\(\int \frac {(a+b x)^6 (A+B x)}{(d+e x)^7} \, dx\) [56]

Optimal result

Integrand size = 20, antiderivative size = 278 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^7} \, dx=\frac {b^6 B x}{e^7}+\frac {(b d-a e)^6 (B d-A e)}{6 e^8 (d+e x)^6}-\frac {(b d-a e)^5 (7 b B d-6 A b e-a B e)}{5 e^8 (d+e x)^5}+\frac {3 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e)}{4 e^8 (d+e x)^4}-\frac {5 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e)}{3 e^8 (d+e x)^3}+\frac {5 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e)}{2 e^8 (d+e x)^2}-\frac {3 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e)}{e^8 (d+e x)}-\frac {b^5 (7 b B d-A b e-6 a B e) \log (d+e x)}{e^8} \] Output:

b^6*B*x/e^7+1/6*(-a*e+b*d)^6*(-A*e+B*d)/e^8/(e*x+d)^6-1/5*(-a*e+b*d)^5*(-6 
*A*b*e-B*a*e+7*B*b*d)/e^8/(e*x+d)^5+3/4*b*(-a*e+b*d)^4*(-5*A*b*e-2*B*a*e+7 
*B*b*d)/e^8/(e*x+d)^4-5/3*b^2*(-a*e+b*d)^3*(-4*A*b*e-3*B*a*e+7*B*b*d)/e^8/ 
(e*x+d)^3+5/2*b^3*(-a*e+b*d)^2*(-3*A*b*e-4*B*a*e+7*B*b*d)/e^8/(e*x+d)^2-3* 
b^4*(-a*e+b*d)*(-2*A*b*e-5*B*a*e+7*B*b*d)/e^8/(e*x+d)-b^5*(-A*b*e-6*B*a*e+ 
7*B*b*d)*ln(e*x+d)/e^8
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(619\) vs. \(2(278)=556\).

Time = 0.20 (sec) , antiderivative size = 619, normalized size of antiderivative = 2.23 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^7} \, dx=-\frac {2 a^6 e^6 (5 A e+B (d+6 e x))+6 a^5 b e^5 \left (2 A e (d+6 e x)+B \left (d^2+6 d e x+15 e^2 x^2\right )\right )+15 a^4 b^2 e^4 \left (A e \left (d^2+6 d e x+15 e^2 x^2\right )+B \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )+20 a^3 b^3 e^3 \left (A e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+2 B \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )+30 a^2 b^4 e^2 \left (A e \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+5 B \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )\right )-6 a b^5 e \left (-10 A e \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )+B d \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )\right )-b^6 \left (A d e \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )-B \left (669 d^7+3594 d^6 e x+7725 d^5 e^2 x^2+8200 d^4 e^3 x^3+4050 d^3 e^4 x^4+360 d^2 e^5 x^5-360 d e^6 x^6-60 e^7 x^7\right )\right )+60 b^5 (7 b B d-A b e-6 a B e) (d+e x)^6 \log (d+e x)}{60 e^8 (d+e x)^6} \] Input:

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^7,x]
 

Output:

-1/60*(2*a^6*e^6*(5*A*e + B*(d + 6*e*x)) + 6*a^5*b*e^5*(2*A*e*(d + 6*e*x) 
+ B*(d^2 + 6*d*e*x + 15*e^2*x^2)) + 15*a^4*b^2*e^4*(A*e*(d^2 + 6*d*e*x + 1 
5*e^2*x^2) + B*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3)) + 20*a^3*b^3 
*e^3*(A*e*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + 2*B*(d^4 + 6*d^3 
*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4)) + 30*a^2*b^4*e^2*(A*e* 
(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4) + 5*B*(d^5 
+ 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5)) 
 - 6*a*b^5*e*(-10*A*e*(d^5 + 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 
 15*d*e^4*x^4 + 6*e^5*x^5) + B*d*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 
 + 2200*d^2*e^3*x^3 + 1350*d*e^4*x^4 + 360*e^5*x^5)) - b^6*(A*d*e*(147*d^5 
 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e^3*x^3 + 1350*d*e^4*x^4 + 36 
0*e^5*x^5) - B*(669*d^7 + 3594*d^6*e*x + 7725*d^5*e^2*x^2 + 8200*d^4*e^3*x 
^3 + 4050*d^3*e^4*x^4 + 360*d^2*e^5*x^5 - 360*d*e^6*x^6 - 60*e^7*x^7)) + 6 
0*b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^6*Log[d + e*x])/(e^8*(d + e*x) 
^6)
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 278, 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 = {86, 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.

\(\displaystyle \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^7} \, dx\)

\(\Big \downarrow \) 86

\(\displaystyle \int \left (\frac {b^5 (6 a B e+A b e-7 b B d)}{e^7 (d+e x)}-\frac {3 b^4 (b d-a e) (5 a B e+2 A b e-7 b B d)}{e^7 (d+e x)^2}+\frac {5 b^3 (b d-a e)^2 (4 a B e+3 A b e-7 b B d)}{e^7 (d+e x)^3}-\frac {5 b^2 (b d-a e)^3 (3 a B e+4 A b e-7 b B d)}{e^7 (d+e x)^4}+\frac {3 b (b d-a e)^4 (2 a B e+5 A b e-7 b B d)}{e^7 (d+e x)^5}+\frac {(a e-b d)^5 (a B e+6 A b e-7 b B d)}{e^7 (d+e x)^6}+\frac {(a e-b d)^6 (A e-B d)}{e^7 (d+e x)^7}+\frac {b^6 B}{e^7}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b^5 \log (d+e x) (-6 a B e-A b e+7 b B d)}{e^8}-\frac {3 b^4 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{e^8 (d+e x)}+\frac {5 b^3 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{2 e^8 (d+e x)^2}-\frac {5 b^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{3 e^8 (d+e x)^3}+\frac {3 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{4 e^8 (d+e x)^4}-\frac {(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{5 e^8 (d+e x)^5}+\frac {(b d-a e)^6 (B d-A e)}{6 e^8 (d+e x)^6}+\frac {b^6 B x}{e^7}\)

