (7 \, B c^{3} d^{6} + 3 \, A b^{3} d^{2} e^{4} - 6 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e + 15 \, {\left (B b^{2} c + A b c^{2}\right )} d^{4} e^{2} - 4 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{3} e^{3}\right )} \log \left (e x + d\right )}{e^{8}} \] Input:
integrate((B*x+A)*(c*x^2+b*x)^3/(e*x+d)^2,x, algorithm="maxima")
Output:
(B*c^3*d^7 + A*b^3*d^3*e^4 - (3*B*b*c^2 + A*c^3)*d^6*e + 3*(B*b^2*c + A*b* c^2)*d^5*e^2 - (B*b^3 + 3*A*b^2*c)*d^4*e^3)/(e^9*x + d*e^8) + 1/60*(10*B*c ^3*e^5*x^6 - 12*(2*B*c^3*d*e^4 - (3*B*b*c^2 + A*c^3)*e^5)*x^5 + 15*(3*B*c^ 3*d^2*e^3 - 2*(3*B*b*c^2 + A*c^3)*d*e^4 + 3*(B*b^2*c + A*b*c^2)*e^5)*x^4 - 20*(4*B*c^3*d^3*e^2 - 3*(3*B*b*c^2 + A*c^3)*d^2*e^3 + 6*(B*b^2*c + A*b*c^ 2)*d*e^4 - (B*b^3 + 3*A*b^2*c)*e^5)*x^3 + 30*(5*B*c^3*d^4*e + A*b^3*e^5 - 4*(3*B*b*c^2 + A*c^3)*d^3*e^2 + 9*(B*b^2*c + A*b*c^2)*d^2*e^3 - 2*(B*b^3 + 3*A*b^2*c)*d*e^4)*x^2 - 60*(6*B*c^3*d^5 + 2*A*b^3*d*e^4 - 5*(3*B*b*c^2 + A*c^3)*d^4*e + 12*(B*b^2*c + A*b*c^2)*d^3*e^2 - 3*(B*b^3 + 3*A*b^2*c)*d^2* e^3)*x)/e^7 + (7*B*c^3*d^6 + 3*A*b^3*d^2*e^4 - 6*(3*B*b*c^2 + A*c^3)*d^5*e + 15*(B*b^2*c + A*b*c^2)*d^4*e^2 - 4*(B*b^3 + 3*A*b^2*c)*d^3*e^3)*log(e*x + d)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 707 vs. \(2 (279) = 558\).
Time = 0.28 (sec) , antiderivative size = 707, normalized size of antiderivative = 2.46 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^2} \, dx=\frac {{\left (10 \, B c^{3} - \frac {12 \, {\left (7 \, B c^{3} d e - 3 \, B b c^{2} e^{2} - A c^{3} e^{2}\right )}}{{\left (e x + d\right )} e} + \frac {45 \, {\left (7 \, B c^{3} d^{2} e^{2} - 6 \, B b c^{2} d e^{3} - 2 \, A c^{3} d e^{3} + B b^{2} c e^{4} + A b c^{2} e^{4}\right )}}{{\left (e x + d\right )}^{2} e^{2}} - \frac {20 \, {\left (35 \, B c^{3} d^{3} e^{3} - 45 \, B b c^{2} d^{2} e^{4} - 15 \, A c^{3} d^{2} e^{4} + 15 \, B b^{2} c d e^{5} + 15 \, A b c^{2} d e^{5} - B b^{3} e^{6} - 3 \, A b^{2} c e^{6}\right )}}{{\left (e x + d\right )}^{3} e^{3}} + \frac {30 \, {\left (35 \, B c^{3} d^{4} e^{4} - 60 \, B b c^{2} d^{3} e^{5} - 20 \, A c^{3} d^{3} e^{5} + 30 \, B b^{2} c d^{2} e^{6} + 30 \, A b c^{2} d^{2} e^{6} - 4 \, B b^{3} d e^{7} - 12 \, A b^{2} c d e^{7} + A b^{3} e^{8}\right )}}{{\left (e x + d\right )}^{4} e^{4}} - \frac {180 \, {\left (7 \, B c^{3} d^{5} e^{5} - 15 \, B b c^{2} d^{4} e^{6} - 5 \, A c^{3} d^{4} e^{6} + 10 \, B b^{2} c d^{3} e^{7} + 10 \, A b c^{2} d^{3} e^{7} - 2 \, B b^{3} d^{2} e^{8} - 6 \, A b^{2} c d^{2} e^{8} + A b^{3} d e^{9}\right )}}{{\left (e x + d\right )}^{5} e^{5}}\right )} {\left (e x + d\right )}^{6}}{60 \, e^{8}} - \frac {{\left (7 \, B c^{3} d^{6} - 18 \, B b