\(\int \frac {(a+b x)^6 (A+B x)}{d+e x} \, dx\) [50]

Optimal result

Integrand size = 20, antiderivative size = 220 \[ \int \frac {(a+b x)^6 (A+B x)}{d+e x} \, dx=\frac {b (b d-a e)^5 (B d-A e) x}{e^7}-\frac {(b d-a e)^4 (B d-A e) (a+b x)^2}{2 e^6}+\frac {(b d-a e)^3 (B d-A e) (a+b x)^3}{3 e^5}-\frac {(b d-a e)^2 (B d-A e) (a+b x)^4}{4 e^4}+\frac {(b d-a e) (B d-A e) (a+b x)^5}{5 e^3}-\frac {(B d-A e) (a+b x)^6}{6 e^2}+\frac {B (a+b x)^7}{7 b e}-\frac {(b d-a e)^6 (B d-A e) \log (d+e x)}{e^8} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(501\) vs. \(2(220)=440\).

Time = 0.16 (sec) , antiderivative size = 501, normalized size of antiderivative = 2.28 \[ \int \frac {(a+b x)^6 (A+B x)}{d+e x} \, dx=\frac {e x \left (420 a^6 B e^6+1260 a^5 b e^5 (-2 B d+2 A e+B e x)+1050 a^4 b^2 e^4 \left (3 A e (-2 d+e x)+B \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+700 a^3 b^3 e^3 \left (2 A e \left (6 d^2-3 d e x+2 e^2 x^2\right )+B \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )+105 a^2 b^4 e^2 \left (5 A e \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+B \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )+42 a b^5 e \left (A e \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+B \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )\right )+b^6 \left (7 A e \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )+B \left (420 d^6-210 d^5 e x+140 d^4 e^2 x^2-105 d^3 e^3 x^3+84 d^2 e^4 x^4-70 d e^5 x^5+60 e^6 x^6\right )\right )\right )-420 (b d-a e)^6 (B d-A e) \log (d+e x)}{420 e^8} \] Input:

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

Output:

(e*x*(420*a^6*B*e^6 + 1260*a^5*b*e^5*(-2*B*d + 2*A*e + B*e*x) + 1050*a^4*b 
^2*e^4*(3*A*e*(-2*d + e*x) + B*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) + 700*a^3*b^ 
3*e^3*(2*A*e*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + B*(-12*d^3 + 6*d^2*e*x - 4*d* 
e^2*x^2 + 3*e^3*x^3)) + 105*a^2*b^4*e^2*(5*A*e*(-12*d^3 + 6*d^2*e*x - 4*d* 
e^2*x^2 + 3*e^3*x^3) + B*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3* 
x^3 + 12*e^4*x^4)) + 42*a*b^5*e*(A*e*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 
 - 15*d*e^3*x^3 + 12*e^4*x^4) + B*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 
 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5)) + b^6*(7*A*e*(-60*d^5 + 30*d 
^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5) + B* 
(420*d^6 - 210*d^5*e*x + 140*d^4*e^2*x^2 - 105*d^3*e^3*x^3 + 84*d^2*e^4*x^ 
4 - 70*d*e^5*x^5 + 60*e^6*x^6))) - 420*(b*d - a*e)^6*(B*d - A*e)*Log[d + e 
*x])/(420*e^8)
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 220, 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.

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

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(b*(b*d - a*e)^5*(B*d - A*e)*x)/e^7 - ((b*d - a*e)^4*(B*d - A*e)*(a + b*x) 
^2)/(2*e^6) + ((b*d - a*e)^3*(B*d - A*e)*(a + b*x)^3)/(3*e^5) - ((b*d - a* 
e)^2*(B*d - A*e)*(a + b*x)^4)/(4*e^4) + ((b*d - a*e)*(B*d - A*e)*(a + b*x) 
^5)/(5*e^3) - ((B*d - A*e)*(a + b*x)^6)/(6*e^2) + (B*(a + b*x)^7)/(7*b*e) 
- ((b*d - a*e)^6*(B*d - A*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. \(775\) vs. \(2(208)=416\).

Time = 0.23 (sec) , antiderivative size = 776, normalized size of antiderivative = 3.53