Input:

Int[((a + b*x)^6*(A + B*x))/(d + e*x)^7,x]
 

Output:

(b^6*B*x)/e^7 + ((b*d - a*e)^6*(B*d - A*e))/(6*e^8*(d + e*x)^6) - ((b*d - 
a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e))/(5*e^8*(d + e*x)^5) + (3*b*(b*d - a*e) 
^4*(7*b*B*d - 5*A*b*e - 2*a*B*e))/(4*e^8*(d + e*x)^4) - (5*b^2*(b*d - a*e) 
^3*(7*b*B*d - 4*A*b*e - 3*a*B*e))/(3*e^8*(d + e*x)^3) + (5*b^3*(b*d - a*e) 
^2*(7*b*B*d - 3*A*b*e - 4*a*B*e))/(2*e^8*(d + e*x)^2) - (3*b^4*(b*d - a*e) 
*(7*b*B*d - 2*A*b*e - 5*a*B*e))/(e^8*(d + e*x)) - (b^5*(7*b*B*d - A*b*e - 
6*a*B*e)*Log[d + e*x])/e^8
 

Defintions of rubi rules used

rule 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 
] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p 
+ 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(804\) vs. \(2(268)=536\).

Time = 0.23 (sec) , antiderivative size = 805, normalized size of antiderivative = 2.90