c^{2} d^{5} e - 6 \, A c^{3} d^{5} e + 15 \, B b^{2} c d^{4} e^{2} + 15 \, A b c^{2} d^{4} e^{2} - 4 \, B b^{3} d^{3} e^{3} - 12 \, A b^{2} c d^{3} e^{3} + 3 \, A b^{3} d^{2} e^{4}\right )} \log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{e^{8}} + \frac {\frac {B c^{3} d^{7} e^{6}}{e x + d} - \frac {3 \, B b c^{2} d^{6} e^{7}}{e x + d} - \frac {A c^{3} d^{6} e^{7}}{e x + d} + \frac {3 \, B b^{2} c d^{5} e^{8}}{e x + d} + \frac {3 \, A b c^{2} d^{5} e^{8}}{e x + d} - \frac {B b^{3} d^{4} e^{9}}{e x + d} - \frac {3 \, A b^{2} c d^{4} e^{9}}{e x + d} + \frac {A b^{3} d^{3} e^{10}}{e x + d}}{e^{14}} \] Input:
integrate((B*x+A)*(c*x^2+b*x)^3/(e*x+d)^2,x, algorithm="giac")
Output:
1/60*(10*B*c^3 - 12*(7*B*c^3*d*e - 3*B*b*c^2*e^2 - A*c^3*e^2)/((e*x + d)*e ) + 45*(7*B*c^3*d^2*e^2 - 6*B*b*c^2*d*e^3 - 2*A*c^3*d*e^3 + B*b^2*c*e^4 + A*b*c^2*e^4)/((e*x + d)^2*e^2) - 20*(35*B*c^3*d^3*e^3 - 45*B*b*c^2*d^2*e^4 - 15*A*c^3*d^2*e^4 + 15*B*b^2*c*d*e^5 + 15*A*b*c^2*d*e^5 - B*b^3*e^6 - 3* A*b^2*c*e^6)/((e*x + d)^3*e^3) + 30*(35*B*c^3*d^4*e^4 - 60*B*b*c^2*d^3*e^5 - 20*A*c^3*d^3*e^5 + 30*B*b^2*c*d^2*e^6 + 30*A*b*c^2*d^2*e^6 - 4*B*b^3*d* e^7 - 12*A*b^2*c*d*e^7 + A*b^3*e^8)/((e*x + d)^4*e^4) - 180*(7*B*c^3*d^5*e ^5 - 15*B*b*c^2*d^4*e^6 - 5*A*c^3*d^4*e^6 + 10*B*b^2*c*d^3*e^7 + 10*A*b*c^ 2*d^3*e^7 - 2*B*b^3*d^2*e^8 - 6*A*b^2*c*d^2*e^8 + A*b^3*d*e^9)/((e*x + d)^ 5*e^5))*(e*x + d)^6/e^8 - (7*B*c^3*d^6 - 18*B*b*c^2*d^5*e - 6*A*c^3*d^5*e + 15*B*b^2*c*d^4*e^2 + 15*A*b*c^2*d^4*e^2 - 4*B*b^3*d^3*e^3 - 12*A*b^2*c*d ^3*e^3 + 3*A*b^3*d^2*e^4)*log(abs(e*x + d)/((e*x + d)^2*abs(e)))/e^8 + (B* c^3*d^7*e^6/(e*x + d) - 3*B*b*c^2*d^6*e^7/(e*x + d) - A*c^3*d^6*e^7/(e*x + d) + 3*B*b^2*c*d^5*e^8/(e*x + d) + 3*A*b*c^2*d^5*e^8/(e*x + d) - B*b^3*d^ 4*e^9/(e*x + d) - 3*A*b^2*c*d^4*e^9/(e*x + d) + A*b^3*d^3*e^10/(e*x + d))/ e^14
Time = 10.79 (sec) , antiderivative size = 997, normalized size of antiderivative = 3.47 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^2} \, dx =\text {Too large to display} \] Input:
int(((b*x + c*x^2)^3*(A + B*x))/(d + e*x)^2,x)
Output:
x^3*((B*b^3 + 3*A*b^2*c)/(3*e^2) + (2*d*((2*d*((A*c^3 + 3*B*b*c^2)/e^2 - ( 2*B*c^3*d)/e^3))/e - (3*b*c*(A*c + B*b))/e^2 + (B*c^3*d^2)/e^4))/(3*e) - ( d^2*((A*c^3 + 3*B*b*c^2)/e^2 - (2*B*c^3*d)/e^3))/(3*e^2)) + x^2*((A*b^3)/( 2*e^2) - (d*((B*b^3 + 3*A*b^2*c)/e^2 + (2*d*((2*d*((A*c^3 + 3*B*b*c^2)/e^2 - (2*B*c^3*d)/e^3))/e - (3*b*c*(A*c + B*b))/e^2 + (B*c^3*d^2)/e^4))/e - ( d^2*((A*c^3 + 3*B*b*c^2)/e^2 - (2*B*c^3*d)/e^3))/e^2))/e + (d^2*((2*d*((A* c^3 + 3*B*b*c^2)/e^2 - (2*B*c^3*d)/e^3))/e - (3*b*c*(A*c + B*b))/e^2 + (B* c^3*d^2)/e^4))/(2*e^2)) + x^5*((A*c^3 + 3*B*b*c^2)/(5*e^2) - (2*B*c^3*d)/( 