\(\int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{13}} \, dx\) [1072]

Optimal result

Integrand size = 20, antiderivative size = 292 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{13}} \, dx=\frac {(b d-a e)^6 (B d-A e)}{12 e^8 (d+e x)^{12}}-\frac {(b d-a e)^5 (7 b B d-6 A b e-a B e)}{11 e^8 (d+e x)^{11}}+\frac {3 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e)}{10 e^8 (d+e x)^{10}}-\frac {5 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e)}{9 e^8 (d+e x)^9}+\frac {5 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e)}{8 e^8 (d+e x)^8}-\frac {3 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e)}{7 e^8 (d+e x)^7}+\frac {b^5 (7 b B d-A b e-6 a B e)}{6 e^8 (d+e x)^6}-\frac {b^6 B}{5 e^8 (d+e x)^5} \]

[Out]

1/12*(-a*e+b*d)^6*(-A*e+B*d)/e^8/(e*x+d)^12-1/11*(-a*e+b*d)^5*(-6*A*b*e-B*a*e+7*B*b*d)/e^8/(e*x+d)^11+3/10*b*(
-a*e+b*d)^4*(-5*A*b*e-2*B*a*e+7*B*b*d)/e^8/(e*x+d)^10-5/9*b^2*(-a*e+b*d)^3*(-4*A*b*e-3*B*a*e+7*B*b*d)/e^8/(e*x
+d)^9+5/8*b^3*(-a*e+b*d)^2*(-3*A*b*e-4*B*a*e+7*B*b*d)/e^8/(e*x+d)^8-3/7*b^4*(-a*e+b*d)*(-2*A*b*e-5*B*a*e+7*B*b
*d)/e^8/(e*x+d)^7+1/6*b^5*(-A*b*e-6*B*a*e+7*B*b*d)/e^8/(e*x+d)^6-1/5*b^6*B/e^8/(e*x+d)^5

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{13}} \, dx=\frac {b^5 (-6 a B e-A b e+7 b B d)}{6 e^8 (d+e x)^6}-\frac {3 b^4 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{7 e^8 (d+e x)^7}+\frac {5 b^3 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{8 e^8 (d+e x)^8}-\frac {5 b^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{9 e^8 (d+e x)^9}+\frac {3 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{10 e^8 (d+e x)^{10}}-\frac {(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{11 e^8 (d+e x)^{11}}+\frac {(b d-a e)^6 (B d-A e)}{12 e^8 (d+e x)^{12}}-\frac {b^6 B}{5 e^8 (d+e x)^5} \]

[In]

Int[((a + b*x)^6*(A + B*x))/(d + e*x)^13,x]

[Out]

((b*d - a*e)^6*(B*d - A*e))/(12*e^8*(d + e*x)^12) - ((b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e))/(11*e^8*(d + e
*x)^11) + (3*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e))/(10*e^8*(d + e*x)^10) - (5*b^2*(b*d - a*e)^3*(7*b*
B*d - 4*A*b*e - 3*a*B*e))/(9*e^8*(d + e*x)^9) + (5*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e))/(8*e^8*(d
+ e*x)^8) - (3*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e))/(7*e^8*(d + e*x)^7) + (b^5*(7*b*B*d - A*b*e - 6*
a*B*e))/(6*e^8*(d + e*x)^6) - (b^6*B)/(5*e^8*(d + e*x)^5)

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((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]))))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b d+a e)^6 (-B d+A e)}{e^7 (d+e x)^{13}}+\frac {(-b d+a e)^5 (-7 b B d+6 A b e+a B e)}{e^7 (d+e x)^{12}}+\frac {3 b (b d-a e)^4 (-7 b B d+5 A b e+2 a B e)}{e^7 (d+e x)^{11}}-\frac {5 b^2 (b d-a e)^3 (-7 b B d+4 A b e+3 a B e)}{e^7 (d+e x)^{10}}+\frac {5 b^3 (b d-a e)^2 (-7 b B d+3 A b e+4 a B e)}{e^7 (d+e x)^9}-\frac {3 b^4 (b d-a e) (-7 b B d+2 A b e+5 a B e)}{e^7 (d+e x)^8}+\frac {b^5 (-7 b B d+A b e+6 a B e)}{e^7 (d+e x)^7}+\frac {b^6 B}{e^7 (d+e x)^6}\right ) \, dx \\ & = \frac {(b d-a e)^6 (B d-A e)}{12 e^8 (d+e x)^{12}}-\frac {(b d-a e)^5 (7 b B d-6 A b e-a B e)}{11 e^8 (d+e x)^{11}}+\frac {3 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e)}{10 e^8 (d+e x)^{10}}-\frac {5 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e)}{9 e^8 (d+e x)^9}+\frac {5 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e)}{8 e^8 (d+e x)^8}-\frac {3 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e)}{7 e^8 (d+e x)^7}+\frac {b^5 (7 b B d-A b e-6 a B e)}{6 e^8 (d+e x)^6}-\frac {b^6 B}{5 e^8 (d+e x)^5} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(600\) vs. \(2(292)=584\).