method	result	size
norman	\(\frac {\left (6 A \,a^{5} b \,e^{6}-15 A \,a^{4} b^{2} d \,e^{5}+20 A \,a^{3} b^{3} d^{2} e^{4}-15 A \,a^{2} b^{4} d^{3} e^{3}+6 A a \,b^{5} d^{4} e^{2}-A \,b^{6} d^{5} e +B \,a^{6} e^{6}-6 B \,a^{5} b d \,e^{5}+15 B \,a^{4} b^{2} d^{2} e^{4}-20 B \,a^{3} b^{3} d^{3} e^{3}+15 B \,a^{2} b^{4} d^{4} e^{2}-6 B a \,b^{5} d^{5} e +b^{6} B \,d^{6}\right ) x}{e^{7}}+\frac {b \left (15 A \,a^{4} b \,e^{5}-20 A \,a^{3} b^{2} d \,e^{4}+15 A \,a^{2} b^{3} d^{2} e^{3}-6 A a \,b^{4} d^{3} e^{2}+A \,b^{5} d^{4} e +6 B \,a^{5} e^{5}-15 B \,a^{4} b d \,e^{4}+20 B \,a^{3} b^{2} d^{2} e^{3}-15 B \,a^{2} b^{3} d^{3} e^{2}+6 B a \,b^{4} d^{4} e -B \,b^{5} d^{5}\right ) x^{2}}{2 e^{6}}+\frac {b^{2} \left (20 A \,a^{3} b \,e^{4}-15 A \,a^{2} b^{2} d \,e^{3}+6 A a \,b^{3} d^{2} e^{2}-A \,b^{4} d^{3} e +15 B \,a^{4} e^{4}-20 B \,a^{3} b d \,e^{3}+15 B \,a^{2} b^{2} d^{2} e^{2}-6 B a \,b^{3} d^{3} e +B \,b^{4} d^{4}\right ) x^{3}}{3 e^{5}}+\frac {b^{3} \left (15 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+A \,b^{3} d^{2} e +20 B \,a^{3} e^{3}-15 B \,a^{2} b d \,e^{2}+6 B a \,b^{2} d^{2} e -b^{3} B \,d^{3}\right ) x^{4}}{4 e^{4}}+\frac {b^{4} \left (6 A a b \,e^{2}-A \,b^{2} d e +15 B \,a^{2} e^{2}-6 B a b d e +b^{2} B \,d^{2}\right ) x^{5}}{5 e^{3}}+\frac {b^{5} \left (A b e +6 B a e -B b d \right ) x^{6}}{6 e^{2}}+\frac {b^{6} B \,x^{7}}{7 e}+\frac {\left (a^{6} A \,e^{7}-6 A \,a^{5} b d \,e^{6}+15 A \,a^{4} b^{2} d^{2} e^{5}-20 A \,a^{3} b^{3} d^{3} e^{4}+15 A \,a^{2} b^{4} d^{4} e^{3}-6 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e -B \,a^{6} d \,e^{6}+6 B \,a^{5} b \,d^{2} e^{5}-15 B \,a^{4} b^{2} d^{3} e^{4}+20 B \,a^{3} b^{3} d^{4} e^{3}-15 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e -b^{6} B \,d^{7}\right ) \ln \left (e x +d \right )}{e^{8}}\)	\(776\)
default	\(\frac {B a \,b^{5} e^{6} x^{6}-\frac {1}{3} A \,b^{6} d^{3} e^{3} x^{3}+5 B \,a^{4} b^{2} e^{6} x^{3}-\frac {15}{2} B \,a^{4} b^{2} d \,e^{5} x^{2}+10 B \,a^{3} b^{3} d^{2} e^{4} x^{2}-\frac {15}{2} B \,a^{2} b^{4} d^{3} e^{3} x^{2}+3 B a \,b^{5} d^{4} e^{2} x^{2}-15 A \,a^{4} b^{2} d \,e^{5} x +20 A \,a^{3} b^{3} d^{2} e^{4} x -15 A \,a^{2} b^{4} d^{3} e^{3} x +6 A a \,b^{5} d^{4} e^{2} x -6 B \,a^{5} b d \,e^{5} x +15 B \,a^{4} b^{2} d^{2} e^{4} x -20 B \,a^{3} b^{3} d^{3} e^{3} x +15 B \,a^{2} b^{4} d^{4} e^{2} x -6 B a \,b^{5} d^{5} e x +\frac {1}{5} B \,b^{6} d^{2} e^{4} x^{5}+\frac {1}{3} B \,b^{6} d^{4} e^{2} x^{3}+\frac {15}{2} A \,a^{4} b^{2} e^{6} x^{2}+B \,a^{6} e^{6} x +b^{6} B \,d^{6} x -\frac {6}{5} B a \,b^{5} d \,e^{5} x^{5}-\frac {3}{2} A a \,b^{5} d \,e^{5} x^{4}-\frac {15}{4} B \,a^{2} b^{4} d \,e^{5} x^{4}+\frac {3}{2} B a \,b^{5} d^{2} e^{4} x^{4}-5 A \,a^{2} b^{4} d \,e^{5} x^{3}+\frac {1}{7} b^{6} B \,x^{7} e^{6}+\frac {1}{6} A \,b^{6} e^{6} x^{6}+\frac {20}{3} A \,a^{3} b^{3} e^{6} x^{3}+3 B \,a^{2} b^{4} e^{6} x^{5}+5 B \,a^{3} b^{3} e^{6} x^{4}-\frac {1}{4} B \,b^{6} d^{3} e^{3} x^{4}+6 A \,a^{5} b \,e^{6} x -A \,b^{6} d^{5} e x +\frac {1}{2} A \,b^{6} d^{4} e^{2} x^{2}+3 B \,a^{5} b \,e^{6} x^{2}-\frac {1}{2} B \,b^{6} d^{5} e \,x^{2}+2 A a \,b^{5} d^{2} e^{4} x^{3}-\frac {20}{3} B \,a^{3} b^{3} d \,e^{5} x^{3}+5 B \,a^{2} b^{4} d^{2} e^{4} x^{3}-2 B a \,b^{5} d^{3} e^{3} x^{3}-10 A \,a^{3} b^{3} d \,e^{5} x^{2}+\frac {15}{2} A \,a^{2} b^{4} d^{2} e^{4} x^{2}-3 A a \,b^{5} d^{3} e^{3} x^{2}-\frac {1}{6} B \,b^{6} d \,e^{5} x^{6}+\frac {6}{5} A a \,b^{5} e^{6} x^{5}-\frac {1}{5} A \,b^{6} d \,e^{5} x^{5}+\frac {1}{4} A \,b^{6} d^{2} e^{4} x^{4}+\frac {15}{4} A \,a^{2} b^{4} e^{6} x^{4}}{e^{7}}+\frac {\left (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}\right ) \ln \left (e x +d \right )}{e^{8}}\)	\(905\)
risch	\(\frac {3 B \,a^{2} b^{4} x^{5}}{e}+\frac {5 B \,a^{3} b^{3} x^{4}}{e}-\frac {B \,b^{6} d^{3} x^{4}}{4 e^{4}}+\frac {5 B \,a^{4} b^{2} x^{3}}{e}+\frac {B \,b^{6} d^{2} x^{5}}{5 e^{3}}+\frac {B \,b^{6} d^{4} x^{3}}{3 e^{5}}+\frac {15 A \,a^{4} b^{2} x^{2}}{2 e}+\frac {b^{6} B \,d^{6} x}{e^{7}}+\frac {20 A \,a^{3} b^{3} x^{3}}{3 e}-\frac {15 B \,a^{2} b^{4} d \,x^{4}}{4 e^{2}}+\frac {3 B a \,b^{5} d^{2} x^{4}}{2 e^{3}}-\frac {5 A \,a^{2} b^{4} d \,x^{3}}{e^{2}}+\frac {2 A a \,b^{5} d^{2} x^{3}}{e^{3}}-\frac {20 B \,a^{3} b^{3} d \,x^{3}}{3 e^{2}}+\frac {5 B \,a^{2} b^{4} d^{2} x^{3}}{e^{3}}-\frac {2 B a \,b^{5} d^{3} x^{3}}{e^{4}}-\frac {10 A \,a^{3} b^{3} d \,x^{2}}{e^{2}}+\frac {15 A \,a^{2} b^{4} d^{2} x^{2}}{2 e^{3}}-\frac {3 A a \,b^{5} d^{3} x^{2}}{e^{4}}+\frac {B \,a^{6} x}{e}+\frac {A \,b^{6} x^{6}}{6 e}+\frac {\ln \left (e x +d \right ) a^{6} A}{e}+\frac {6 A a \,b^{5} d^{4} x}{e^{5}}-\frac {6 B \,a^{5} b d x}{e^{2}}+\frac {15 B \,a^{4} b^{2} d^{2} x}{e^{3}}-\frac {20 B \,a^{3} b^{3} d^{3} x}{e^{4}}+\frac {15 B \,a^{2} b^{4} d^{4} x}{e^{5}}-\frac {6 B a \,b^{5} d^{5} x}{e^{6}}-\frac {6 B a \,b^{5} d \,x^{5}}{5 e^{2}}-\frac {3 A a \,b^{5} d \,x^{4}}{2 e^{2}}-\frac {15 B \,a^{4} b^{2} d \,x^{2}}{2 e^{2}}+\frac {10 B \,a^{3} b^{3} d^{2} x^{2}}{e^{3}}-\frac {15 B \,a^{2} b^{4} d^{3} x^{2}}{2 e^{4}}+\frac {3 B a \,b^{5} d^{4} x^{2}}{e^{5}}-\frac {15 A \,a^{4} b^{2} d x}{e^{2}}+\frac {20 A \,a^{3} b^{3} d^{2} x}{e^{3}}+\frac {b^{6} B \,x^{7}}{7 e}-\frac {15 A \,a^{2} b^{4} d^{3} x}{e^{4}}+\frac {\ln \left (e x +d \right ) A \,b^{6} d^{6}}{e^{7}}-\frac {\ln \left (e x +d \right ) B \,a^{6} d}{e^{2}}-\frac {\ln \left (e x +d \right ) b^{6} B \,d^{7}}{e^{8}}+\frac {B a \,b^{5} x^{6}}{e}-\frac {A \,b^{6} d^{3} x^{3}}{3 e^{4}}+\frac {6 A \,a^{5} b x}{e}-\frac {A \,b^{6} d^{5} x}{e^{6}}+\frac {A \,b^{6} d^{4} x^{2}}{2 e^{5}}+\frac {3 B \,a^{5} b \,x^{2}}{e}-\frac {B \,b^{6} d^{5} x^{2}}{2 e^{6}}-\frac {B \,b^{6} d \,x^{6}}{6 e^{2}}+\frac {6 A a \,b^{5} x^{5}}{5 e}-\frac {A \,b^{6} d \,x^{5}}{5 e^{2}}+\frac {A \,b^{6} d^{2} x^{4}}{4 e^{3}}+\frac {15 A \,a^{2} b^{4} x^{4}}{4 e}-\frac {6 \ln \left (e x +d \right ) A \,a^{5} b d}{e^{2}}+\frac {15 \ln \left (e x +d \right ) A \,a^{4} b^{2} d^{2}}{e^{3}}-\frac {20 \ln \left (e x +d \right ) A \,a^{3} b^{3} d^{3}}{e^{4}}+\frac {15 \ln \left (e x +d \right ) A \,a^{2} b^{4} d^{4}}{e^{5}}-\frac {6 \ln \left (e x +d \right ) A a \,b^{5} d^{5}}{e^{6}}+\frac {6 \ln \left (e x +d \right ) B \,a^{5} b \,d^{2}}{e^{3}}-\frac {15 \ln \left (e x +d \right ) B \,a^{4} b^{2} d^{3}}{e^{4}}+\frac {20 \ln \left (e x +d \right ) B \,a^{3} b^{3} d^{4}}{e^{5}}-\frac {15 \ln \left (e x +d \right ) B \,a^{2} b^{4} d^{5}}{e^{6}}+\frac {6 \ln \left (e x +d \right ) B a \,b^{5} d^{6}}{e^{7}}\)	\(989\)
parallelrisch	\(\frac {-8400 A \ln \left (e x +d \right ) a^{3} b^{3} d^{3} e^{4}+6300 A \ln \left (e x +d \right ) a^{2} b^{4} d^{4} e^{3}-2520 A \ln \left (e x +d \right ) a \,b^{5} d^{5} e^{2}+2520 B \ln \left (e x +d \right ) a^{5} b \,d^{2} e^{5}-6300 B \ln \left (e x +d \right ) a^{4} b^{2} d^{3} e^{4}+8400 B \ln \left (e x +d \right ) a^{3} b^{3} d^{4} e^{3}-6300 B \ln \left (e x +d \right ) a^{2} b^{4} d^{5} e^{2}+2520 B \ln \left (e x +d \right ) a \,b^{5} d^{6} e -6300 A x \,a^{4} b^{2} d \,e^{6}+420 B x \,a^{6} e^{7}+70 A \,x^{6} b^{6} e^{7}+420 A \ln \left (e x +d \right ) a^{6} e^{7}-420 B \ln \left (e x +d \right ) b^{6} d^{7}+60 B \,x^{7} b^{6} e^{7}-6300 A x \,a^{2} b^{4} d^{3} e^{4}+2520 A x a \,b^{5} d^{4} e^{3}-2520 B x \,a^{5} b d \,e^{6}+6300 B x \,a^{4} b^{2} d^{2} e^{5}-8400 B x \,a^{3} b^{3} d^{3} e^{4}+6300 B x \,a^{2} b^{4} d^{4} e^{3}-2520 B x a \,b^{5} d^{5} e^{2}-4200 A \,x^{2} a^{3} b^{3} d \,e^{6}+3150 A \,x^{2} a^{2} b^{4} d^{2} e^{5}-1260 A \,x^{2} a \,b^{5} d^{3} e^{4}-3150 B \,x^{2} a^{4} b^{2} d \,e^{6}+4200 B \,x^{2} a^{3} b^{3} d^{2} e^{5}-3150 B \,x^{2} a^{2} b^{4} d^{3} e^{4}+1260 B \,x^{2} a \,b^{5} d^{4} e^{3}-2100 A \,x^{3} a^{2} b^{4} d \,e^{6}+840 A \,x^{3} a \,b^{5} d^{2} e^{5}-2800 B \,x^{3} a^{3} b^{3} d \,e^{6}+2100 B \,x^{3} a^{2} b^{4} d^{2} e^{5}-840 B \,x^{3} a \,b^{5} d^{3} e^{4}-630 A \,x^{4} a \,b^{5} d \,e^{6}-1575 B \,x^{4} a^{2} b^{4} d \,e^{6}+630 B \,x^{4} a \,b^{5} d^{2} e^{5}+8400 A x \,a^{3} b^{3} d^{2} e^{5}+420 B x \,b^{6} d^{6} e +3150 A \,x^{2} a^{4} b^{2} e^{7}+210 A \,x^{2} b^{6} d^{4} e^{3}+1260 B \,x^{2} a^{5} b \,e^{7}-210 B \,x^{2} b^{6} d^{5} e^{2}+2800 A \,x^{3} a^{3} b^{3} e^{7}-140 A \,x^{3} b^{6} d^{3} e^{4}+2100 B \,x^{3} a^{4} b^{2} e^{7}+140 B \,x^{3} b^{6} d^{4} e^{3}+1575 A \,x^{4} a^{2} b^{4} e^{7}+105 A \,x^{4} b^{6} d^{2} e^{5}+2100 B \,x^{4} a^{3} b^{3} e^{7}-105 B \,x^{4} b^{6} d^{3} e^{4}+504 A \,x^{5} a \,b^{5} e^{7}-84 A \,x^{5} b^{6} d \,e^{6}+1260 B \,x^{5} a^{2} b^{4} e^{7}+2520 A x \,a^{5} b \,e^{7}-420 A x \,b^{6} d^{5} e^{2}+84 B \,x^{5} b^{6} d^{2} e^{5}+420 B \,x^{6} a \,b^{5} e^{7}-70 B \,x^{6} b^{6} d \,e^{6}+420 A \ln \left (e x +d \right ) b^{6} d^{6} e -420 B \ln \left (e x +d \right ) a^{6} d \,e^{6}-504 B \,x^{5} a \,b^{5} d \,e^{6}-2520 A \ln \left (e x +d \right ) a^{5} b d \,e^{6}+6300 A \ln \left (e x +d \right ) a^{4} b^{2} d^{2} e^{5}}{420 e^{8}}\)	\(990\)