method	result	size
default	\(\frac {b^{6} B x}{e^{7}}-\frac {a^{6} A \,e^{7}-6 A \,a^{5} b d \,e^{6}+15 A \,a^{4} b^{2} d^{2} e^{5}-20 A \,a^{3} b^{3} d^{3} e^{4}+15 A \,a^{2} b^{4} d^{4} e^{3}-6 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e -B \,a^{6} d \,e^{6}+6 B \,a^{5} b \,d^{2} e^{5}-15 B \,a^{4} b^{2} d^{3} e^{4}+20 B \,a^{3} b^{3} d^{4} e^{3}-15 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e -b^{6} B \,d^{7}}{6 e^{8} \left (e x +d \right )^{6}}-\frac {3 b^{4} \left (2 A a b \,e^{2}-2 A \,b^{2} d e +5 B \,a^{2} e^{2}-12 B a b d e +7 b^{2} B \,d^{2}\right )}{e^{8} \left (e x +d \right )}-\frac {5 b^{3} \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +4 B \,a^{3} e^{3}-15 B \,a^{2} b d \,e^{2}+18 B a \,b^{2} d^{2} e -7 b^{3} B \,d^{3}\right )}{2 e^{8} \left (e x +d \right )^{2}}-\frac {6 A \,a^{5} b \,e^{6}-30 A \,a^{4} b^{2} d \,e^{5}+60 A \,a^{3} b^{3} d^{2} e^{4}-60 A \,a^{2} b^{4} d^{3} e^{3}+30 A a \,b^{5} d^{4} e^{2}-6 A \,b^{6} d^{5} e +B \,a^{6} e^{6}-12 B \,a^{5} b d \,e^{5}+45 B \,a^{4} b^{2} d^{2} e^{4}-80 B \,a^{3} b^{3} d^{3} e^{3}+75 B \,a^{2} b^{4} d^{4} e^{2}-36 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}}{5 e^{8} \left (e x +d \right )^{5}}-\frac {3 b \left (5 A \,a^{4} b \,e^{5}-20 A \,a^{3} b^{2} d \,e^{4}+30 A \,a^{2} b^{3} d^{2} e^{3}-20 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e +2 B \,a^{5} e^{5}-15 B \,a^{4} b d \,e^{4}+40 B \,a^{3} b^{2} d^{2} e^{3}-50 B \,a^{2} b^{3} d^{3} e^{2}+30 B a \,b^{4} d^{4} e -7 B \,b^{5} d^{5}\right )}{4 e^{8} \left (e x +d \right )^{4}}+\frac {b^{5} \left (A b e +6 B a e -7 B b d \right ) \ln \left (e x +d \right )}{e^{8}}-\frac {5 b^{2} \left (4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e +3 B \,a^{4} e^{4}-16 B \,a^{3} b d \,e^{3}+30 B \,a^{2} b^{2} d^{2} e^{2}-24 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right )}{3 e^{8} \left (e x +d \right )^{3}}\)	\(805\)
norman	\(\frac {\frac {b^{6} B \,x^{7}}{e}-\frac {10 a^{6} A \,e^{7}+12 A \,a^{5} b d \,e^{6}+15 A \,a^{4} b^{2} d^{2} e^{5}+20 A \,a^{3} b^{3} d^{3} e^{4}+30 A \,a^{2} b^{4} d^{4} e^{3}+60 A a \,b^{5} d^{5} e^{2}-147 A \,b^{6} d^{6} e +2 B \,a^{6} d \,e^{6}+6 B \,a^{5} b \,d^{2} e^{5}+15 B \,a^{4} b^{2} d^{3} e^{4}+40 B \,a^{3} b^{3} d^{4} e^{3}+150 B \,a^{2} b^{4} d^{5} e^{2}-882 B a \,b^{5} d^{6} e +1029 b^{6} B \,d^{7}}{60 e^{8}}-\frac {3 \left (2 A a \,b^{5} e^{2}-2 A \,b^{6} d e +5 B \,a^{2} b^{4} e^{2}-12 B a \,b^{5} d e +14 b^{6} B \,d^{2}\right ) x^{5}}{e^{3}}-\frac {5 \left (3 A \,a^{2} b^{4} e^{3}+6 A a \,b^{5} d \,e^{2}-9 A \,b^{6} d^{2} e +4 B \,a^{3} b^{3} e^{3}+15 B \,a^{2} b^{4} d \,e^{2}-54 B a \,b^{5} d^{2} e +63 b^{6} B \,d^{3}\right ) x^{4}}{2 e^{4}}-\frac {5 \left (4 A \,a^{3} b^{3} e^{4}+6 A \,a^{2} b^{4} d \,e^{3}+12 A a \,b^{5} d^{2} e^{2}-22 A \,b^{6} d^{3} e +3 B \,a^{4} b^{2} e^{4}+8 B \,a^{3} b^{3} d \,e^{3}+30 B \,a^{2} b^{4} d^{2} e^{2}-132 B a \,b^{5} d^{3} e +154 b^{6} B \,d^{4}\right ) x^{3}}{3 e^{5}}-\frac {\left (15 A \,a^{4} b^{2} e^{5}+20 A \,a^{3} b^{3} d \,e^{4}+30 A \,a^{2} b^{4} d^{2} e^{3}+60 A a \,b^{5} d^{3} e^{2}-125 A \,b^{6} d^{4} e +6 B \,a^{5} b \,e^{5}+15 B \,a^{4} b^{2} d \,e^{4}+40 B \,a^{3} b^{3} d^{2} e^{3}+150 B \,a^{2} b^{4} d^{3} e^{2}-750 B a \,b^{5} d^{4} e +875 b^{6} B \,d^{5}\right ) x^{2}}{4 e^{6}}-\frac {\left (12 A \,a^{5} b \,e^{6}+15 A \,a^{4} b^{2} d \,e^{5}+20 A \,a^{3} b^{3} d^{2} e^{4}+30 A \,a^{2} b^{4} d^{3} e^{3}+60 A a \,b^{5} d^{4} e^{2}-137 A \,b^{6} d^{5} e +2 B \,a^{6} e^{6}+6 B \,a^{5} b d \,e^{5}+15 B \,a^{4} b^{2} d^{2} e^{4}+40 B \,a^{3} b^{3} d^{3} e^{3}+150 B \,a^{2} b^{4} d^{4} e^{2}-822 B a \,b^{5} d^{5} e +959 b^{6} B \,d^{6}\right ) x}{10 e^{7}}}{\left (e x +d \right )^{6}}+\frac {b^{5} \left (A b e +6 B a e -7 B b d \right ) \ln \left (e x +d \right )}{e^{8}}\)	\(806\)
risch	\(\frac {b^{6} B x}{e^{7}}+\frac {\left (-6 A a \,b^{5} e^{6}+6 A \,b^{6} d \,e^{5}-15 B \,a^{2} b^{4} e^{6}+36 B a \,b^{5} d \,e^{5}-21 b^{6} B \,d^{2} e^{4}\right ) x^{5}-\frac {5 e^{3} b^{3} \left (3 A \,a^{2} b \,e^{3}+6 A a \,b^{2} d \,e^{2}-9 A \,b^{3} d^{2} e +4 B \,a^{3} e^{3}+15 B \,a^{2} b d \,e^{2}-54 B a \,b^{2} d^{2} e +35 b^{3} B \,d^{3}\right ) x^{4}}{2}-\frac {5 b^{2} e^{2} \left (4 A \,a^{3} b \,e^{4}+6 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-22 A \,b^{4} d^{3} e +3 B \,a^{4} e^{4}+8 B \,a^{3} b d \,e^{3}+30 B \,a^{2} b^{2} d^{2} e^{2}-132 B a \,b^{3} d^{3} e +91 B \,b^{4} d^{4}\right ) x^{3}}{3}-\frac {b e \left (15 A \,a^{4} b \,e^{5}+20 A \,a^{3} b^{2} d \,e^{4}+30 A \,a^{2} b^{3} d^{2} e^{3}+60 A a \,b^{4} d^{3} e^{2}-125 A \,b^{5} d^{4} e +6 B \,a^{5} e^{5}+15 B \,a^{4} b d \,e^{4}+40 B \,a^{3} b^{2} d^{2} e^{3}+150 B \,a^{2} b^{3} d^{3} e^{2}-750 B a \,b^{4} d^{4} e +539 B \,b^{5} d^{5}\right ) x^{2}}{4}+\left (-\frac {6}{5} A \,a^{5} b \,e^{6}-\frac {3}{2} A \,a^{4} b^{2} d \,e^{5}-2 A \,a^{3} b^{3} d^{2} e^{4}-3 A \,a^{2} b^{4} d^{3} e^{3}-6 A a \,b^{5} d^{4} e^{2}+\frac {137}{10} A \,b^{6} d^{5} e -\frac {1}{5} B \,a^{6} e^{6}-\frac {3}{5} B \,a^{5} b d \,e^{5}-\frac {3}{2} B \,a^{4} b^{2} d^{2} e^{4}-4 B \,a^{3} b^{3} d^{3} e^{3}-15 B \,a^{2} b^{4} d^{4} e^{2}+\frac {411}{5} B a \,b^{5} d^{5} e -\frac {609}{10} b^{6} B \,d^{6}\right ) x -\frac {10 a^{6} A \,e^{7}+12 A \,a^{5} b d \,e^{6}+15 A \,a^{4} b^{2} d^{2} e^{5}+20 A \,a^{3} b^{3} d^{3} e^{4}+30 A \,a^{2} b^{4} d^{4} e^{3}+60 A a \,b^{5} d^{5} e^{2}-147 A \,b^{6} d^{6} e +2 B \,a^{6} d \,e^{6}+6 B \,a^{5} b \,d^{2} e^{5}+15 B \,a^{4} b^{2} d^{3} e^{4}+40 B \,a^{3} b^{3} d^{4} e^{3}+150 B \,a^{2} b^{4} d^{5} e^{2}-882 B a \,b^{5} d^{6} e +669 b^{6} B \,d^{7}}{60 e}}{e^{7} \left (e x +d \right )^{6}}+\frac {b^{6} \ln \left (e x +d \right ) A}{e^{7}}+\frac {6 b^{5} \ln \left (e x +d \right ) B a}{e^{7}}-\frac {7 b^{6} \ln \left (e x +d \right ) B d}{e^{8}}\)	\(810\)
parallelrisch	\(\text {Expression too large to display}\)	\(1287\)