Time = 0.19 (sec) , antiderivative size = 600, normalized size of antiderivative = 2.05 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{13}} \, dx=-\frac {210 a^6 e^6 (11 A e+B (d+12 e x))+252 a^5 b e^5 \left (5 A e (d+12 e x)+B \left (d^2+12 d e x+66 e^2 x^2\right )\right )+210 a^4 b^2 e^4 \left (3 A e \left (d^2+12 d e x+66 e^2 x^2\right )+B \left (d^3+12 d^2 e x+66 d e^2 x^2+220 e^3 x^3\right )\right )+140 a^3 b^3 e^3 \left (2 A e \left (d^3+12 d^2 e x+66 d e^2 x^2+220 e^3 x^3\right )+B \left (d^4+12 d^3 e x+66 d^2 e^2 x^2+220 d e^3 x^3+495 e^4 x^4\right )\right )+15 a^2 b^4 e^2 \left (7 A e \left (d^4+12 d^3 e x+66 d^2 e^2 x^2+220 d e^3 x^3+495 e^4 x^4\right )+5 B \left (d^5+12 d^4 e x+66 d^3 e^2 x^2+220 d^2 e^3 x^3+495 d e^4 x^4+792 e^5 x^5\right )\right )+30 a b^5 e \left (A e \left (d^5+12 d^4 e x+66 d^3 e^2 x^2+220 d^2 e^3 x^3+495 d e^4 x^4+792 e^5 x^5\right )+B \left (d^6+12 d^5 e x+66 d^4 e^2 x^2+220 d^3 e^3 x^3+495 d^2 e^4 x^4+792 d e^5 x^5+924 e^6 x^6\right )\right )+b^6 \left (5 A e \left (d^6+12 d^5 e x+66 d^4 e^2 x^2+220 d^3 e^3 x^3+495 d^2 e^4 x^4+792 d e^5 x^5+924 e^6 x^6\right )+7 B \left (d^7+12 d^6 e x+66 d^5 e^2 x^2+220 d^4 e^3 x^3+495 d^3 e^4 x^4+792 d^2 e^5 x^5+924 d e^6 x^6+792 e^7 x^7\right )\right )}{27720 e^8 (d+e x)^{12}} \]

[In]

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^13,x]

[Out]

-1/27720*(210*a^6*e^6*(11*A*e + B*(d + 12*e*x)) + 252*a^5*b*e^5*(5*A*e*(d + 12*e*x) + B*(d^2 + 12*d*e*x + 66*e
^2*x^2)) + 210*a^4*b^2*e^4*(3*A*e*(d^2 + 12*d*e*x + 66*e^2*x^2) + B*(d^3 + 12*d^2*e*x + 66*d*e^2*x^2 + 220*e^3
*x^3)) + 140*a^3*b^3*e^3*(2*A*e*(d^3 + 12*d^2*e*x + 66*d*e^2*x^2 + 220*e^3*x^3) + B*(d^4 + 12*d^3*e*x + 66*d^2
*e^2*x^2 + 220*d*e^3*x^3 + 495*e^4*x^4)) + 15*a^2*b^4*e^2*(7*A*e*(d^4 + 12*d^3*e*x + 66*d^2*e^2*x^2 + 220*d*e^
3*x^3 + 495*e^4*x^4) + 5*B*(d^5 + 12*d^4*e*x + 66*d^3*e^2*x^2 + 220*d^2*e^3*x^3 + 495*d*e^4*x^4 + 792*e^5*x^5)
) + 30*a*b^5*e*(A*e*(d^5 + 12*d^4*e*x + 66*d^3*e^2*x^2 + 220*d^2*e^3*x^3 + 495*d*e^4*x^4 + 792*e^5*x^5) + B*(d
^6 + 12*d^5*e*x + 66*d^4*e^2*x^2 + 220*d^3*e^3*x^3 + 495*d^2*e^4*x^4 + 792*d*e^5*x^5 + 924*e^6*x^6)) + b^6*(5*
A*e*(d^6 + 12*d^5*e*x + 66*d^4*e^2*x^2 + 220*d^3*e^3*x^3 + 495*d^2*e^4*x^4 + 792*d*e^5*x^5 + 924*e^6*x^6) + 7*
B*(d^7 + 12*d^6*e*x + 66*d^5*e^2*x^2 + 220*d^4*e^3*x^3 + 495*d^3*e^4*x^4 + 792*d^2*e^5*x^5 + 924*d*e^6*x^6 + 7
92*e^7*x^7)))/(e^8*(d + e*x)^12)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(788\) vs. \(2(276)=552\).