Input:

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

Output:

(6*A*a^5*b*e^6-15*A*a^4*b^2*d*e^5+20*A*a^3*b^3*d^2*e^4-15*A*a^2*b^4*d^3*e^ 
3+6*A*a*b^5*d^4*e^2-A*b^6*d^5*e+B*a^6*e^6-6*B*a^5*b*d*e^5+15*B*a^4*b^2*d^2 
*e^4-20*B*a^3*b^3*d^3*e^3+15*B*a^2*b^4*d^4*e^2-6*B*a*b^5*d^5*e+B*b^6*d^6)/ 
e^7*x+1/2*b/e^6*(15*A*a^4*b*e^5-20*A*a^3*b^2*d*e^4+15*A*a^2*b^3*d^2*e^3-6* 
A*a*b^4*d^3*e^2+A*b^5*d^4*e+6*B*a^5*e^5-15*B*a^4*b*d*e^4+20*B*a^3*b^2*d^2* 
e^3-15*B*a^2*b^3*d^3*e^2+6*B*a*b^4*d^4*e-B*b^5*d^5)*x^2+1/3*b^2/e^5*(20*A* 
a^3*b*e^4-15*A*a^2*b^2*d*e^3+6*A*a*b^3*d^2*e^2-A*b^4*d^3*e+15*B*a^4*e^4-20 
*B*a^3*b*d*e^3+15*B*a^2*b^2*d^2*e^2-6*B*a*b^3*d^3*e+B*b^4*d^4)*x^3+1/4*b^3 
/e^4*(15*A*a^2*b*e^3-6*A*a*b^2*d*e^2+A*b^3*d^2*e+20*B*a^3*e^3-15*B*a^2*b*d 
*e^2+6*B*a*b^2*d^2*e-B*b^3*d^3)*x^4+1/5*b^4/e^3*(6*A*a*b*e^2-A*b^2*d*e+15* 
B*a^2*e^2-6*B*a*b*d*e+B*b^2*d^2)*x^5+1/6*b^5/e^2*(A*b*e+6*B*a*e-B*b*d)*x^6 
+1/7*b^6*B/e*x^7+(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*ln(e*x+d)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 763 vs. \(2 (208) = 416\).

