\(\int \frac {(a+b x)^6 (A+B x)}{(d+e x)^4} \, dx\) [53]

Optimal result

Integrand size = 20, antiderivative size = 279 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^4} \, dx=-\frac {5 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e) x}{e^7}+\frac {(b d-a e)^6 (B d-A e)}{3 e^8 (d+e x)^3}-\frac {(b d-a e)^5 (7 b B d-6 A b e-a B e)}{2 e^8 (d+e x)^2}+\frac {3 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e)}{e^8 (d+e x)}+\frac {3 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e) (d+e x)^2}{2 e^8}-\frac {b^5 (7 b B d-A b e-6 a B e) (d+e x)^3}{3 e^8}+\frac {b^6 B (d+e x)^4}{4 e^8}+\frac {5 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e) \log (d+e x)}{e^8} \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^4} \, dx=\frac {12 b^3 e \left (20 a^3 B e^3+12 a b^2 d e (5 B d-2 A e)+15 a^2 b e^2 (-4 B d+A e)+10 b^3 d^2 (-2 B d+A e)\right ) x-6 b^4 e^2 \left (-15 a^2 B e^2-6 a b e (-4 B d+A e)+2 b^2 d (-5 B d+2 A e)\right ) x^2+4 b^5 e^3 (-4 b B d+A b e+6 a B e) x^3+3 b^6 B e^4 x^4+\frac {4 (b d-a e)^6 (B d-A e)}{(d+e x)^3}-\frac {6 (b d-a e)^5 (7 b B d-6 A b e-a B e)}{(d+e x)^2}+\frac {36 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e)}{d+e x}+60 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e) \log (d+e x)}{12 e^8} \] Input:

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^4,x]
 

Output:

(12*b^3*e*(20*a^3*B*e^3 + 12*a*b^2*d*e*(5*B*d - 2*A*e) + 15*a^2*b*e^2*(-4* 
B*d + A*e) + 10*b^3*d^2*(-2*B*d + A*e))*x - 6*b^4*e^2*(-15*a^2*B*e^2 - 6*a 
*b*e*(-4*B*d + A*e) + 2*b^2*d*(-5*B*d + 2*A*e))*x^2 + 4*b^5*e^3*(-4*b*B*d 
+ A*b*e + 6*a*B*e)*x^3 + 3*b^6*B*e^4*x^4 + (4*(b*d - a*e)^6*(B*d - A*e))/( 
d + e*x)^3 - (6*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e))/(d + e*x)^2 + ( 
36*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e))/(d + e*x) + 60*b^2*(b*d 
- a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B*e)*Log[d + e*x])/(12*e^8)
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 279, 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.

\(\displaystyle \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^4} \, dx\)

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

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

Input:

Int[((a + b*x)^6*(A + B*x))/(d + e*x)^4,x]
 

Output:

(-5*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*x)/e^7 + ((b*d - a*e)^ 
6*(B*d - A*e))/(3*e^8*(d + e*x)^3) - ((b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a 
*B*e))/(2*e^8*(d + e*x)^2) + (3*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B 
*e))/(e^8*(d + e*x)) + (3*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d 
 + e*x)^2)/(2*e^8) - (b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^3)/(3*e^8) 
 + (b^6*B*(d + e*x)^4)/(4*e^8) + (5*b^2*(b*d - a*e)^3*(7*b*B*d - 4*A*b*e - 
 3*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. \(794\) vs. \(2(269)=538\).

Time = 0.24 (sec) , antiderivative size = 795, normalized size of antiderivative = 2.85