Time = 0.71 (sec) , antiderivative size = 789, normalized size of antiderivative = 2.70

method	result	size
risch	\(\frac {-\frac {b^{6} B \,x^{7}}{5 e}-\frac {b^{5} \left (5 A b e +30 B a e +7 B b d \right ) x^{6}}{30 e^{2}}-\frac {b^{4} \left (30 A a b \,e^{2}+5 A \,b^{2} d e +75 B \,a^{2} e^{2}+30 B a b d e +7 b^{2} B \,d^{2}\right ) x^{5}}{35 e^{3}}-\frac {b^{3} \left (105 A \,a^{2} b \,e^{3}+30 A a \,b^{2} d \,e^{2}+5 A \,b^{3} d^{2} e +140 B \,a^{3} e^{3}+75 B \,a^{2} b d \,e^{2}+30 B a \,b^{2} d^{2} e +7 b^{3} B \,d^{3}\right ) x^{4}}{56 e^{4}}-\frac {b^{2} \left (280 A \,a^{3} b \,e^{4}+105 A \,a^{2} b^{2} d \,e^{3}+30 A a \,b^{3} d^{2} e^{2}+5 A \,b^{4} d^{3} e +210 B \,a^{4} e^{4}+140 B \,a^{3} b d \,e^{3}+75 B \,a^{2} b^{2} d^{2} e^{2}+30 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right ) x^{3}}{126 e^{5}}-\frac {b \left (630 A \,a^{4} b \,e^{5}+280 A \,a^{3} b^{2} d \,e^{4}+105 A \,a^{2} b^{3} d^{2} e^{3}+30 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e +252 B \,a^{5} e^{5}+210 B \,a^{4} b d \,e^{4}+140 B \,a^{3} b^{2} d^{2} e^{3}+75 B \,a^{2} b^{3} d^{3} e^{2}+30 B a \,b^{4} d^{4} e +7 B \,b^{5} d^{5}\right ) x^{2}}{420 e^{6}}-\frac {\left (1260 A \,a^{5} b \,e^{6}+630 A \,a^{4} b^{2} d \,e^{5}+280 A \,a^{3} b^{3} d^{2} e^{4}+105 A \,a^{2} b^{4} d^{3} e^{3}+30 A a \,b^{5} d^{4} e^{2}+5 A \,b^{6} d^{5} e +210 B \,a^{6} e^{6}+252 B \,a^{5} b d \,e^{5}+210 B \,a^{4} b^{2} d^{2} e^{4}+140 B \,a^{3} b^{3} d^{3} e^{3}+75 B \,a^{2} b^{4} d^{4} e^{2}+30 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}\right ) x}{2310 e^{7}}-\frac {2310 A \,a^{6} e^{7}+1260 A \,a^{5} b d \,e^{6}+630 A \,a^{4} b^{2} d^{2} e^{5}+280 A \,a^{3} b^{3} d^{3} e^{4}+105 A \,a^{2} b^{4} d^{4} e^{3}+30 A a \,b^{5} d^{5} e^{2}+5 A \,b^{6} d^{6} e +210 B \,a^{6} d \,e^{6}+252 B \,a^{5} b \,d^{2} e^{5}+210 B \,a^{4} b^{2} d^{3} e^{4}+140 B \,a^{3} b^{3} d^{4} e^{3}+75 B \,a^{2} b^{4} d^{5} e^{2}+30 B a \,b^{5} d^{6} e +7 b^{6} B \,d^{7}}{27720 e^{8}}}{\left (e x +d \right )^{12}}\)	\(789\)
default	\(-\frac {3 b \left (5 A \,a^{4} b \,e^{5}-20 A \,a^{3} b^{2} d \,e^{4}+30 A \,a^{2} b^{3} d^{2} e^{3}-20 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e +2 B \,a^{5} e^{5}-15 B \,a^{4} b d \,e^{4}+40 B \,a^{3} b^{2} d^{2} e^{3}-50 B \,a^{2} b^{3} d^{3} e^{2}+30 B a \,b^{4} d^{4} e -7 B \,b^{5} d^{5}\right )}{10 e^{8} \left (e x +d \right )^{10}}-\frac {5 b^{3} \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +4 B \,a^{3} e^{3}-15 B \,a^{2} b d \,e^{2}+18 B a \,b^{2} d^{2} e -7 b^{3} B \,d^{3}\right )}{8 