Input:

int((b*x+a)^6*(B*x+A)/(e*x+d)^7,x,method=_RETURNVERBOSE)
 

Output:

b^6*B*x/e^7-1/6*(A*a^6*e^7-6*A*a^5*b*d*e^6+15*A*a^4*b^2*d^2*e^5-20*A*a^3*b 
^3*d^3*e^4+15*A*a^2*b^4*d^4*e^3-6*A*a*b^5*d^5*e^2+A*b^6*d^6*e-B*a^6*d*e^6+ 
6*B*a^5*b*d^2*e^5-15*B*a^4*b^2*d^3*e^4+20*B*a^3*b^3*d^4*e^3-15*B*a^2*b^4*d 
^5*e^2+6*B*a*b^5*d^6*e-B*b^6*d^7)/e^8/(e*x+d)^6-3*b^4/e^8*(2*A*a*b*e^2-2*A 
*b^2*d*e+5*B*a^2*e^2-12*B*a*b*d*e+7*B*b^2*d^2)/(e*x+d)-5/2/e^8*b^3*(3*A*a^ 
2*b*e^3-6*A*a*b^2*d*e^2+3*A*b^3*d^2*e+4*B*a^3*e^3-15*B*a^2*b*d*e^2+18*B*a* 
b^2*d^2*e-7*B*b^3*d^3)/(e*x+d)^2-1/5/e^8*(6*A*a^5*b*e^6-30*A*a^4*b^2*d*e^5 
+60*A*a^3*b^3*d^2*e^4-60*A*a^2*b^4*d^3*e^3+30*A*a*b^5*d^4*e^2-6*A*b^6*d^5* 
e+B*a^6*e^6-12*B*a^5*b*d*e^5+45*B*a^4*b^2*d^2*e^4-80*B*a^3*b^3*d^3*e^3+75* 
B*a^2*b^4*d^4*e^2-36*B*a*b^5*d^5*e+7*B*b^6*d^6)/(e*x+d)^5-3/4*b/e^8*(5*A*a 
^4*b*e^5-20*A*a^3*b^2*d*e^4+30*A*a^2*b^3*d^2*e^3-20*A*a*b^4*d^3*e^2+5*A*b^ 
5*d^4*e+2*B*a^5*e^5-15*B*a^4*b*d*e^4+40*B*a^3*b^2*d^2*e^3-50*B*a^2*b^3*d^3 
*e^2+30*B*a*b^4*d^4*e-7*B*b^5*d^5)/(e*x+d)^4+b^5/e^8*(A*b*e+6*B*a*e-7*B*b* 
d)*ln(e*x+d)-5/3*b^2/e^8*(4*A*a^3*b*e^4-12*A*a^2*b^2*d*e^3+12*A*a*b^3*d^2* 
e^2-4*A*b^4*d^3*e+3*B*a^4*e^4-16*B*a^3*b*d*e^3+30*B*a^2*b^2*d^2*e^2-24*B*a 
*b^3*d^3*e+7*B*b^4*d^4)/(e*x+d)^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1063 vs. \(2 (268) = 536\).

Time = 0.12 (sec) , antiderivative size = 1063, normalized size of antiderivative = 3.82 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^7} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^7,x, algorithm="fricas")
 

Output:

1/60*(60*B*b^6*e^7*x^7 + 360*B*b^6*d*e^6*x^6 - 669*B*b^6*d^7 - 10*A*a^6*e^ 
7 + 147*(6*B*a*b^5 + A*b^6)*d^6*e - 30*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 
 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 - 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^ 
3*e^4 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 - 2*(B*a^6 + 6*A*a^5*b)*d*e^6 
- 180*(2*B*b^6*d^2*e^5 - 2*(6*B*a*b^5 + A*b^6)*d*e^6 + (5*B*a^2*b^4 + 2*A* 
a*b^5)*e^7)*x^5 - 150*(27*B*b^6*d^3*e^4 - 9*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 
3*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 - 
 100*(82*B*b^6*d^4*e^3 - 22*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 6*(5*B*a^2*b^4 + 
 2*A*a*b^5)*d^2*e^5 + 2*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + (3*B*a^4*b^2 + 
 4*A*a^3*b^3)*e^7)*x^3 - 15*(515*B*b^6*d^5*e^2 - 125*(6*B*a*b^5 + A*b^6)*d 
^4*e^3 + 30*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 + 10*(4*B*a^3*b^3 + 3*A*a^2* 
b^4)*d^2*e^5 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 + 3*(2*B*a^5*b + 5*A*a^ 
4*b^2)*e^7)*x^2 - 6*(599*B*b^6*d^6*e - 137*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 3 
0*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 + 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e 
^4 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 + 3*(2*B*a^5*b + 5*A*a^4*b^2)*d 
*e^6 + 2*(B*a^6 + 6*A*a^5*b)*e^7)*x - 60*(7*B*b^6*d^7 - (6*B*a*b^5 + A*b^6 
)*d^6*e + (7*B*b^6*d*e^6 - (6*B*a*b^5 + A*b^6)*e^7)*x^6 + 6*(7*B*b^6*d^2*e 
^5 - (6*B*a*b^5 + A*b^6)*d*e^6)*x^5 + 15*(7*B*b^6*d^3*e^4 - (6*B*a*b^5 + A 
*b^6)*d^2*e^5)*x^4 + 20*(7*B*b^6*d^4*e^3 - (6*B*a*b^5 + A*b^6)*d^3*e^4)*x^ 
3 + 15*(7*B*b^6*d^5*e^2 - (6*B*a*b^5 + A*b^6)*d^4*e^3)*x^2 + 6*(7*B*b^6...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^7} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**6*(B*x+A)/(e*x+d)**7,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 824 vs. \(2 (268) = 536\).

Time = 0.08 (sec) , antiderivative size = 824, normalized size of antiderivative = 2.96 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^7} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^7,x, algorithm="maxima")
 

Output:

B*b^6*x/e^7 - 1/60*(669*B*b^6*d^7 + 10*A*a^6*e^7 - 147*(6*B*a*b^5 + A*b^6) 
*d^6*e + 30*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + 10*(4*B*a^3*b^3 + 3*A*a^2* 
b^4)*d^4*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 3*(2*B*a^5*b + 5*A* 
a^4*b^2)*d^2*e^5 + 2*(B*a^6 + 6*A*a^5*b)*d*e^6 + 180*(7*B*b^6*d^2*e^5 - 2* 
(6*B*a*b^5 + A*b^6)*d*e^6 + (5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 150*(35*B 
*b^6*d^3*e^4 - 9*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 3*(5*B*a^2*b^4 + 2*A*a*b^5) 
*d*e^6 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 100*(91*B*b^6*d^4*e^3 - 22 
*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 + 2*(4* 
B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 15 
*(539*B*b^6*d^5*e^2 - 125*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 30*(5*B*a^2*b^4 + 
2*A*a*b^5)*d^3*e^4 + 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 5*(3*B*a^4*b 
^2 + 4*A*a^3*b^3)*d*e^6 + 3*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 6*(609*B* 
b^6*d^6*e - 137*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 30*(5*B*a^2*b^4 + 2*A*a*b^5) 
*d^4*e^3 + 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 5*(3*B*a^4*b^2 + 4*A*a 
^3*b^3)*d^2*e^5 + 3*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + 2*(B*a^6 + 6*A*a^5*b 
)*e^7)*x)/(e^14*x^6 + 6*d*e^13*x^5 + 15*d^2*e^12*x^4 + 20*d^3*e^11*x^3 + 1 
5*d^4*e^10*x^2 + 6*d^5*e^9*x + d^6*e^8) - (7*B*b^6*d - (6*B*a*b^5 + A*b^6) 
*e)*log(e*x + d)/e^8
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 826 vs. \(2 (268) = 536\).

Time = 0.12 (sec) , antiderivative size = 826, normalized size of antiderivative = 2.97 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^7} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^7,x, algorithm="giac")
 

Output:

B*b^6*x/e^7 - (7*B*b^6*d - 6*B*a*b^5*e - A*b^6*e)*log(abs(e*x + d))/e^8 - 
1/60*(669*B*b^6*d^7 - 882*B*a*b^5*d^6*e - 147*A*b^6*d^6*e + 150*B*a^2*b^4* 
d^5*e^2 + 60*A*a*b^5*d^5*e^2 + 40*B*a^3*b^3*d^4*e^3 + 30*A*a^2*b^4*d^4*e^3 
 + 15*B*a^4*b^2*d^3*e^4 + 20*A*a^3*b^3*d^3*e^4 + 6*B*a^5*b*d^2*e^5 + 15*A* 
a^4*b^2*d^2*e^5 + 2*B*a^6*d*e^6 + 12*A*a^5*b*d*e^6 + 10*A*a^6*e^7 + 180*(7 
*B*b^6*d^2*e^5 - 12*B*a*b^5*d*e^6 - 2*A*b^6*d*e^6 + 5*B*a^2*b^4*e^7 + 2*A* 
a*b^5*e^7)*x^5 + 150*(35*B*b^6*d^3*e^4 - 54*B*a*b^5*d^2*e^5 - 9*A*b^6*d^2* 
e^5 + 15*B*a^2*b^4*d*e^6 + 6*A*a*b^5*d*e^6 + 4*B*a^3*b^3*e^7 + 3*A*a^2*b^4 
*e^7)*x^4 + 100*(91*B*b^6*d^4*e^3 - 132*B*a*b^5*d^3*e^4 - 22*A*b^6*d^3*e^4 
 + 30*B*a^2*b^4*d^2*e^5 + 12*A*a*b^5*d^2*e^5 + 8*B*a^3*b^3*d*e^6 + 6*A*a^2 
*b^4*d*e^6 + 3*B*a^4*b^2*e^7 + 4*A*a^3*b^3*e^7)*x^3 + 15*(539*B*b^6*d^5*e^ 
2 - 750*B*a*b^5*d^4*e^3 - 125*A*b^6*d^4*e^3 + 150*B*a^2*b^4*d^3*e^4 + 60*A 
*a*b^5*d^3*e^4 + 40*B*a^3*b^3*d^2*e^5 + 30*A*a^2*b^4*d^2*e^5 + 15*B*a^4*b^ 
2*d*e^6 + 20*A*a^3*b^3*d*e^6 + 6*B*a^5*b*e^7 + 15*A*a^4*b^2*e^7)*x^2 + 6*( 
609*B*b^6*d^6*e - 822*B*a*b^5*d^5*e^2 - 137*A*b^6*d^5*e^2 + 150*B*a^2*b^4* 
d^4*e^3 + 60*A*a*b^5*d^4*e^3 + 40*B*a^3*b^3*d^3*e^4 + 30*A*a^2*b^4*d^3*e^4 
 + 15*B*a^4*b^2*d^2*e^5 + 20*A*a^3*b^3*d^2*e^5 + 6*B*a^5*b*d*e^6 + 15*A*a^ 
4*b^2*d*e^6 + 2*B*a^6*e^7 + 12*A*a^5*b*e^7)*x)/((e*x + d)^6*e^8)
 

Mupad [B] (verification not implemented)

Time = 1.06 (sec) , antiderivative size = 875, normalized size of antiderivative = 3.15 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^7} \, dx=\frac {\ln \left (d+e\,x\right )\,\left (A\,b^6\,e-7\,B\,b^6\,d+6\,B\,a\,b^5\,e\right )}{e^8}-\frac {x^3\,\left (5\,B\,a^4\,b^2\,e^6+\frac {40\,B\,a^3\,b^3\,d\,e^5}{3}+\frac {20\,A\,a^3\,b^3\,e^6}{3}+50\,B\,a^2\,b^4\,d^2\,e^4+10\,A\,a^2\,b^4\,d\,e^5-220\,B\,a\,b^5\,d^3\,e^3+20\,A\,a\,b^5\,d^2\,e^4+\frac {455\,B\,b^6\,d^4\,e^2}{3}-\frac {110\,A\,b^6\,d^3\,e^3}{3}\right )+\frac {2\,B\,a^6\,d\,e^6+10\,A\,a^6\,e^7+6\,B\,a^5\,b\,d^2\,e^5+12\,A\,a^5\,b\,d\,e^6+15\,B\,a^4\,b^2\,d^3\,e^4+15\,A\,a^4\,b^2\,d^2\,e^5+40\,B\,a^3\,b^3\,d^4\,e^3+20\,A\,a^3\,b^3\,d^3\,e^4+150\,B\,a^2\,b^4\,d^5\,e^2+30\,A\,a^2\,b^4\,d^4\,e^3-882\,B\,a\,b^5\,d^6\,e+60\,A\,a\,b^5\,d^5\,e^2+669\,B\,b^6\,d^7-147\,A\,b^6\,d^6\,e}{60\,e}+x\,\left (\frac {B\,a^6\,e^6}{5}+\frac {3\,B\,a^5\,b\,d\,e^5}{5}+\frac {6\,A\,a^5\,b\,e^6}{5}+\frac {3\,B\,a^4\,b^2\,d^2\,e^4}{2}+\frac {3\,A\,a^4\,b^2\,d\,e^5}{2}+4\,B\,a^3\,b^3\,d^3\,e^3+2\,A\,a^3\,b^3\,d^2\,e^4+15\,B\,a^2\,b^4\,d^4\,e^2+3\,A\,a^2\,b^4\,d^3\,e^3-\frac {411\,B\,a\,b^5\,d^5\,e}{5}+6\,A\,a\,b^5\,d^4\,e^2+\frac {609\,B\,b^6\,d^6}{10}-\frac {137\,A\,b^6\,d^5\,e}{10}\right )+x^5\,\left (15\,B\,a^2\,b^4\,e^6-36\,B\,a\,b^5\,d\,e^5+6\,A\,a\,b^5\,e^6+21\,B\,b^6\,d^2\,e^4-6\,A\,b^6\,d\,e^5\right )+x^2\,\left (\frac {3\,B\,a^5\,b\,e^6}{2}+\frac {15\,B\,a^4\,b^2\,d\,e^5}{4}+\frac {15\,A\,a^4\,b^2\,e^6}{4}+10\,B\,a^3\,b^3\,d^2\,e^4+5\,A\,a^3\,b^3\,d\,e^5+\frac {75\,B\,a^2\,b^4\,d^3\,e^3}{2}+\frac {15\,A\,a^2\,b^4\,d^2\,e^4}{2}-\frac {375\,B\,a\,b^5\,d^4\,e^2}{2}+15\,A\,a\,b^5\,d^3\,e^3+\frac {539\,B\,b^6\,d^5\,e}{4}-\frac {125\,A\,b^6\,d^4\,e^2}{4}\right )+x^4\,\left (10\,B\,a^3\,b^3\,e^6+\frac {75\,B\,a^2\,b^4\,d\,e^5}{2}+\frac {15\,A\,a^2\,b^4\,e^6}{2}-135\,B\,a\,b^5\,d^2\,e^4+15\,A\,a\,b^5\,d\,e^5+\frac {175\,B\,b^6\,d^3\,e^3}{2}-\frac {45\,A\,b^6\,d^2\,e^4}{2}\right )}{d^6\,e^7+6\,d^5\,e^8\,x+15\,d^4\,e^9\,x^2+20\,d^3\,e^{10}\,x^3+15\,d^2\,e^{11}\,x^4+6\,d\,e^{12}\,x^5+e^{13}\,x^6}+\frac {B\,b^6\,x}{e^7} \] Input:

int(((A + B*x)*(a + b*x)^6)/(d + e*x)^7,x)
 

Output:

(log(d + e*x)*(A*b^6*e - 7*B*b^6*d + 6*B*a*b^5*e))/e^8 - (x^3*((20*A*a^3*b 
^3*e^6)/3 + 5*B*a^4*b^2*e^6 - (110*A*b^6*d^3*e^3)/3 + (455*B*b^6*d^4*e^2)/ 
3 + 20*A*a*b^5*d^2*e^4 + 10*A*a^2*b^4*d*e^5 - 220*B*a*b^5*d^3*e^3 + (40*B* 
a^3*b^3*d*e^5)/3 + 50*B*a^2*b^4*d^2*e^4) + (10*A*a^6*e^7 + 669*B*b^6*d^7 - 
 147*A*b^6*d^6*e + 2*B*a^6*d*e^6 + 60*A*a*b^5*d^5*e^2 + 6*B*a^5*b*d^2*e^5 
+ 30*A*a^2*b^4*d^4*e^3 + 20*A*a^3*b^3*d^3*e^4 + 15*A*a^4*b^2*d^2*e^5 + 150 
*B*a^2*b^4*d^5*e^2 + 40*B*a^3*b^3*d^4*e^3 + 15*B*a^4*b^2*d^3*e^4 + 12*A*a^ 
5*b*d*e^6 - 882*B*a*b^5*d^6*e)/(60*e) + x*((B*a^6*e^6)/5 + (609*B*b^6*d^6) 
/10 + (6*A*a^5*b*e^6)/5 - (137*A*b^6*d^5*e)/10 + 6*A*a*b^5*d^4*e^2 + (3*A* 
a^4*b^2*d*e^5)/2 + 3*A*a^2*b^4*d^3*e^3 + 2*A*a^3*b^3*d^2*e^4 + 15*B*a^2*b^ 
4*d^4*e^2 + 4*B*a^3*b^3*d^3*e^3 + (3*B*a^4*b^2*d^2*e^4)/2 - (411*B*a*b^5*d 
^5*e)/5 + (3*B*a^5*b*d*e^5)/5) + x^5*(6*A*a*b^5*e^6 - 6*A*b^6*d*e^5 + 15*B 
*a^2*b^4*e^6 + 21*B*b^6*d^2*e^4 - 36*B*a*b^5*d*e^5) + x^2*((3*B*a^5*b*e^6) 
/2 + (539*B*b^6*d^5*e)/4 + (15*A*a^4*b^2*e^6)/4 - (125*A*b^6*d^4*e^2)/4 + 
15*A*a*b^5*d^3*e^3 + 5*A*a^3*b^3*d*e^5 - (375*B*a*b^5*d^4*e^2)/2 + (15*B*a 
^4*b^2*d*e^5)/4 + (15*A*a^2*b^4*d^2*e^4)/2 + (75*B*a^2*b^4*d^3*e^3)/2 + 10 
*B*a^3*b^3*d^2*e^4) + x^4*((15*A*a^2*b^4*e^6)/2 + 10*B*a^3*b^3*e^6 - (45*A 
*b^6*d^2*e^4)/2 + (175*B*b^6*d^3*e^3)/2 - 135*B*a*b^5*d^2*e^4 + (75*B*a^2* 
b^4*d*e^5)/2 + 15*A*a*b^5*d*e^5))/(d^6*e^7 + e^13*x^6 + 6*d^5*e^8*x + 6*d* 
e^12*x^5 + 15*d^4*e^9*x^2 + 20*d^3*e^10*x^3 + 15*d^2*e^11*x^4) + (B*b^6...
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 735, normalized size of antiderivative = 2.64 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^7} \, dx=\frac {6300 \,\mathrm {log}\left (e x +d \right ) a \,b^{6} d^{5} e^{3} x^{2}-2520 \,\mathrm {log}\left (e x +d \right ) b^{7} d^{3} e^{5} x^{5}-6300 \,\mathrm {log}\left (e x +d \right ) b^{7} d^{6} e^{2} x^{2}-420 a \,b^{6} d \,e^{7} x^{6}+420 \,\mathrm {log}\left (e x +d \right ) a \,b^{6} d \,e^{7} x^{6}+8400 \,\mathrm {log}\left (e x +d \right ) a \,b^{6} d^{4} e^{4} x^{3}-10 a^{7} d \,e^{7}-420 \,\mathrm {log}\left (e x +d \right ) b^{7} d^{2} e^{6} x^{6}+210 a^{2} b^{5} e^{8} x^{6}-14 a^{6} b \,d^{2} e^{6}-6300 \,\mathrm {log}\left (e x +d \right ) b^{7} d^{4} e^{4} x^{4}-21 a^{5} b^{2} d^{3} e^{5}-35 a^{4} b^{3} d^{4} e^{4}-70 a^{3} b^{4} d^{5} e^{3}+609 a \,b^{6} d^{7} e -420 \,\mathrm {log}\left (e x +d \right ) b^{7} d^{8}-3234 b^{7} d^{7} e x -6825 b^{7} d^{6} e^{2} x^{2}-7000 b^{7} d^{5} e^{3} x^{3}-3150 b^{7} d^{4} e^{4} x^{4}+420 b^{7} d^{2} e^{6} x^{6}+60 b^{7} d \,e^{7} x^{7}+2520 \,\mathrm {log}\left (e x +d \right ) a \,b^{6} d^{6} e^{2} x +6300 \,\mathrm {log}\left (e x +d \right ) a \,b^{6} d^{3} e^{5} x^{4}+420 \,\mathrm {log}\left (e x +d \right ) a \,b^{6} d^{7} e -2520 \,\mathrm {log}\left (e x +d \right ) b^{7} d^{7} e x -84 a^{6} b d \,e^{7} x -126 a^{5} b^{2} d^{2} e^{6} x -315 a^{5} b^{2} d \,e^{7} x^{2}-210 a^{4} b^{3} d^{3} e^{5} x -525 a^{4} b^{3} d^{2} e^{6} x^{2}-700 a^{4} b^{3} d \,e^{7} x^{3}-420 a^{3} b^{4} d^{4} e^{4} x -1050 a^{3} b^{4} d^{3} e^{5} x^{2}-1400 a^{3} b^{4} d^{2} e^{6} x^{3}-1050 a^{3} b^{4} d \,e^{7} x^{4}+3234 a \,b^{6} d^{6} e^{2} x +6825 a \,b^{6} d^{5} e^{3} x^{2}+7000 a \,b^{6} d^{4} e^{4} x^{3}+3150 a \,b^{6} d^{3} e^{5} x^{4}-609 b^{7} d^{8}-8400 \,\mathrm {log}\left (e x +d \right ) b^{7} d^{5} e^{3} x^{3}+2520 \,\mathrm {log}\left (e x +d \right ) a \,b^{6} d^{2} e^{6} x^{5}}{60 d \,e^{8} \left (e^{6} x^{6}+6 d \,e^{5} x^{5}+15 d^{2} e^{4} x^{4}+20 d^{3} e^{3} x^{3}+15 d^{4} e^{2} x^{2}+6 d^{5} e x +d^{6}\right )} \] Input:

int((b*x+a)^6*(B*x+A)/(e*x+d)^7,x)
                                                                                    
                                                                                    
 

Output:

(420*log(d + e*x)*a*b**6*d**7*e + 2520*log(d + e*x)*a*b**6*d**6*e**2*x + 6 
300*log(d + e*x)*a*b**6*d**5*e**3*x**2 + 8400*log(d + e*x)*a*b**6*d**4*e** 
4*x**3 + 6300*log(d + e*x)*a*b**6*d**3*e**5*x**4 + 2520*log(d + e*x)*a*b** 
6*d**2*e**6*x**5 + 420*log(d + e*x)*a*b**6*d*e**7*x**6 - 420*log(d + e*x)* 
b**7*d**8 - 2520*log(d + e*x)*b**7*d**7*e*x - 6300*log(d + e*x)*b**7*d**6* 
e**2*x**2 - 8400*log(d + e*x)*b**7*d**5*e**3*x**3 - 6300*log(d + e*x)*b**7 
*d**4*e**4*x**4 - 2520*log(d + e*x)*b**7*d**3*e**5*x**5 - 420*log(d + e*x) 
*b**7*d**2*e**6*x**6 - 10*a**7*d*e**7 - 14*a**6*b*d**2*e**6 - 84*a**6*b*d* 
e**7*x - 21*a**5*b**2*d**3*e**5 - 126*a**5*b**2*d**2*e**6*x - 315*a**5*b** 
2*d*e**7*x**2 - 35*a**4*b**3*d**4*e**4 - 210*a**4*b**3*d**3*e**5*x - 525*a 
**4*b**3*d**2*e**6*x**2 - 700*a**4*b**3*d*e**7*x**3 - 70*a**3*b**4*d**5*e* 
*3 - 420*a**3*b**4*d**4*e**4*x - 1050*a**3*b**4*d**3*e**5*x**2 - 1400*a**3 
*b**4*d**2*e**6*x**3 - 1050*a**3*b**4*d*e**7*x**4 + 210*a**2*b**5*e**8*x** 
6 + 609*a*b**6*d**7*e + 3234*a*b**6*d**6*e**2*x + 6825*a*b**6*d**5*e**3*x* 
*2 + 7000*a*b**6*d**4*e**4*x**3 + 3150*a*b**6*d**3*e**5*x**4 - 420*a*b**6* 
d*e**7*x**6 - 609*b**7*d**8 - 3234*b**7*d**7*e*x - 6825*b**7*d**6*e**2*x** 
2 - 7000*b**7*d**5*e**3*x**3 - 3150*b**7*d**4*e**4*x**4 + 420*b**7*d**2*e* 
*6*x**6 + 60*b**7*d*e**7*x**7)/(60*d*e**8*(d**6 + 6*d**5*e*x + 15*d**4*e** 
2*x**2 + 20*d**3*e**3*x**3 + 15*d**2*e**4*x**4 + 6*d*e**5*x**5 + e**6*x**6 
))