method	result	size
norman	\(\frac {-\frac {2 a^{6} A \,e^{7}+6 A \,a^{5} b d \,e^{6}+30 A \,a^{4} b^{2} d^{2} e^{5}-220 A \,a^{3} b^{3} d^{3} e^{4}+660 A \,a^{2} b^{4} d^{4} e^{3}-660 A a \,b^{5} d^{5} e^{2}+220 A \,b^{6} d^{6} e +B \,a^{6} d \,e^{6}+12 B \,a^{5} b \,d^{2} e^{5}-165 B \,a^{4} b^{2} d^{3} e^{4}+880 B \,a^{3} b^{3} d^{4} e^{3}-1650 B \,a^{2} b^{4} d^{5} e^{2}+1320 B a \,b^{5} d^{6} e -385 b^{6} B \,d^{7}}{6 e^{8}}-\frac {3 \left (5 A \,a^{4} b^{2} e^{5}-20 A \,a^{3} b^{3} d \,e^{4}+60 A \,a^{2} b^{4} d^{2} e^{3}-60 A a \,b^{5} d^{3} e^{2}+20 A \,b^{6} d^{4} e +2 B \,a^{5} b \,e^{5}-15 B \,a^{4} b^{2} d \,e^{4}+80 B \,a^{3} b^{3} d^{2} e^{3}-150 B \,a^{2} b^{4} d^{3} e^{2}+120 B a \,b^{5} d^{4} e -35 b^{6} B \,d^{5}\right ) x^{2}}{e^{6}}-\frac {\left (6 A \,a^{5} b \,e^{6}+30 A \,a^{4} b^{2} d \,e^{5}-180 A \,a^{3} b^{3} d^{2} e^{4}+540 A \,a^{2} b^{4} d^{3} e^{3}-540 A a \,b^{5} d^{4} e^{2}+180 A \,b^{6} d^{5} e +B \,a^{6} e^{6}+12 B \,a^{5} b d \,e^{5}-135 B \,a^{4} b^{2} d^{2} e^{4}+720 B \,a^{3} b^{3} d^{3} e^{3}-1350 B \,a^{2} b^{4} d^{4} e^{2}+1080 B a \,b^{5} d^{5} e -315 b^{6} B \,d^{6}\right ) x}{2 e^{7}}+\frac {5 b^{3} \left (12 A \,a^{2} b \,e^{3}-12 A a \,b^{2} d \,e^{2}+4 A \,b^{3} d^{2} e +16 B \,a^{3} e^{3}-30 B \,a^{2} b d \,e^{2}+24 B a \,b^{2} d^{2} e -7 b^{3} B \,d^{3}\right ) x^{4}}{4 e^{4}}+\frac {b^{4} \left (12 A a b \,e^{2}-4 A \,b^{2} d e +30 B \,a^{2} e^{2}-24 B a b d e +7 b^{2} B \,d^{2}\right ) x^{5}}{4 e^{3}}+\frac {b^{5} \left (4 A b e +24 B a e -7 B b d \right ) x^{6}}{12 e^{2}}+\frac {b^{6} B \,x^{7}}{4 e}}{\left (e x +d \right )^{3}}+\frac {5 b^{2} \left (4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e +3 B \,a^{4} e^{4}-16 B \,a^{3} b d \,e^{3}+30 B \,a^{2} b^{2} d^{2} e^{2}-24 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right ) \ln \left (e x +d \right )}{e^{8}}\)	\(795\)
default	\(\frac {b^{3} \left (\frac {1}{4} b^{3} B \,x^{4} e^{3}+\frac {1}{3} A \,b^{3} e^{3} x^{3}+2 B a \,b^{2} e^{3} x^{3}-\frac {4}{3} B \,b^{3} d \,e^{2} x^{3}+3 A a \,b^{2} e^{3} x^{2}-2 A \,b^{3} d \,e^{2} x^{2}+\frac {15}{2} B \,a^{2} b \,e^{3} x^{2}-12 B a \,b^{2} d \,e^{2} x^{2}+5 B \,b^{3} d^{2} e \,x^{2}+15 A \,a^{2} b \,e^{3} x -24 A a \,b^{2} d \,e^{2} x +10 A \,b^{3} d^{2} e x +20 B \,a^{3} e^{3} x -60 B \,a^{2} b d \,e^{2} x +60 B a \,b^{2} d^{2} e x -20 b^{3} B \,d^{3} x \right )}{e^{7}}-\frac {3 b \left (5 A \,a^{4} b \,e^{5}-20 A \,a^{3} b^{2} d \,e^{4}+30 A \,a^{2} b^{3} d^{2} e^{3}-20 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e +2 B \,a^{5} e^{5}-15 B \,a^{4} b d \,e^{4}+40 B \,a^{3} b^{2} d^{2} e^{3}-50 B \,a^{2} b^{3} d^{3} e^{2}+30 B a \,b^{4} d^{4} e -7 B \,b^{5} d^{5}\right )}{e^{8} \left (e x +d \right )}-\frac {6 A \,a^{5} b \,e^{6}-30 A \,a^{4} b^{2} d \,e^{5}+60 A \,a^{3} b^{3} d^{2} e^{4}-60 A \,a^{2} b^{4} d^{3} e^{3}+30 A a \,b^{5} d^{4} e^{2}-6 A \,b^{6} d^{5} e +B \,a^{6} e^{6}-12 B \,a^{5} b d \,e^{5}+45 B \,a^{4} b^{2} d^{2} e^{4}-80 B \,a^{3} b^{3} d^{3} e^{3}+75 B \,a^{2} b^{4} d^{4} e^{2}-36 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}}{2 e^{8} \left (e x +d \right )^{2}}+\frac {5 b^{2} \left (4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e +3 B \,a^{4} e^{4}-16 B \,a^{3} b d \,e^{3}+30 B \,a^{2} b^{2} d^{2} e^{2}-24 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right ) \ln \left (e x +d \right )}{e^{8}}-\frac {a^{6} A \,e^{7}-6 A \,a^{5} b d \,e^{6}+15 A \,a^{4} b^{2} d^{2} e^{5}-20 A \,a^{3} b^{3} d^{3} e^{4}+15 A \,a^{2} b^{4} d^{4} e^{3}-6 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e -B \,a^{6} d \,e^{6}+6 B \,a^{5} b \,d^{2} e^{5}-15 B \,a^{4} b^{2} d^{3} e^{4}+20 B \,a^{3} b^{3} d^{4} e^{3}-15 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e -b^{6} B \,d^{7}}{3 e^{8} \left (e x +d \right )^{3}}\)	\(825\)
risch	\(\frac {\left (-15 A \,a^{4} b^{2} e^{6}+60 A \,a^{3} b^{3} d \,e^{5}-90 A \,a^{2} b^{4} d^{2} e^{4}+60 A a \,b^{5} d^{3} e^{3}-15 A \,b^{6} d^{4} e^{2}-6 B \,a^{5} b \,e^{6}+45 B \,a^{4} b^{2} d \,e^{5}-120 B \,a^{3} b^{3} d^{2} e^{4}+150 B \,a^{2} b^{4} d^{3} e^{3}-90 B a \,b^{5} d^{4} e^{2}+21 b^{6} B \,d^{5} e \right ) x^{2}+\left (-3 A \,a^{5} b \,e^{6}-15 A \,a^{4} b^{2} d \,e^{5}+90 A \,a^{3} b^{3} d^{2} e^{4}-150 A \,a^{2} b^{4} d^{3} e^{3}+105 A a \,b^{5} d^{4} e^{2}-27 A \,b^{6} d^{5} e -\frac {1}{2} B \,a^{6} e^{6}-6 B \,a^{5} b d \,e^{5}+\frac {135}{2} B \,a^{4} b^{2} d^{2} e^{4}-200 B \,a^{3} b^{3} d^{3} e^{3}+\frac {525}{2} B \,a^{2} b^{4} d^{4} e^{2}-162 B a \,b^{5} d^{5} e +\frac {77}{2} b^{6} B \,d^{6}\right ) x -\frac {2 a^{6} A \,e^{7}+6 A \,a^{5} b d \,e^{6}+30 A \,a^{4} b^{2} d^{2} e^{5}-220 A \,a^{3} b^{3} d^{3} e^{4}+390 A \,a^{2} b^{4} d^{4} e^{3}-282 A a \,b^{5} d^{5} e^{2}+74 A \,b^{6} d^{6} e +B \,a^{6} d \,e^{6}+12 B \,a^{5} b \,d^{2} e^{5}-165 B \,a^{4} b^{2} d^{3} e^{4}+520 B \,a^{3} b^{3} d^{4} e^{3}-705 B \,a^{2} b^{4} d^{5} e^{2}+444 B a \,b^{5} d^{6} e -107 b^{6} B \,d^{7}}{6 e}}{e^{7} \left (e x +d \right )^{3}}-\frac {12 b^{5} B a d \,x^{2}}{e^{5}}-\frac {24 b^{5} A a d x}{e^{5}}-\frac {60 b^{4} B \,a^{2} d x}{e^{5}}+\frac {60 b^{5} B a \,d^{2} x}{e^{6}}+\frac {3 b^{5} A a \,x^{2}}{e^{4}}-\frac {2 b^{6} A d \,x^{2}}{e^{5}}+\frac {15 b^{4} B \,a^{2} x^{2}}{2 e^{4}}+\frac {5 b^{6} B \,d^{2} x^{2}}{e^{6}}+\frac {15 b^{4} A \,a^{2} x}{e^{4}}+\frac {10 b^{6} A \,d^{2} x}{e^{6}}+\frac {20 b^{3} B \,a^{3} x}{e^{4}}-\frac {20 b^{6} B \,d^{3} x}{e^{7}}+\frac {20 b^{3} \ln \left (e x +d \right ) A \,a^{3}}{e^{4}}-\frac {20 b^{6} \ln \left (e x +d \right ) A \,d^{3}}{e^{7}}+\frac {15 b^{2} \ln \left (e x +d \right ) B \,a^{4}}{e^{4}}+\frac {35 b^{6} \ln \left (e x +d \right ) B \,d^{4}}{e^{8}}+\frac {2 b^{5} B a \,x^{3}}{e^{4}}-\frac {4 b^{6} B d \,x^{3}}{3 e^{5}}-\frac {60 b^{4} \ln \left (e x +d \right ) A \,a^{2} d}{e^{5}}+\frac {60 b^{5} \ln \left (e x +d \right ) A a \,d^{2}}{e^{6}}-\frac {80 b^{3} \ln \left (e x +d \right ) B \,a^{3} d}{e^{5}}+\frac {150 b^{4} \ln \left (e x +d \right ) B \,a^{2} d^{2}}{e^{6}}-\frac {120 b^{5} \ln \left (e x +d \right ) B a \,d^{3}}{e^{7}}+\frac {b^{6} B \,x^{4}}{4 e^{4}}+\frac {b^{6} A \,x^{3}}{3 e^{4}}\)	\(886\)
parallelrisch	\(\text {Expression too large to display}\)	\(1527\)