e^{8} \left (e x +d \right )^{8}}-\frac {6 A \,a^{5} b \,e^{6}-30 A \,a^{4} b^{2} d \,e^{5}+60 A \,a^{3} b^{3} d^{2} e^{4}-60 A \,a^{2} b^{4} d^{3} e^{3}+30 A a \,b^{5} d^{4} e^{2}-6 A \,b^{6} d^{5} e +B \,a^{6} e^{6}-12 B \,a^{5} b d \,e^{5}+45 B \,a^{4} b^{2} d^{2} e^{4}-80 B \,a^{3} b^{3} d^{3} e^{3}+75 B \,a^{2} b^{4} d^{4} e^{2}-36 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}}{11 e^{8} \left (e x +d \right )^{11}}-\frac {3 b^{4} \left (2 A a b \,e^{2}-2 A \,b^{2} d e +5 B \,a^{2} e^{2}-12 B a b d e +7 b^{2} B \,d^{2}\right )}{7 e^{8} \left (e x +d \right )^{7}}-\frac {b^{6} B}{5 e^{8} \left (e x +d \right )^{5}}-\frac {5 b^{2} \left (4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e +3 B \,a^{4} e^{4}-16 B \,a^{3} b d \,e^{3}+30 B \,a^{2} b^{2} d^{2} e^{2}-24 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right )}{9 e^{8} \left (e x +d \right )^{9}}-\frac {b^{5} \left (A b e +6 B a e -7 B b d \right )}{6 e^{8} \left (e x +d \right )^{6}}-\frac {A \,a^{6} e^{7}-6 A \,a^{5} b d \,e^{6}+15 A \,a^{4} b^{2} d^{2} e^{5}-20 A \,a^{3} b^{3} d^{3} e^{4}+15 A \,a^{2} b^{4} d^{4} e^{3}-6 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e -B \,a^{6} d \,e^{6}+6 B \,a^{5} b \,d^{2} e^{5}-15 B \,a^{4} b^{2} d^{3} e^{4}+20 B \,a^{3} b^{3} d^{4} e^{3}-15 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e -b^{6} B \,d^{7}}{12 e^{8} \left (e x +d \right )^{12}}\)	\(814\)
norman	\(\frac {-\frac {b^{6} B \,x^{7}}{5 e}-\frac {\left (5 A \,b^{6} e^{5}+30 B a \,b^{5} e^{5}+7 b^{6} B d \,e^{4}\right ) x^{6}}{30 e^{6}}-\frac {\left (30 A a \,b^{5} e^{6}+5 A \,b^{6} d \,e^{5}+75 B \,a^{2} b^{4} e^{6}+30 B a \,b^{5} d \,e^{5}+7 b^{6} B \,d^{2} e^{4}\right ) x^{5}}{35 e^{7}}-\frac {\left (105 A \,a^{2} b^{4} e^{7}+30 A a \,b^{5} d \,e^{6}+5 A \,b^{6} d^{2} e^{5}+140 B \,a^{3} b^{3} e^{7}+75 B \,a^{2} b^{4} d \,e^{6}+30 B a \,b^{5} d^{2} e^{5}+7 b^{6} B \,d^{3} e^{4}\right ) x^{4}}{56 e^{8}}-\frac {\left (280 A \,a^{3} b^{3} e^{8}+105 A \,a^{2} b^{4} d \,e^{7}+30 A a \,b^{5} d^{2} e^{6}+5 A \,b^{6} d^{3} e^{5}+210 B \,a^{4} b^{2} e^{8}+140 B \,a^{3} b^{3} d \,e^{7}+75 B \,a^{2} b^{4} d^{2} e^{6}+30 B a \,b^{5} d^{3} e^{5}+7 b^{6} B \,d^{4} e^{4}\right ) x^{3}}{126 e^{9}}-\frac {\left (630 A \,a^{4} b^{2} e^{9}+280 A \,a^{3} b^{3} d \,e^{8}+105 A \,a^{2} b^{4} d^{2} e^{7}+30 A a \,b^{5} d^{3} e^{6}+5 A \,b^{6} d^{4} e^{5}+252 B \,a^{5} b \,e^{9}+210 B \,a^{4} b^{2} d \,e^{8}+140 B \,a^{3} b^{3} d^{2} e^{7}+75 B \,a^{2} b^{4} d^{3} e^{6}+30 B a \,b^{5} d^{4} e^{5}+7 B \,b^{6} d^{5} e^{4}\right ) x^{2}}{420 