Time = 0.08 (sec) , antiderivative size = 763, normalized size of antiderivative = 3.47 \[ \int \frac {(a+b x)^6 (A+B x)}{d+e x} \, dx =\text {Too large to display} \] Input:

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

Output:

1/420*(60*B*b^6*e^7*x^7 - 70*(B*b^6*d*e^6 - (6*B*a*b^5 + A*b^6)*e^7)*x^6 + 
 84*(B*b^6*d^2*e^5 - (6*B*a*b^5 + A*b^6)*d*e^6 + 3*(5*B*a^2*b^4 + 2*A*a*b^ 
5)*e^7)*x^5 - 105*(B*b^6*d^3*e^4 - (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 - 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 140*(B 
*b^6*d^4*e^3 - (6*B*a*b^5 + A*b^6)*d^3*e^4 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d 
^2*e^5 - 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^ 
3)*e^7)*x^3 - 210*(B*b^6*d^5*e^2 - (6*B*a*b^5 + A*b^6)*d^4*e^3 + 3*(5*B*a^ 
2*b^4 + 2*A*a*b^5)*d^3*e^4 - 5*(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 + 42 
0*(B*b^6*d^6*e - (6*B*a*b^5 + A*b^6)*d^5*e^2 + 3*(5*B*a^2*b^4 + 2*A*a*b^5) 
*d^4*e^3 - 5*(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 + (B*a^6 + 6*A*a^5*b)*e 
^7)*x - 420*(B*b^6*d^7 - A*a^6*e^7 - (6*B*a*b^5 + A*b^6)*d^6*e + 3*(5*B*a^ 
2*b^4 + 2*A*a*b^5)*d^5*e^2 - 5*(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 + ( 
B*a^6 + 6*A*a^5*b)*d*e^6)*log(e*x + d))/e^8
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 736 vs. \(2 (189) = 378\).

Time = 0.87 (sec) , antiderivative size = 736, normalized size of antiderivative = 3.35 \[ \int \frac {(a+b x)^6 (A+B x)}{d+e x} \, dx=\frac {B b^{6} x^{7}}{7 e} + x^{6} \left (\frac {A b^{6}}{6 e} + \frac {B a b^{5}}{e} - \frac {B b^{6} d}{6 e^{2}}\right ) + x^{5} \cdot \left (\frac {6 A a b^{5}}{5 e} - \frac {A b^{6} d}{5 e^{2}} + \frac {3 B a^{2} b^{4}}{e} - \frac {6 B a b^{5} d}{5 e^{2}} + \frac {B b^{6} d^{2}}{5 e^{3}}\right ) + x^{4} \cdot \left (\frac {15 A a^{2} b^{4}}{4 e} - \frac {3 A a b^{5} d}{2 e^{2}} + \frac {A b^{6} d^{2}}{4 e^{3}} + \frac {5 B a^{3} b^{3}}{e} - \frac {15 B a^{2} b^{4} d}{4 e^{2}} + \frac {3 B a b^{5} d^{2}}{2 e^{3}} - \frac {B b^{6} d^{3}}{4 e^{4}}\right ) + x^{3} \cdot \left (\frac {20 A a^{3} b^{3}}{3 e} - \frac {5 A a^{2} b^{4} d}{e^{2}} + \frac {2 A a b^{5} d^{2}}{e^{3}} - \frac {A b^{6} d^{3}}{3 e^{4}} + \frac {5 B a^{4} b^{2}}{e} - \frac {20 B a^{3} b^{3} d}{3 e^{2}} + \frac {5 B a^{2} b^{4} d^{2}}{e^{3}} - \frac {2 B a b^{5} d^{3}}{e^{4}} + \frac {B b^{6} d^{4}}{3 e^{5}}\right ) + x^{2} \cdot \left (\frac {15 A a^{4} b^{2}}{2 e} - \frac {10 A a^{3} b^{3} d}{e^{2}} + \frac {15 A a^{2} b^{4} d^{2}}{2 e^{3}} - \frac {3 A a b^{5} d^{3}}{e^{4}} + \frac {A b^{6} d^{4}}{2 e^{5}} + \frac {3 B a^{5} b}{e} - \frac {15 B a^{4} b^{2} d}{2 e^{2}} + \frac {10 B a^{3} b^{3} d^{2}}{e^{3}} - \frac {15 B a^{2} b^{4} d^{3}}{2 e^{4}} + \frac {3 B a b^{5} d^{4}}{e^{5}} - \frac {B b^{6} d^{5}}{2 e^{6}}\right ) + x \left (\frac {6 A a^{5} b}{e} - \frac {15 A a^{4} b^{2} d}{e^{2}} + \frac {20 A a^{3} b^{3} d^{2}}{e^{3}} - \frac {15 A a^{2} b^{4} d^{3}}{e^{4}} + \frac {6 A a b^{5} d^{4}}{e^{5}} - \frac {A b^{6} d^{5}}{e^{6}} + \frac {B a^{6}}{e} - \frac {6 B a^{5} b d}{e^{2}} + \frac {15 B a^{4} b^{2} d^{2}}{e^{3}} - \frac {20 B a^{3} b^{3} d^{3}}{e^{4}} + \frac {15 B a^{2} b^{4} d^{4}}{e^{5}} - \frac {6 B a b^{5} d^{5}}{e^{6}} + \frac {B b^{6} d^{6}}{e^{7}}\right ) - \frac {\left (- A e + B d\right ) \left (a e - b d\right )^{6} \log {\left (d + e x \right )}}{e^{8}} \] Input:

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

Output:

B*b**6*x**7/(7*e) + x**6*(A*b**6/(6*e) + B*a*b**5/e - B*b**6*d/(6*e**2)) + 
 x**5*(6*A*a*b**5/(5*e) - A*b**6*d/(5*e**2) + 3*B*a**2*b**4/e - 6*B*a*b**5 
*d/(5*e**2) + B*b**6*d**2/(5*e**3)) + x**4*(15*A*a**2*b**4/(4*e) - 3*A*a*b 
**5*d/(2*e**2) + A*b**6*d**2/(4*e**3) + 5*B*a**3*b**3/e - 15*B*a**2*b**4*d 
/(4*e**2) + 3*B*a*b**5*d**2/(2*e**3) - B*b**6*d**3/(4*e**4)) + x**3*(20*A* 
a**3*b**3/(3*e) - 5*A*a**2*b**4*d/e**2 + 2*A*a*b**5*d**2/e**3 - A*b**6*d** 
3/(3*e**4) + 5*B*a**4*b**2/e - 20*B*a**3*b**3*d/(3*e**2) + 5*B*a**2*b**4*d 
**2/e**3 - 2*B*a*b**5*d**3/e**4 + B*b**6*d**4/(3*e**5)) + x**2*(15*A*a**4* 
b**2/(2*e) - 10*A*a**3*b**3*d/e**2 + 15*A*a**2*b**4*d**2/(2*e**3) - 3*A*a* 
b**5*d**3/e**4 + A*b**6*d**4/(2*e**5) + 3*B*a**5*b/e - 15*B*a**4*b**2*d/(2 
*e**2) + 10*B*a**3*b**3*d**2/e**3 - 15*B*a**2*b**4*d**3/(2*e**4) + 3*B*a*b 
**5*d**4/e**5 - B*b**6*d**5/(2*e**6)) + x*(6*A*a**5*b/e - 15*A*a**4*b**2*d 
/e**2 + 20*A*a**3*b**3*d**2/e**3 - 15*A*a**2*b**4*d**3/e**4 + 6*A*a*b**5*d 
**4/e**5 - A*b**6*d**5/e**6 + B*a**6/e - 6*B*a**5*b*d/e**2 + 15*B*a**4*b** 
2*d**2/e**3 - 20*B*a**3*b**3*d**3/e**4 + 15*B*a**2*b**4*d**4/e**5 - 6*B*a* 
b**5*d**5/e**6 + B*b**6*d**6/e**7) - (-A*e + B*d)*(a*e - b*d)**6*log(d + e 
*x)/e**8
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 762 vs. \(2 (208) = 416\).

Time = 0.06 (sec) , antiderivative size = 762, normalized size of antiderivative = 3.46 \[ \int \frac {(a+b x)^6 (A+B x)}{d+e x} \, dx =\text {Too large to display} \] Input:

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

Output:

1/420*(60*B*b^6*e^6*x^7 - 70*(B*b^6*d*e^5 - (6*B*a*b^5 + A*b^6)*e^6)*x^6 + 
 84*(B*b^6*d^2*e^4 - (6*B*a*b^5 + A*b^6)*d*e^5 + 3*(5*B*a^2*b^4 + 2*A*a*b^ 
5)*e^6)*x^5 - 105*(B*b^6*d^3*e^3 - (6*B*a*b^5 + A*b^6)*d^2*e^4 + 3*(5*B*a^ 
2*b^4 + 2*A*a*b^5)*d*e^5 - 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^6)*x^4 + 140*(B 
*b^6*d^4*e^2 - (6*B*a*b^5 + A*b^6)*d^3*e^3 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d 
^2*e^4 - 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^5 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^ 
3)*e^6)*x^3 - 210*(B*b^6*d^5*e - (6*B*a*b^5 + A*b^6)*d^4*e^2 + 3*(5*B*a^2* 
b^4 + 2*A*a*b^5)*d^3*e^3 - 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^4 + 5*(3*B* 
a^4*b^2 + 4*A*a^3*b^3)*d*e^5 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*e^6)*x^2 + 420* 
(B*b^6*d^6 - (6*B*a*b^5 + A*b^6)*d^5*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e 
^2 - 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3) 
*d^2*e^4 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^5 + (B*a^6 + 6*A*a^5*b)*e^6)*x) 
/e^7 - (B*b^6*d^7 - A*a^6*e^7 - (6*B*a*b^5 + A*b^6)*d^6*e + 3*(5*B*a^2*b^4 
 + 2*A*a*b^5)*d^5*e^2 - 5*(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 + (B*a^6 
 + 6*A*a^5*b)*d*e^6)*log(e*x + d)/e^8
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 910 vs. \(2 (208) = 416\).

Time = 0.12 (sec) , antiderivative size = 910, normalized size of antiderivative = 4.14 \[ \int \frac {(a+b x)^6 (A+B x)}{d+e x} \, dx =\text {Too large to display} \] Input:

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

Output:

1/420*(60*B*b^6*e^6*x^7 - 70*B*b^6*d*e^5*x^6 + 420*B*a*b^5*e^6*x^6 + 70*A* 
b^6*e^6*x^6 + 84*B*b^6*d^2*e^4*x^5 - 504*B*a*b^5*d*e^5*x^5 - 84*A*b^6*d*e^ 
5*x^5 + 1260*B*a^2*b^4*e^6*x^5 + 504*A*a*b^5*e^6*x^5 - 105*B*b^6*d^3*e^3*x 
^4 + 630*B*a*b^5*d^2*e^4*x^4 + 105*A*b^6*d^2*e^4*x^4 - 1575*B*a^2*b^4*d*e^ 
5*x^4 - 630*A*a*b^5*d*e^5*x^4 + 2100*B*a^3*b^3*e^6*x^4 + 1575*A*a^2*b^4*e^ 
6*x^4 + 140*B*b^6*d^4*e^2*x^3 - 840*B*a*b^5*d^3*e^3*x^3 - 140*A*b^6*d^3*e^ 
3*x^3 + 2100*B*a^2*b^4*d^2*e^4*x^3 + 840*A*a*b^5*d^2*e^4*x^3 - 2800*B*a^3* 
b^3*d*e^5*x^3 - 2100*A*a^2*b^4*d*e^5*x^3 + 2100*B*a^4*b^2*e^6*x^3 + 2800*A 
*a^3*b^3*e^6*x^3 - 210*B*b^6*d^5*e*x^2 + 1260*B*a*b^5*d^4*e^2*x^2 + 210*A* 
b^6*d^4*e^2*x^2 - 3150*B*a^2*b^4*d^3*e^3*x^2 - 1260*A*a*b^5*d^3*e^3*x^2 + 
4200*B*a^3*b^3*d^2*e^4*x^2 + 3150*A*a^2*b^4*d^2*e^4*x^2 - 3150*B*a^4*b^2*d 
*e^5*x^2 - 4200*A*a^3*b^3*d*e^5*x^2 + 1260*B*a^5*b*e^6*x^2 + 3150*A*a^4*b^ 
2*e^6*x^2 + 420*B*b^6*d^6*x - 2520*B*a*b^5*d^5*e*x - 420*A*b^6*d^5*e*x + 6 
300*B*a^2*b^4*d^4*e^2*x + 2520*A*a*b^5*d^4*e^2*x - 8400*B*a^3*b^3*d^3*e^3* 
x - 6300*A*a^2*b^4*d^3*e^3*x + 6300*B*a^4*b^2*d^2*e^4*x + 8400*A*a^3*b^3*d 
^2*e^4*x - 2520*B*a^5*b*d*e^5*x - 6300*A*a^4*b^2*d*e^5*x + 420*B*a^6*e^6*x 
 + 2520*A*a^5*b*e^6*x)/e^7 - (B*b^6*d^7 - 6*B*a*b^5*d^6*e - A*b^6*d^6*e + 
15*B*a^2*b^4*d^5*e^2 + 6*A*a*b^5*d^5*e^2 - 20*B*a^3*b^3*d^4*e^3 - 15*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 + B*a^6*d*e^6 + 6*A*a^5*b*d*e^6 - A*a^6*e^7...
 