Input:

int((b*x+a)^6*(B*x+A)/(e*x+d)^4,x,method=_RETURNVERBOSE)
 

Output:

(-1/6*(2*A*a^6*e^7+6*A*a^5*b*d*e^6+30*A*a^4*b^2*d^2*e^5-220*A*a^3*b^3*d^3* 
e^4+660*A*a^2*b^4*d^4*e^3-660*A*a*b^5*d^5*e^2+220*A*b^6*d^6*e+B*a^6*d*e^6+ 
12*B*a^5*b*d^2*e^5-165*B*a^4*b^2*d^3*e^4+880*B*a^3*b^3*d^4*e^3-1650*B*a^2* 
b^4*d^5*e^2+1320*B*a*b^5*d^6*e-385*B*b^6*d^7)/e^8-3*(5*A*a^4*b^2*e^5-20*A* 
a^3*b^3*d*e^4+60*A*a^2*b^4*d^2*e^3-60*A*a*b^5*d^3*e^2+20*A*b^6*d^4*e+2*B*a 
^5*b*e^5-15*B*a^4*b^2*d*e^4+80*B*a^3*b^3*d^2*e^3-150*B*a^2*b^4*d^3*e^2+120 
*B*a*b^5*d^4*e-35*B*b^6*d^5)/e^6*x^2-1/2*(6*A*a^5*b*e^6+30*A*a^4*b^2*d*e^5 
-180*A*a^3*b^3*d^2*e^4+540*A*a^2*b^4*d^3*e^3-540*A*a*b^5*d^4*e^2+180*A*b^6 
*d^5*e+B*a^6*e^6+12*B*a^5*b*d*e^5-135*B*a^4*b^2*d^2*e^4+720*B*a^3*b^3*d^3* 
e^3-1350*B*a^2*b^4*d^4*e^2+1080*B*a*b^5*d^5*e-315*B*b^6*d^6)/e^7*x+5/4*b^3 
*(12*A*a^2*b*e^3-12*A*a*b^2*d*e^2+4*A*b^3*d^2*e+16*B*a^3*e^3-30*B*a^2*b*d* 
e^2+24*B*a*b^2*d^2*e-7*B*b^3*d^3)/e^4*x^4+1/4*b^4*(12*A*a*b*e^2-4*A*b^2*d* 
e+30*B*a^2*e^2-24*B*a*b*d*e+7*B*b^2*d^2)/e^3*x^5+1/12*b^5*(4*A*b*e+24*B*a* 
e-7*B*b*d)/e^2*x^6+1/4*b^6*B/e*x^7)/(e*x+d)^3+5*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)*ln(e*x+d)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1225 vs. \(2 (269) = 538\).

Time = 0.10 (sec) , antiderivative size = 1225, normalized size of antiderivative = 4.39 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^4} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^4,x, algorithm="fricas")
 

Output:

1/12*(3*B*b^6*e^7*x^7 + 214*B*b^6*d^7 - 4*A*a^6*e^7 - 148*(6*B*a*b^5 + A*b 
^6)*d^6*e + 282*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 260*(4*B*a^3*b^3 + 3*A 
*a^2*b^4)*d^4*e^3 + 110*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - 12*(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 - (7*B*b^6*d*e^6 - 
4*(6*B*a*b^5 + A*b^6)*e^7)*x^6 + 3*(7*B*b^6*d^2*e^5 - 4*(6*B*a*b^5 + A*b^6 
)*d*e^6 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 - 15*(7*B*b^6*d^3*e^4 - 4*( 
6*B*a*b^5 + A*b^6)*d^2*e^5 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 - 4*(4*B*a^ 
3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 - 2*(278*B*b^6*d^4*e^3 - 146*(6*B*a*b^5 + A* 
b^6)*d^3*e^4 + 189*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 - 90*(4*B*a^3*b^3 + 3 
*A*a^2*b^4)*d*e^6)*x^3 - 6*(68*B*b^6*d^5*e^2 - 26*(6*B*a*b^5 + A*b^6)*d^4* 
e^3 + 9*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 + 30*(4*B*a^3*b^3 + 3*A*a^2*b^4) 
*d^2*e^5 - 30*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 + 6*(2*B*a^5*b + 5*A*a^4*b 
^2)*e^7)*x^2 + 6*(37*B*b^6*d^6*e - 34*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 81*(5* 
B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - 90*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 
45*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 - 6*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 
 - (B*a^6 + 6*A*a^5*b)*e^7)*x + 60*(7*B*b^6*d^7 - 4*(6*B*a*b^5 + A*b^6)*d^ 
6*e + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)* 
d^4*e^3 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + (7*B*b^6*d^4*e^3 - 4*(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 - 4*(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 + 3*(7*...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 867 vs. \(2 (284) = 568\).