e^{10}}-\frac {\left (1260 A \,a^{5} b \,e^{10}+630 A \,a^{4} b^{2} d \,e^{9}+280 A \,a^{3} b^{3} d^{2} e^{8}+105 A \,a^{2} b^{4} d^{3} e^{7}+30 A a \,b^{5} d^{4} e^{6}+5 A \,b^{6} d^{5} e^{5}+210 B \,a^{6} e^{10}+252 B \,a^{5} b d \,e^{9}+210 B \,a^{4} b^{2} d^{2} e^{8}+140 B \,a^{3} b^{3} d^{3} e^{7}+75 B \,a^{2} b^{4} d^{4} e^{6}+30 B a \,b^{5} d^{5} e^{5}+7 b^{6} B \,d^{6} e^{4}\right ) x}{2310 e^{11}}-\frac {2310 A \,a^{6} e^{11}+1260 A \,a^{5} b d \,e^{10}+630 A \,a^{4} b^{2} d^{2} e^{9}+280 A \,a^{3} b^{3} d^{3} e^{8}+105 A \,a^{2} b^{4} d^{4} e^{7}+30 A a \,b^{5} d^{5} e^{6}+5 A \,b^{6} d^{6} e^{5}+210 B \,a^{6} d \,e^{10}+252 B \,a^{5} b \,d^{2} e^{9}+210 B \,a^{4} b^{2} d^{3} e^{8}+140 B \,a^{3} b^{3} d^{4} e^{7}+75 B \,a^{2} b^{4} d^{5} e^{6}+30 B a \,b^{5} d^{6} e^{5}+7 B \,b^{6} d^{7} e^{4}}{27720 e^{12}}}{\left (e x +d \right )^{12}}\)	\(858\)
gosper	\(-\frac {5544 B \,x^{7} b^{6} e^{7}+4620 A \,x^{6} b^{6} e^{7}+27720 B \,x^{6} a \,b^{5} e^{7}+6468 B \,x^{6} b^{6} d \,e^{6}+23760 A \,x^{5} a \,b^{5} e^{7}+3960 A \,x^{5} b^{6} d \,e^{6}+59400 B \,x^{5} a^{2} b^{4} e^{7}+23760 B \,x^{5} a \,b^{5} d \,e^{6}+5544 B \,x^{5} b^{6} d^{2} e^{5}+51975 A \,x^{4} a^{2} b^{4} e^{7}+14850 A \,x^{4} a \,b^{5} d \,e^{6}+2475 A \,x^{4} b^{6} d^{2} e^{5}+69300 B \,x^{4} a^{3} b^{3} e^{7}+37125 B \,x^{4} a^{2} b^{4} d \,e^{6}+14850 B \,x^{4} a \,b^{5} d^{2} e^{5}+3465 B \,x^{4} b^{6} d^{3} e^{4}+61600 A \,x^{3} a^{3} b^{3} e^{7}+23100 A \,x^{3} a^{2} b^{4} d \,e^{6}+6600 A \,x^{3} a \,b^{5} d^{2} e^{5}+1100 A \,x^{3} b^{6} d^{3} e^{4}+46200 B \,x^{3} a^{4} b^{2} e^{7}+30800 B \,x^{3} a^{3} b^{3} d \,e^{6}+16500 B \,x^{3} a^{2} b^{4} d^{2} e^{5}+6600 B \,x^{3} a \,b^{5} d^{3} e^{4}+1540 B \,x^{3} b^{6} d^{4} e^{3}+41580 A \,x^{2} a^{4} b^{2} e^{7}+18480 A \,x^{2} a^{3} b^{3} d \,e^{6}+6930 A \,x^{2} a^{2} b^{4} d^{2} e^{5}+1980 A \,x^{2} a \,b^{5} d^{3} e^{4}+330 A \,x^{2} b^{6} d^{4} e^{3}+16632 B \,x^{2} a^{5} b \,e^{7}+13860 B \,x^{2} a^{4} b^{2} d \,e^{6}+9240 B \,x^{2} a^{3} b^{3} d^{2} e^{5}+4950 B \,x^{2} a^{2} b^{4} d^{3} e^{4}+1980 B \,x^{2} a \,b^{5} d^{4} e^{3}+462 B \,x^{2} b^{6} d^{5} e^{2}+15120 A x \,a^{5} b \,e^{7}+7560 A x \,a^{4} b^{2} d \,e^{6}+3360 A x \,a^{3} b^{3} d^{2} e^{5}+1260 A x \,a^{2} b^{4} d^{3} e^{4}+360 A x a \,b^{5} d^{4} e^{3}+60 A x \,b^{6} d^{5} e^{2}+2520 B x \,a^{6} e^{7}+3024 B x \,a^{5} b d \,e^{6}+2520 B x \,a^{4} b^{2} d^{2} e^{5}+1680 B x \,a^{3} b^{3} d^{3} e^{4}+900 B x \,a^{2} b^{4} d^{4} e^{3}+360 B x a \,b^{5} d^{5} e^{2}+84 B x \,b^{6} d^{6} e +2310 A \,a^{6} e^{7}+1260 