5*e^3)) - x*((2*d*((A*b^3)/e^2 - (2*d*((B*b^3 + 3*A*b^2*c)/e^2 + (2*d*((2* d*((A*c^3 + 3*B*b*c^2)/e^2 - (2*B*c^3*d)/e^3))/e - (3*b*c*(A*c + B*b))/e^2 + (B*c^3*d^2)/e^4))/e - (d^2*((A*c^3 + 3*B*b*c^2)/e^2 - (2*B*c^3*d)/e^3)) /e^2))/e + (d^2*((2*d*((A*c^3 + 3*B*b*c^2)/e^2 - (2*B*c^3*d)/e^3))/e - (3* b*c*(A*c + B*b))/e^2 + (B*c^3*d^2)/e^4))/e^2))/e + (d^2*((B*b^3 + 3*A*b^2* c)/e^2 + (2*d*((2*d*((A*c^3 + 3*B*b*c^2)/e^2 - (2*B*c^3*d)/e^3))/e - (3*b* c*(A*c + B*b))/e^2 + (B*c^3*d^2)/e^4))/e - (d^2*((A*c^3 + 3*B*b*c^2)/e^2 - (2*B*c^3*d)/e^3))/e^2))/e^2) - x^4*((d*((A*c^3 + 3*B*b*c^2)/e^2 - (2*B*c^ 3*d)/e^3))/(2*e) - (3*b*c*(A*c + B*b))/(4*e^2) + (B*c^3*d^2)/(4*e^4)) + (l og(d + e*x)*(7*B*c^3*d^6 - 6*A*c^3*d^5*e + 3*A*b^3*d^2*e^4 - 4*B*b^3*d^3*e ^3 + 15*A*b*c^2*d^4*e^2 - 12*A*b^2*c*d^3*e^3 + 15*B*b^2*c*d^4*e^2 - 18*B*b *c^2*d^5*e))/e^8 + (B*c^3*d^7 - A*c^3*d^6*e + A*b^3*d^3*e^4 - B*b^3*d^4...
Time = 0.26 (sec) , antiderivative size = 865, normalized size of antiderivative = 3.01 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^2} \, dx =\text {Too large to display} \] Input:
int((B*x+A)*(c*x^2+b*x)^3/(e*x+d)^2,x)
Output:
(180*log(d + e*x)*a*b**3*d**3*e**4 + 180*log(d + e*x)*a*b**3*d**2*e**5*x - 720*log(d + e*x)*a*b**2*c*d**4*e**3 - 720*log(d + e*x)*a*b**2*c*d**3*e**4 *x + 900*log(d + e*x)*a*b*c**2*d**5*e**2 + 900*log(d + e*x)*a*b*c**2*d**4* e**3*x - 360*log(d + e*x)*a*c**3*d**6*e - 360*log(d + e*x)*a*c**3*d**5*e** 2*x - 240*log(d + e*x)*b**4*d**4*e**3 - 240*log(d + e*x)*b**4*d**3*e**4*x + 900*log(d + e*x)*b**3*c*d**5*e**2 + 900*log(d + e*x)*b**3*c*d**4*e**3*x - 1080*log(d + e*x)*b**2*c**2*d**6*e - 1080*log(d + e*x)*b**2*c**2*d**5*e* *2*x + 420*log(d + e*x)*b*c**3*d**7 + 420*log(d + e*x)*b*c**3*d**6*e*x - 1 80*a*b**3*d**2*e**5*x - 90*a*b**3*d*e**6*x**2 + 30*a*b**3*e**7*x**3 + 720* a*b**2*c*d**3*e**4*x + 360*a*b**2*c*d**2*e**5*x**2 - 120*a*b**2*c*d*e**6*x **3 + 60*a*b**2*c*e**7*x**4 - 900*a*b*c**2*d**4*e**3*x - 450*a*b*c**2*d**3 *e**4*x**2 + 150*a*b*c**2*d**2*e**5*x**3 - 75*a*b*c**2*d*e**6*x**4 + 45*a* b*c**2*e**7*x**5 + 360*a*c**3*d**5*e**2*x + 180*a*c**3*d**4*e**3*x**2 - 60 *a*c**3*d**3*e**4*x**3 + 30*a*c**3*d**2*e**5*x**4 - 18*a*c**3*d*e**6*x**5 + 12*a*c**3*e**7*x**6 + 240*b**4*d**3*e**4*x + 120*b**4*d**2*e**5*x**2 - 4 0*b**4*d*e**6*x**3 + 20*b**4*e**7*x**4 - 900*b**3*c*d**4*e**3*x - 450*b**3 *c*d**3*e**4*x**2 + 150*b**3*c*d**2*e**5*x**3 - 75*b**3*c*d*e**6*x**4 + 45 *b**3*c*e**7*x**5 + 1080*b**2*c**2*d**5*e**2*x + 540*b**2*c**2*d**4*e**3*x **2 - 180*b**2*c**2*d**3*e**4*x**3 + 90*b**2*c**2*d**2*e**5*x**4 - 54*b**2 *c**2*d*e**6*x**5 + 36*b**2*c**2*e**7*x**6 - 420*b*c**3*d**6*e*x - 210*...