Time = 29.91 (sec) , antiderivative size = 867, normalized size of antiderivative = 3.11 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^4} \, dx=\frac {B b^{6} x^{4}}{4 e^{4}} + \frac {5 b^{2} \left (a e - b d\right )^{3} \cdot \left (4 A b e + 3 B a e - 7 B b d\right ) \log {\left (d + e x \right )}}{e^{8}} + x^{3} \left (\frac {A b^{6}}{3 e^{4}} + \frac {2 B a b^{5}}{e^{4}} - \frac {4 B b^{6} d}{3 e^{5}}\right ) + x^{2} \cdot \left (\frac {3 A a b^{5}}{e^{4}} - \frac {2 A b^{6} d}{e^{5}} + \frac {15 B a^{2} b^{4}}{2 e^{4}} - \frac {12 B a b^{5} d}{e^{5}} + \frac {5 B b^{6} d^{2}}{e^{6}}\right ) + x \left (\frac {15 A a^{2} b^{4}}{e^{4}} - \frac {24 A a b^{5} d}{e^{5}} + \frac {10 A b^{6} d^{2}}{e^{6}} + \frac {20 B a^{3} b^{3}}{e^{4}} - \frac {60 B a^{2} b^{4} d}{e^{5}} + \frac {60 B a b^{5} d^{2}}{e^{6}} - \frac {20 B b^{6} d^{3}}{e^{7}}\right ) + \frac {- 2 A a^{6} e^{7} - 6 A a^{5} b d e^{6} - 30 A a^{4} b^{2} d^{2} e^{5} + 220 A a^{3} b^{3} d^{3} e^{4} - 390 A a^{2} b^{4} d^{4} e^{3} + 282 A a b^{5} d^{5} e^{2} - 74 A b^{6} d^{6} e - B a^{6} d e^{6} - 12 B a^{5} b d^{2} e^{5} + 165 B a^{4} b^{2} d^{3} e^{4} - 520 B a^{3} b^{3} d^{4} e^{3} + 705 B a^{2} b^{4} d^{5} e^{2} - 444 B a b^{5} d^{6} e + 107 B b^{6} d^{7} + x^{2} \left (- 90 A a^{4} b^{2} e^{7} + 360 A a^{3} b^{3} d e^{6} - 540 A a^{2} b^{4} d^{2} e^{5} + 360 A a b^{5} d^{3} e^{4} - 90 A b^{6} d^{4} e^{3} - 36 B a^{5} b e^{7} + 270 B a^{4} b^{2} d e^{6} - 720 B a^{3} b^{3} d^{2} e^{5} + 900 B a^{2} b^{4} d^{3} e^{4} - 540 B a b^{5} d^{4} e^{3} + 126 B b^{6} d^{5} e^{2}\right ) + x \left (- 18 A a^{5} b e^{7} - 90 A a^{4} b^{2} d e^{6} + 540 A a^{3} b^{3} d^{2} e^{5} - 900 A a^{2} b^{4} d^{3} e^{4} + 630 A a b^{5} d^{4} e^{3} - 162 A b^{6} d^{5} e^{2} - 3 B a^{6} e^{7} - 36 B a^{5} b d e^{6} + 405 B a^{4} b^{2} d^{2} e^{5} - 1200 B a^{3} b^{3} d^{3} e^{4} + 1575 B a^{2} b^{4} d^{4} e^{3} - 972 B a b^{5} d^{5} e^{2} + 231 B b^{6} d^{6} e\right )}{6 d^{3} e^{8} + 18 d^{2} e^{9} x + 18 d e^{10} x^{2} + 6 e^{11} x^{3}} \] Input:

integrate((b*x+a)**6*(B*x+A)/(e*x+d)**4,x)
 

Output:

B*b**6*x**4/(4*e**4) + 5*b**2*(a*e - b*d)**3*(4*A*b*e + 3*B*a*e - 7*B*b*d) 
*log(d + e*x)/e**8 + x**3*(A*b**6/(3*e**4) + 2*B*a*b**5/e**4 - 4*B*b**6*d/ 
(3*e**5)) + x**2*(3*A*a*b**5/e**4 - 2*A*b**6*d/e**5 + 15*B*a**2*b**4/(2*e* 
*4) - 12*B*a*b**5*d/e**5 + 5*B*b**6*d**2/e**6) + x*(15*A*a**2*b**4/e**4 - 
24*A*a*b**5*d/e**5 + 10*A*b**6*d**2/e**6 + 20*B*a**3*b**3/e**4 - 60*B*a**2 
*b**4*d/e**5 + 60*B*a*b**5*d**2/e**6 - 20*B*b**6*d**3/e**7) + (-2*A*a**6*e 
**7 - 6*A*a**5*b*d*e**6 - 30*A*a**4*b**2*d**2*e**5 + 220*A*a**3*b**3*d**3* 
e**4 - 390*A*a**2*b**4*d**4*e**3 + 282*A*a*b**5*d**5*e**2 - 74*A*b**6*d**6 
*e - B*a**6*d*e**6 - 12*B*a**5*b*d**2*e**5 + 165*B*a**4*b**2*d**3*e**4 - 5 
20*B*a**3*b**3*d**4*e**3 + 705*B*a**2*b**4*d**5*e**2 - 444*B*a*b**5*d**6*e 
 + 107*B*b**6*d**7 + x**2*(-90*A*a**4*b**2*e**7 + 360*A*a**3*b**3*d*e**6 - 
 540*A*a**2*b**4*d**2*e**5 + 360*A*a*b**5*d**3*e**4 - 90*A*b**6*d**4*e**3 
- 36*B*a**5*b*e**7 + 270*B*a**4*b**2*d*e**6 - 720*B*a**3*b**3*d**2*e**5 + 
900*B*a**2*b**4*d**3*e**4 - 540*B*a*b**5*d**4*e**3 + 126*B*b**6*d**5*e**2) 
 + x*(-18*A*a**5*b*e**7 - 90*A*a**4*b**2*d*e**6 + 540*A*a**3*b**3*d**2*e** 
5 - 900*A*a**2*b**4*d**3*e**4 + 630*A*a*b**5*d**4*e**3 - 162*A*b**6*d**5*e 
**2 - 3*B*a**6*e**7 - 36*B*a**5*b*d*e**6 + 405*B*a**4*b**2*d**2*e**5 - 120 
0*B*a**3*b**3*d**3*e**4 + 1575*B*a**2*b**4*d**4*e**3 - 972*B*a*b**5*d**5*e 
**2 + 231*B*b**6*d**6*e))/(6*d**3*e**8 + 18*d**2*e**9*x + 18*d*e**10*x**2 
+ 6*e**11*x**3)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 793 vs. \(2 (269) = 538\).