A \,a^{5} b d \,e^{6}+630 A \,a^{4} b^{2} d^{2} e^{5}+280 A \,a^{3} b^{3} d^{3} e^{4}+105 A \,a^{2} b^{4} d^{4} e^{3}+30 A a \,b^{5} d^{5} e^{2}+5 A \,b^{6} d^{6} e +210 B \,a^{6} d \,e^{6}+252 B \,a^{5} b \,d^{2} e^{5}+210 B \,a^{4} b^{2} d^{3} e^{4}+140 B \,a^{3} b^{3} d^{4} e^{3}+75 B \,a^{2} b^{4} d^{5} e^{2}+30 B a \,b^{5} d^{6} e +7 b^{6} B \,d^{7}}{27720 e^{8} \left (e x +d \right )^{12}}\)	\(913\)
parallelrisch	\(-\frac {5544 B \,b^{6} x^{7} e^{11}+4620 A \,b^{6} e^{11} x^{6}+27720 B a \,b^{5} e^{11} x^{6}+6468 B \,b^{6} d \,e^{10} x^{6}+23760 A a \,b^{5} e^{11} x^{5}+3960 A \,b^{6} d \,e^{10} x^{5}+59400 B \,a^{2} b^{4} e^{11} x^{5}+23760 B a \,b^{5} d \,e^{10} x^{5}+5544 B \,b^{6} d^{2} e^{9} x^{5}+51975 A \,a^{2} b^{4} e^{11} x^{4}+14850 A a \,b^{5} d \,e^{10} x^{4}+2475 A \,b^{6} d^{2} e^{9} x^{4}+69300 B \,a^{3} b^{3} e^{11} x^{4}+37125 B \,a^{2} b^{4} d \,e^{10} x^{4}+14850 B a \,b^{5} d^{2} e^{9} x^{4}+3465 B \,b^{6} d^{3} e^{8} x^{4}+61600 A \,a^{3} b^{3} e^{11} x^{3}+23100 A \,a^{2} b^{4} d \,e^{10} x^{3}+6600 A a \,b^{5} d^{2} e^{9} x^{3}+1100 A \,b^{6} d^{3} e^{8} x^{3}+46200 B \,a^{4} b^{2} e^{11} x^{3}+30800 B \,a^{3} b^{3} d \,e^{10} x^{3}+16500 B \,a^{2} b^{4} d^{2} e^{9} x^{3}+6600 B a \,b^{5} d^{3} e^{8} x^{3}+1540 B \,b^{6} d^{4} e^{7} x^{3}+41580 A \,a^{4} b^{2} e^{11} x^{2}+18480 A \,a^{3} b^{3} d \,e^{10} x^{2}+6930 A \,a^{2} b^{4} d^{2} e^{9} x^{2}+1980 A a \,b^{5} d^{3} e^{8} x^{2}+330 A \,b^{6} d^{4} e^{7} x^{2}+16632 B \,a^{5} b \,e^{11} x^{2}+13860 B \,a^{4} b^{2} d \,e^{10} x^{2}+9240 B \,a^{3} b^{3} d^{2} e^{9} x^{2}+4950 B \,a^{2} b^{4} d^{3} e^{8} x^{2}+1980 B a \,b^{5} d^{4} e^{7} x^{2}+462 B \,b^{6} d^{5} e^{6} x^{2}+15120 A \,a^{5} b \,e^{11} x +7560 A \,a^{4} b^{2} d \,e^{10} x +3360 A \,a^{3} b^{3} d^{2} e^{9} x +1260 A \,a^{2} b^{4} d^{3} e^{8} x +360 A a \,b^{5} d^{4} e^{7} x +60 A \,b^{6} d^{5} e^{6} x +2520 B \,a^{6} e^{11} x +3024 B \,a^{5} b d \,e^{10} x +2520 B \,a^{4} b^{2} d^{2} e^{9} x +1680 B \,a^{3} b^{3} d^{3} e^{8} x +900 B \,a^{2} b^{4} d^{4} e^{7} x +360 B a \,b^{5} d^{5} e^{6} x +84 B \,b^{6} d^{6} e^{5} x +2310 A \,a^{6} e^{11}+1260 A \,a^{5} b d \,e^{10}+630 A \,a^{4} b^{2} d^{2} e^{9}+280 A \,a^{3} b^{3} d^{3} e^{8}+105 A \,a^{2} b^{4} d^{4} e^{7}+30 A a \,b^{5} d^{5} e^{6}+5 A \,b^{6} d^{6} e^{5}+210 B \,a^{6} d \,e^{10}+252 B \,a^{5} b \,d^{2} e^{9}+210 B \,a^{4} b^{2} d^{3} e^{8}+140 B \,a^{3} b^{3} d^{4} e^{7}+75 B \,a^{2} b^{4} d^{5} e^{6}+30 B a \,b^{5} d^{6} e^{5}+7 B \,b^{6} d^{7} e^{4}}{27720 e^{12} \left (e x +d \right )^{12}}\)	\(922\)