Mupad [B] (verification not implemented)

Time = 0.94 (sec) , antiderivative size = 769, normalized size of antiderivative = 3.50 \[ \int \frac {(a+b x)^6 (A+B x)}{d+e x} \, dx=x\,\left (\frac {B\,a^6+6\,A\,b\,a^5}{e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e}-\frac {B\,b^6\,d}{e^2}\right )}{e}-\frac {3\,a\,b^4\,\left (2\,A\,b+5\,B\,a\right )}{e}\right )}{e}+\frac {5\,a^2\,b^3\,\left (3\,A\,b+4\,B\,a\right )}{e}\right )}{e}-\frac {5\,a^3\,b^2\,\left (4\,A\,b+3\,B\,a\right )}{e}\right )}{e}+\frac {3\,a^4\,b\,\left (5\,A\,b+2\,B\,a\right )}{e}\right )}{e}\right )-x^3\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e}-\frac {B\,b^6\,d}{e^2}\right )}{e}-\frac {3\,a\,b^4\,\left (2\,A\,b+5\,B\,a\right )}{e}\right )}{e}+\frac {5\,a^2\,b^3\,\left (3\,A\,b+4\,B\,a\right )}{e}\right )}{3\,e}-\frac {5\,a^3\,b^2\,\left (4\,A\,b+3\,B\,a\right )}{3\,e}\right )+x^4\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e}-\frac {B\,b^6\,d}{e^2}\right )}{e}-\frac {3\,a\,b^4\,\left (2\,A\,b+5\,B\,a\right )}{e}\right )}{4\,e}+\frac {5\,a^2\,b^3\,\left (3\,A\,b+4\,B\,a\right )}{4\,e}\right )-x^5\,\left (\frac {d\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e}-\frac {B\,b^6\,d}{e^2}\right )}{5\,e}-\frac {3\,a\,b^4\,\left (2\,A\,b+5\,B\,a\right )}{5\,e}\right )+x^6\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{6\,e}-\frac {B\,b^6\,d}{6\,e^2}\right )+x^2\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e}-\frac {B\,b^6\,d}{e^2}\right )}{e}-\frac {3\,a\,b^4\,\left (2\,A\,b+5\,B\,a\right )}{e}\right )}{e}+\frac {5\,a^2\,b^3\,\left (3\,A\,b+4\,B\,a\right )}{e}\right )}{e}-\frac {5\,a^3\,b^2\,\left (4\,A\,b+3\,B\,a\right )}{e}\right )}{2\,e}+\frac {3\,a^4\,b\,\left (5\,A\,b+2\,B\,a\right )}{2\,e}\right )+\frac {\ln \left (d+e\,x\right )\,\left (-B\,a^6\,d\,e^6+A\,a^6\,e^7+6\,B\,a^5\,b\,d^2\,e^5-6\,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+20\,B\,a^3\,b^3\,d^4\,e^3-20\,A\,a^3\,b^3\,d^3\,e^4-15\,B\,a^2\,b^4\,d^5\,e^2+15\,A\,a^2\,b^4\,d^4\,e^3+6\,B\,a\,b^5\,d^6\,e-6\,A\,a\,b^5\,d^5\,e^2-B\,b^6\,d^7+A\,b^6\,d^6\,e\right )}{e^8}+\frac {B\,b^6\,x^7}{7\,e} \] Input:

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

Output:

x*((B*a^6 + 6*A*a^5*b)/e - (d*((d*((d*((d*((d*((A*b^6 + 6*B*a*b^5)/e - (B* 
b^6*d)/e^2))/e - (3*a*b^4*(2*A*b + 5*B*a))/e))/e + (5*a^2*b^3*(3*A*b + 4*B 
*a))/e))/e - (5*a^3*b^2*(4*A*b + 3*B*a))/e))/e + (3*a^4*b*(5*A*b + 2*B*a)) 
/e))/e) - x^3*((d*((d*((d*((A*b^6 + 6*B*a*b^5)/e - (B*b^6*d)/e^2))/e - (3* 
a*b^4*(2*A*b + 5*B*a))/e))/e + (5*a^2*b^3*(3*A*b + 4*B*a))/e))/(3*e) - (5* 
a^3*b^2*(4*A*b + 3*B*a))/(3*e)) + x^4*((d*((d*((A*b^6 + 6*B*a*b^5)/e - (B* 
b^6*d)/e^2))/e - (3*a*b^4*(2*A*b + 5*B*a))/e))/(4*e) + (5*a^2*b^3*(3*A*b + 
 4*B*a))/(4*e)) - x^5*((d*((A*b^6 + 6*B*a*b^5)/e - (B*b^6*d)/e^2))/(5*e) - 
 (3*a*b^4*(2*A*b + 5*B*a))/(5*e)) + x^6*((A*b^6 + 6*B*a*b^5)/(6*e) - (B*b^ 
6*d)/(6*e^2)) + x^2*((d*((d*((d*((d*((A*b^6 + 6*B*a*b^5)/e - (B*b^6*d)/e^2 
))/e - (3*a*b^4*(2*A*b + 5*B*a))/e))/e + (5*a^2*b^3*(3*A*b + 4*B*a))/e))/e 
 - (5*a^3*b^2*(4*A*b + 3*B*a))/e))/(2*e) + (3*a^4*b*(5*A*b + 2*B*a))/(2*e) 
) + (log(d + e*x)*(A*a^6*e^7 - B*b^6*d^7 + A*b^6*d^6*e - B*a^6*d*e^6 - 6*A 
*a*b^5*d^5*e^2 + 6*B*a^5*b*d^2*e^5 + 15*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 - 15*B*a^2*b^4*d^5*e^2 + 20*B*a^3*b^3*d^4*e^ 
3 - 15*B*a^4*b^2*d^3*e^4 - 6*A*a^5*b*d*e^6 + 6*B*a*b^5*d^6*e))/e^8 + (B*b^ 
6*x^7)/(7*e)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 538, normalized size of antiderivative = 2.45 \[ \int \frac {(a+b x)^6 (A+B x)}{d+e x} \, dx=\frac {-2940 \,\mathrm {log}\left (e x +d \right ) a^{6} b d \,e^{6}+8820 \,\mathrm {log}\left (e x +d \right ) a^{5} b^{2} d^{2} e^{5}-14700 \,\mathrm {log}\left (e x +d \right ) a^{4} b^{3} d^{3} e^{4}+14700 \,\mathrm {log}\left (e x +d \right ) a^{3} b^{4} d^{4} e^{3}-8820 \,\mathrm {log}\left (e x +d \right ) a^{2} b^{5} d^{5} e^{2}+2940 \,\mathrm {log}\left (e x +d \right ) a \,b^{6} d^{6} e -8820 a^{5} b^{2} d \,e^{6} x +14700 a^{4} b^{3} d^{2} e^{5} x -7350 a^{4} b^{3} d \,e^{6} x^{2}-14700 a^{3} b^{4} d^{3} e^{4} x +7350 a^{3} b^{4} d^{2} e^{5} x^{2}-4900 a^{3} b^{4} d \,e^{6} x^{3}+8820 a^{2} b^{5} d^{4} e^{3} x -4410 a^{2} b^{5} d^{3} e^{4} x^{2}+2940 a^{2} b^{5} d^{2} e^{5} x^{3}-2205 a^{2} b^{5} d \,e^{6} x^{4}-2940 a \,b^{6} d^{5} e^{2} x +1470 a \,b^{6} d^{4} e^{3} x^{2}-980 a \,b^{6} d^{3} e^{4} x^{3}+735 a \,b^{6} d^{2} e^{5} x^{4}-588 a \,b^{6} d \,e^{6} x^{5}+420 \,\mathrm {log}\left (e x +d \right ) a^{7} e^{7}-420 \,\mathrm {log}\left (e x +d \right ) b^{7} d^{7}+60 b^{7} e^{7} x^{7}+420 b^{7} d^{6} e x -210 b^{7} d^{5} e^{2} x^{2}+140 b^{7} d^{4} e^{3} x^{3}-105 b^{7} d^{3} e^{4} x^{4}+84 b^{7} d^{2} e^{5} x^{5}-70 b^{7} d \,e^{6} x^{6}+2940 a^{6} b \,e^{7} x +4410 a^{5} b^{2} e^{7} x^{2}+4900 a^{4} b^{3} e^{7} x^{3}+3675 a^{3} b^{4} e^{7} x^{4}+1764 a^{2} b^{5} e^{7} x^{5}+490 a \,b^{6} e^{7} x^{6}}{420 e^{8}} \] Input:

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

Output:

(420*log(d + e*x)*a**7*e**7 - 2940*log(d + e*x)*a**6*b*d*e**6 + 8820*log(d 
 + e*x)*a**5*b**2*d**2*e**5 - 14700*log(d + e*x)*a**4*b**3*d**3*e**4 + 147 
00*log(d + e*x)*a**3*b**4*d**4*e**3 - 8820*log(d + e*x)*a**2*b**5*d**5*e** 
2 + 2940*log(d + e*x)*a*b**6*d**6*e - 420*log(d + e*x)*b**7*d**7 + 2940*a* 
*6*b*e**7*x - 8820*a**5*b**2*d*e**6*x + 4410*a**5*b**2*e**7*x**2 + 14700*a 
**4*b**3*d**2*e**5*x - 7350*a**4*b**3*d*e**6*x**2 + 4900*a**4*b**3*e**7*x* 
*3 - 14700*a**3*b**4*d**3*e**4*x + 7350*a**3*b**4*d**2*e**5*x**2 - 4900*a* 
*3*b**4*d*e**6*x**3 + 3675*a**3*b**4*e**7*x**4 + 8820*a**2*b**5*d**4*e**3* 
x - 4410*a**2*b**5*d**3*e**4*x**2 + 2940*a**2*b**5*d**2*e**5*x**3 - 2205*a 
**2*b**5*d*e**6*x**4 + 1764*a**2*b**5*e**7*x**5 - 2940*a*b**6*d**5*e**2*x 
+ 1470*a*b**6*d**4*e**3*x**2 - 980*a*b**6*d**3*e**4*x**3 + 735*a*b**6*d**2 
*e**5*x**4 - 588*a*b**6*d*e**6*x**5 + 490*a*b**6*e**7*x**6 + 420*b**7*d**6 
*e*x - 210*b**7*d**5*e**2*x**2 + 140*b**7*d**4*e**3*x**3 - 105*b**7*d**3*e 
**4*x**4 + 84*b**7*d**2*e**5*x**5 - 70*b**7*d*e**6*x**6 + 60*b**7*e**7*x** 
7)/(420*e**8)