Time = 0.07 (sec) , antiderivative size = 793, normalized size of antiderivative = 2.84 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^4} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^4,x, algorithm="maxima")
 

Output:

1/6*(107*B*b^6*d^7 - 2*A*a^6*e^7 - 74*(6*B*a*b^5 + A*b^6)*d^6*e + 141*(5*B 
*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 130*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 
55*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - 6*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e 
^5 - (B*a^6 + 6*A*a^5*b)*d*e^6 + 18*(7*B*b^6*d^5*e^2 - 5*(6*B*a*b^5 + A*b^ 
6)*d^4*e^3 + 10*(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 - (2*B*a^5*b + 5*A* 
a^4*b^2)*e^7)*x^2 + 3*(77*B*b^6*d^6*e - 54*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 1 
05*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - 100*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3 
*e^4 + 45*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 - 6*(2*B*a^5*b + 5*A*a^4*b^2 
)*d*e^6 - (B*a^6 + 6*A*a^5*b)*e^7)*x)/(e^11*x^3 + 3*d*e^10*x^2 + 3*d^2*e^9 
*x + d^3*e^8) + 1/12*(3*B*b^6*e^3*x^4 - 4*(4*B*b^6*d*e^2 - (6*B*a*b^5 + A* 
b^6)*e^3)*x^3 + 6*(10*B*b^6*d^2*e - 4*(6*B*a*b^5 + A*b^6)*d*e^2 + 3*(5*B*a 
^2*b^4 + 2*A*a*b^5)*e^3)*x^2 - 12*(20*B*b^6*d^3 - 10*(6*B*a*b^5 + A*b^6)*d 
^2*e + 12*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^2 - 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)* 
e^3)*x)/e^7 + 5*(7*B*b^6*d^4 - 4*(6*B*a*b^5 + A*b^6)*d^3*e + 6*(5*B*a^2*b^ 
4 + 2*A*a*b^5)*d^2*e^2 - 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^3 + (3*B*a^4*b^ 
2 + 4*A*a^3*b^3)*e^4)*log(e*x + d)/e^8
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 848 vs. \(2 (269) = 538\).

Time = 0.13 (sec) , antiderivative size = 848, normalized size of antiderivative = 3.04 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^4} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^4,x, algorithm="giac")
 

Output:

5*(7*B*b^6*d^4 - 24*B*a*b^5*d^3*e - 4*A*b^6*d^3*e + 30*B*a^2*b^4*d^2*e^2 + 
 12*A*a*b^5*d^2*e^2 - 16*B*a^3*b^3*d*e^3 - 12*A*a^2*b^4*d*e^3 + 3*B*a^4*b^ 
2*e^4 + 4*A*a^3*b^3*e^4)*log(abs(e*x + d))/e^8 + 1/6*(107*B*b^6*d^7 - 444* 
B*a*b^5*d^6*e - 74*A*b^6*d^6*e + 705*B*a^2*b^4*d^5*e^2 + 282*A*a*b^5*d^5*e 
^2 - 520*B*a^3*b^3*d^4*e^3 - 390*A*a^2*b^4*d^4*e^3 + 165*B*a^4*b^2*d^3*e^4 
 + 220*A*a^3*b^3*d^3*e^4 - 12*B*a^5*b*d^2*e^5 - 30*A*a^4*b^2*d^2*e^5 - B*a 
^6*d*e^6 - 6*A*a^5*b*d*e^6 - 2*A*a^6*e^7 + 18*(7*B*b^6*d^5*e^2 - 30*B*a*b^ 
5*d^4*e^3 - 5*A*b^6*d^4*e^3 + 50*B*a^2*b^4*d^3*e^4 + 20*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 - 2*B*a^5*b*e^7 - 5*A*a^4*b^2*e^7)*x^2 + 3*(77*B*b^6*d^6*e - 3 
24*B*a*b^5*d^5*e^2 - 54*A*b^6*d^5*e^2 + 525*B*a^2*b^4*d^4*e^3 + 210*A*a*b^ 
5*d^4*e^3 - 400*B*a^3*b^3*d^3*e^4 - 300*A*a^2*b^4*d^3*e^4 + 135*B*a^4*b^2* 
d^2*e^5 + 180*A*a^3*b^3*d^2*e^5 - 12*B*a^5*b*d*e^6 - 30*A*a^4*b^2*d*e^6 - 
B*a^6*e^7 - 6*A*a^5*b*e^7)*x)/((e*x + d)^3*e^8) + 1/12*(3*B*b^6*e^12*x^4 - 
 16*B*b^6*d*e^11*x^3 + 24*B*a*b^5*e^12*x^3 + 4*A*b^6*e^12*x^3 + 60*B*b^6*d 
^2*e^10*x^2 - 144*B*a*b^5*d*e^11*x^2 - 24*A*b^6*d*e^11*x^2 + 90*B*a^2*b^4* 
e^12*x^2 + 36*A*a*b^5*e^12*x^2 - 240*B*b^6*d^3*e^9*x + 720*B*a*b^5*d^2*e^1 
0*x + 120*A*b^6*d^2*e^10*x - 720*B*a^2*b^4*d*e^11*x - 288*A*a*b^5*d*e^11*x 
 + 240*B*a^3*b^3*e^12*x + 180*A*a^2*b^4*e^12*x)/e^16
 

Mupad [B] (verification not implemented)