[In]

int((b*x+a)^6*(B*x+A)/(e*x+d)^13,x,method=_RETURNVERBOSE)

[Out]

(-1/5*b^6*B/e*x^7-1/30*b^5/e^2*(5*A*b*e+30*B*a*e+7*B*b*d)*x^6-1/35*b^4/e^3*(30*A*a*b*e^2+5*A*b^2*d*e+75*B*a^2*
e^2+30*B*a*b*d*e+7*B*b^2*d^2)*x^5-1/56*b^3/e^4*(105*A*a^2*b*e^3+30*A*a*b^2*d*e^2+5*A*b^3*d^2*e+140*B*a^3*e^3+7
5*B*a^2*b*d*e^2+30*B*a*b^2*d^2*e+7*B*b^3*d^3)*x^4-1/126*b^2/e^5*(280*A*a^3*b*e^4+105*A*a^2*b^2*d*e^3+30*A*a*b^
3*d^2*e^2+5*A*b^4*d^3*e+210*B*a^4*e^4+140*B*a^3*b*d*e^3+75*B*a^2*b^2*d^2*e^2+30*B*a*b^3*d^3*e+7*B*b^4*d^4)*x^3
-1/420*b/e^6*(630*A*a^4*b*e^5+280*A*a^3*b^2*d*e^4+105*A*a^2*b^3*d^2*e^3+30*A*a*b^4*d^3*e^2+5*A*b^5*d^4*e+252*B
*a^5*e^5+210*B*a^4*b*d*e^4+140*B*a^3*b^2*d^2*e^3+75*B*a^2*b^3*d^3*e^2+30*B*a*b^4*d^4*e+7*B*b^5*d^5)*x^2-1/2310
/e^7*(1260*A*a^5*b*e^6+630*A*a^4*b^2*d*e^5+280*A*a^3*b^3*d^2*e^4+105*A*a^2*b^4*d^3*e^3+30*A*a*b^5*d^4*e^2+5*A*
b^6*d^5*e+210*B*a^6*e^6+252*B*a^5*b*d*e^5+210*B*a^4*b^2*d^2*e^4+140*B*a^3*b^3*d^3*e^3+75*B*a^2*b^4*d^4*e^2+30*
B*a*b^5*d^5*e+7*B*b^6*d^6)*x-1/27720/e^8*(2310*A*a^6*e^7+1260*A*a^5*b*d*e^6+630*A*a^4*b^2*d^2*e^5+280*A*a^3*b^
3*d^3*e^4+105*A*a^2*b^4*d^4*e^3+30*A*a*b^5*d^5*e^2+5*A*b^6*d^6*e+210*B*a^6*d*e^6+252*B*a^5*b*d^2*e^5+210*B*a^4
*b^2*d^3*e^4+140*B*a^3*b^3*d^4*e^3+75*B*a^2*b^4*d^5*e^2+30*B*a*b^5*d^6*e+7*B*b^6*d^7))/(e*x+d)^12

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 894 vs. \(2 (276) = 552\).

Time = 0.26 (sec) , antiderivative size = 894, normalized size of antiderivative = 3.06 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{13}} \, dx=-\frac {5544 \, B b^{6} e^{7} x^{7} + 7 \, B b^{6} d^{7} + 2310 \, A a^{6} e^{7} + 5 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 15 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} + 35 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 70 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} + 126 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + 210 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} + 924 \, {\left (7 \, B b^{6} d e^{6} + 5 \, {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 792 \, {\left (7 \, B b^{6} d^{2} e^{5} + 5 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} + 15 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} + 495 \, {\left (7 \, B b^{6} d^{3} e^{4} + 5 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + 15 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} + 35 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 220 \, {\left (7 \, B b^{6} d^{4} e^{3} + 5 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + 15 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} + 35 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} + 70 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} + 66 \, {\left (7 \, B b^{6} d^{5} e^{2} + 5 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 15 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} + 35 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} + 70 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} + 126 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 12 \, {\left (7 \, B b^{6} d^{6} e + 5 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 15 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} + 35 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + 70 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} + 126 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + 210 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x}{27720 \, {\left (e^{20} x^{12} + 12 \, d e^{19} x^{11} + 66 \, d^{2} e^{18} x^{10} + 220 \, d^{3} e^{17} x^{9} + 495 \, d^{4} e^{16} x^{8} + 792 \, d^{5} e^{15} x^{7} + 924 \, d^{6} e^{14} x^{6} + 792 \, d^{7} e^{13} x^{5} + 495 \, d^{8} e^{12} x^{4} + 220 \, d^{9} e^{11} x^{3} + 66 \, d^{10} e^{10} x^{2} + 12 \, d^{11} e^{9} x + d^{12} e^{8}\right )}} \]

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^13,x, algorithm="fricas")

[Out]