Time = 0.97 (sec) , antiderivative size = 907, normalized size of antiderivative = 3.25 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^4} \, dx=x^3\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{3\,e^4}-\frac {4\,B\,b^6\,d}{3\,e^5}\right )-x^2\,\left (\frac {2\,d\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e^4}-\frac {4\,B\,b^6\,d}{e^5}\right )}{e}-\frac {3\,a\,b^4\,\left (2\,A\,b+5\,B\,a\right )}{2\,e^4}+\frac {3\,B\,b^6\,d^2}{e^6}\right )-\frac {\frac {B\,a^6\,d\,e^6+2\,A\,a^6\,e^7+12\,B\,a^5\,b\,d^2\,e^5+6\,A\,a^5\,b\,d\,e^6-165\,B\,a^4\,b^2\,d^3\,e^4+30\,A\,a^4\,b^2\,d^2\,e^5+520\,B\,a^3\,b^3\,d^4\,e^3-220\,A\,a^3\,b^3\,d^3\,e^4-705\,B\,a^2\,b^4\,d^5\,e^2+390\,A\,a^2\,b^4\,d^4\,e^3+444\,B\,a\,b^5\,d^6\,e-282\,A\,a\,b^5\,d^5\,e^2-107\,B\,b^6\,d^7+74\,A\,b^6\,d^6\,e}{6\,e}+x\,\left (\frac {B\,a^6\,e^6}{2}+6\,B\,a^5\,b\,d\,e^5+3\,A\,a^5\,b\,e^6-\frac {135\,B\,a^4\,b^2\,d^2\,e^4}{2}+15\,A\,a^4\,b^2\,d\,e^5+200\,B\,a^3\,b^3\,d^3\,e^3-90\,A\,a^3\,b^3\,d^2\,e^4-\frac {525\,B\,a^2\,b^4\,d^4\,e^2}{2}+150\,A\,a^2\,b^4\,d^3\,e^3+162\,B\,a\,b^5\,d^5\,e-105\,A\,a\,b^5\,d^4\,e^2-\frac {77\,B\,b^6\,d^6}{2}+27\,A\,b^6\,d^5\,e\right )+x^2\,\left (6\,B\,a^5\,b\,e^6-45\,B\,a^4\,b^2\,d\,e^5+15\,A\,a^4\,b^2\,e^6+120\,B\,a^3\,b^3\,d^2\,e^4-60\,A\,a^3\,b^3\,d\,e^5-150\,B\,a^2\,b^4\,d^3\,e^3+90\,A\,a^2\,b^4\,d^2\,e^4+90\,B\,a\,b^5\,d^4\,e^2-60\,A\,a\,b^5\,d^3\,e^3-21\,B\,b^6\,d^5\,e+15\,A\,b^6\,d^4\,e^2\right )}{d^3\,e^7+3\,d^2\,e^8\,x+3\,d\,e^9\,x^2+e^{10}\,x^3}-x\,\left (\frac {6\,d^2\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e^4}-\frac {4\,B\,b^6\,d}{e^5}\right )}{e^2}-\frac {4\,d\,\left (\frac {4\,d\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e^4}-\frac {4\,B\,b^6\,d}{e^5}\right )}{e}-\frac {3\,a\,b^4\,\left (2\,A\,b+5\,B\,a\right )}{e^4}+\frac {6\,B\,b^6\,d^2}{e^6}\right )}{e}-\frac {5\,a^2\,b^3\,\left (3\,A\,b+4\,B\,a\right )}{e^4}+\frac {4\,B\,b^6\,d^3}{e^7}\right )+\frac {\ln \left (d+e\,x\right )\,\left (15\,B\,a^4\,b^2\,e^4-80\,B\,a^3\,b^3\,d\,e^3+20\,A\,a^3\,b^3\,e^4+150\,B\,a^2\,b^4\,d^2\,e^2-60\,A\,a^2\,b^4\,d\,e^3-120\,B\,a\,b^5\,d^3\,e+60\,A\,a\,b^5\,d^2\,e^2+35\,B\,b^6\,d^4-20\,A\,b^6\,d^3\,e\right )}{e^8}+\frac {B\,b^6\,x^4}{4\,e^4} \] Input:

int(((A + B*x)*(a + b*x)^6)/(d + e*x)^4,x)
 

Output:

x^3*((A*b^6 + 6*B*a*b^5)/(3*e^4) - (4*B*b^6*d)/(3*e^5)) - x^2*((2*d*((A*b^ 
6 + 6*B*a*b^5)/e^4 - (4*B*b^6*d)/e^5))/e - (3*a*b^4*(2*A*b + 5*B*a))/(2*e^ 
4) + (3*B*b^6*d^2)/e^6) - ((2*A*a^6*e^7 - 107*B*b^6*d^7 + 74*A*b^6*d^6*e + 
 B*a^6*d*e^6 - 282*A*a*b^5*d^5*e^2 + 12*B*a^5*b*d^2*e^5 + 390*A*a^2*b^4*d^ 
4*e^3 - 220*A*a^3*b^3*d^3*e^4 + 30*A*a^4*b^2*d^2*e^5 - 705*B*a^2*b^4*d^5*e 
^2 + 520*B*a^3*b^3*d^4*e^3 - 165*B*a^4*b^2*d^3*e^4 + 6*A*a^5*b*d*e^6 + 444 
*B*a*b^5*d^6*e)/(6*e) + x*((B*a^6*e^6)/2 - (77*B*b^6*d^6)/2 + 3*A*a^5*b*e^ 
6 + 27*A*b^6*d^5*e - 105*A*a*b^5*d^4*e^2 + 15*A*a^4*b^2*d*e^5 + 150*A*a^2* 
b^4*d^3*e^3 - 90*A*a^3*b^3*d^2*e^4 - (525*B*a^2*b^4*d^4*e^2)/2 + 200*B*a^3 
*b^3*d^3*e^3 - (135*B*a^4*b^2*d^2*e^4)/2 + 162*B*a*b^5*d^5*e + 6*B*a^5*b*d 
*e^5) + x^2*(6*B*a^5*b*e^6 - 21*B*b^6*d^5*e + 15*A*a^4*b^2*e^6 + 15*A*b^6* 
d^4*e^2 - 60*A*a*b^5*d^3*e^3 - 60*A*a^3*b^3*d*e^5 + 90*B*a*b^5*d^4*e^2 - 4 
5*B*a^4*b^2*d*e^5 + 90*A*a^2*b^4*d^2*e^4 - 150*B*a^2*b^4*d^3*e^3 + 120*B*a 
^3*b^3*d^2*e^4))/(d^3*e^7 + e^10*x^3 + 3*d^2*e^8*x + 3*d*e^9*x^2) - x*((6* 
d^2*((A*b^6 + 6*B*a*b^5)/e^4 - (4*B*b^6*d)/e^5))/e^2 - (4*d*((4*d*((A*b^6 
+ 6*B*a*b^5)/e^4 - (4*B*b^6*d)/e^5))/e - (3*a*b^4*(2*A*b + 5*B*a))/e^4 + ( 
6*B*b^6*d^2)/e^6))/e - (5*a^2*b^3*(3*A*b + 4*B*a))/e^4 + (4*B*b^6*d^3)/e^7 
) + (log(d + e*x)*(35*B*b^6*d^4 - 20*A*b^6*d^3*e + 20*A*a^3*b^3*e^4 + 15*B 
*a^4*b^2*e^4 + 60*A*a*b^5*d^2*e^2 - 60*A*a^2*b^4*d*e^3 - 80*B*a^3*b^3*d*e^ 
3 + 150*B*a^2*b^4*d^2*e^2 - 120*B*a*b^5*d^3*e))/e^8 + (B*b^6*x^4)/(4*e^...
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 840, normalized size of antiderivative = 3.01 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^4} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^6*(B*x+A)/(e*x+d)^4,x)
 

Output:

(420*log(d + e*x)*a**4*b**3*d**4*e**4 + 1260*log(d + e*x)*a**4*b**3*d**3*e 
**5*x + 1260*log(d + e*x)*a**4*b**3*d**2*e**6*x**2 + 420*log(d + e*x)*a**4 
*b**3*d*e**7*x**3 - 1680*log(d + e*x)*a**3*b**4*d**5*e**3 - 5040*log(d + e 
*x)*a**3*b**4*d**4*e**4*x - 5040*log(d + e*x)*a**3*b**4*d**3*e**5*x**2 - 1 
680*log(d + e*x)*a**3*b**4*d**2*e**6*x**3 + 2520*log(d + e*x)*a**2*b**5*d* 
*6*e**2 + 7560*log(d + e*x)*a**2*b**5*d**5*e**3*x + 7560*log(d + e*x)*a**2 
*b**5*d**4*e**4*x**2 + 2520*log(d + e*x)*a**2*b**5*d**3*e**5*x**3 - 1680*l 
og(d + e*x)*a*b**6*d**7*e - 5040*log(d + e*x)*a*b**6*d**6*e**2*x - 5040*lo 
g(d + e*x)*a*b**6*d**5*e**3*x**2 - 1680*log(d + e*x)*a*b**6*d**4*e**4*x**3 
 + 420*log(d + e*x)*b**7*d**8 + 1260*log(d + e*x)*b**7*d**7*e*x + 1260*log 
(d + e*x)*b**7*d**6*e**2*x**2 + 420*log(d + e*x)*b**7*d**5*e**3*x**3 - 4*a 
**7*d*e**7 - 14*a**6*b*d**2*e**6 - 42*a**6*b*d*e**7*x + 84*a**5*b**2*e**8* 
x**3 + 350*a**4*b**3*d**4*e**4 + 630*a**4*b**3*d**3*e**5*x - 420*a**4*b**3 
*d*e**7*x**3 - 1400*a**3*b**4*d**5*e**3 - 2520*a**3*b**4*d**4*e**4*x + 168 
0*a**3*b**4*d**2*e**6*x**3 + 420*a**3*b**4*d*e**7*x**4 + 2100*a**2*b**5*d* 
*6*e**2 + 3780*a**2*b**5*d**5*e**3*x - 2520*a**2*b**5*d**3*e**5*x**3 - 630 
*a**2*b**5*d**2*e**6*x**4 + 126*a**2*b**5*d*e**7*x**5 - 1400*a*b**6*d**7*e 
 - 2520*a*b**6*d**6*e**2*x + 1680*a*b**6*d**4*e**4*x**3 + 420*a*b**6*d**3* 
e**5*x**4 - 84*a*b**6*d**2*e**6*x**5 + 28*a*b**6*d*e**7*x**6 + 350*b**7*d* 
*8 + 630*b**7*d**7*e*x - 420*b**7*d**5*e**3*x**3 - 105*b**7*d**4*e**4*x...