-1/27720*(5544*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 2310*A*a^6*e^7 + 5*(6*B*a*b^5 + A*b^6)*d^6*e + 15*(5*B*a^2*b^4 +
2*A*a*b^5)*d^5*e^2 + 35*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 70*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 126*(2*
B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 + 210*(B*a^6 + 6*A*a^5*b)*d*e^6 + 924*(7*B*b^6*d*e^6 + 5*(6*B*a*b^5 + A*b^6)*e^
7)*x^6 + 792*(7*B*b^6*d^2*e^5 + 5*(6*B*a*b^5 + A*b^6)*d*e^6 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 495*(7*B
*b^6*d^3*e^4 + 5*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 + 35*(4*B*a^3*b^3 + 3*A*a^2*
b^4)*e^7)*x^4 + 220*(7*B*b^6*d^4*e^3 + 5*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 +
35*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 70*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 66*(7*B*b^6*d^5*e^2 + 5*(6*B*
a*b^5 + A*b^6)*d^4*e^3 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 + 35*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 70*(3
*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 + 126*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 12*(7*B*b^6*d^6*e + 5*(6*B*a*b^5 +
A*b^6)*d^5*e^2 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 + 35*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 70*(3*B*a^4*b
^2 + 4*A*a^3*b^3)*d^2*e^5 + 126*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + 210*(B*a^6 + 6*A*a^5*b)*e^7)*x)/(e^20*x^12 +
12*d*e^19*x^11 + 66*d^2*e^18*x^10 + 220*d^3*e^17*x^9 + 495*d^4*e^16*x^8 + 792*d^5*e^15*x^7 + 924*d^6*e^14*x^6
+ 792*d^7*e^13*x^5 + 495*d^8*e^12*x^4 + 220*d^9*e^11*x^3 + 66*d^10*e^10*x^2 + 12*d^11*e^9*x + d^12*e^8)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{13}} \, dx=\text {Timed out} \]

[In]

integrate((b*x+a)**6*(B*x+A)/(e*x+d)**13,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 894 vs. \(2 (276) = 552\).

Time = 0.23 (sec) , antiderivative size = 894, normalized size of antiderivative = 3.06 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{13}} \, dx=-\frac {5544 \, B b^{6} e^{7} x^{7} + 7 \, B b^{6} d^{7} + 2310 \, A a^{6} e^{7} + 5 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 15 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} + 35 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 70 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} + 126 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + 210 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} + 924 \, {\left (7 \, B b^{6} d e^{6} + 5 \, {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 792 \, {\left (7 \, B b^{6} d^{2} e^{5} + 5 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} + 15 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} + 495 \, {\left (7 \, B b^{6} d^{3} e^{4} + 5 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + 15 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} + 35 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 220 \, {\left (7 \, B b^{6} d^{4} e^{3} + 5 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + 15 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} + 35 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} + 70 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} + 66 \, {\left (7 \, B b^{6} d^{5} e^{2} + 5 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 15 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} + 35 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} + 70 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} + 126 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 12 \, {\left (7 \, B b^{6} d^{6} e + 5 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 15 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} + 35 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + 70 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} + 126 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + 210 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x}{27720 \, {\left (e^{20} x^{12} + 12 \, d e^{19} x^{11} + 66 \, d^{2} e^{18} x^{10} + 220 \, d^{3} e^{17} x^{9} + 495 \, d^{4} e^{16} x^{8} + 792 \, d^{5} e^{15} x^{7} + 924 \, d^{6} e^{14} x^{6} + 792 \, d^{7} e^{13} x^{5} + 495 \, d^{8} e^{12} x^{4} + 220 \, d^{9} e^{11} x^{3} + 66 \, d^{10} e^{10} x^{2} + 12 \, d^{11} e^{9} x + d^{12} e^{8}\right )}} \]

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^13,x, algorithm="maxima")

[Out]

-1/27720*(5544*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 2310*A*a^6*e^7 + 5*(6*B*a*b^5 + A*b^6)*d^6*e + 15*(5*B*a^2*b^4 +
2*A*a*b^5)*d^5*e^2 + 35*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 70*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 126*(2*
B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 + 210*(B*a^6 + 6*A*a^5*b)*d*e^6 + 924*(7*B*b^6*d*e^6 + 5*(6*B*a*b^5 + A*b^6)*e^
7)*x^6 + 792*(7*B*b^6*d^2*e^5 + 5*(6*B*a*b^5 + A*b^6)*d*e^6 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 495*(7*B
*b^6*d^3*e^4 + 5*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 + 35*(4*B*a^3*b^3 + 3*A*a^2*
b^4)*e^7)*x^4 + 220*(7*B*b^6*d^4*e^3 + 5*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 +
35*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 70*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 66*(7*B*b^6*d^5*e^2 + 5*(6*B*
a*b^5 + A*b^6)*d^4*e^3 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 + 35*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 70*(3
*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 + 126*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 12*(7*B*b^6*d^6*e + 5*(6*B*a*b^5 +
A*b^6)*d^5*e^2 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 + 35*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 70*(3*B*a^4*b
^2 + 4*A*a^3*b^3)*d^2*e^5 + 126*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + 210*(B*a^6 + 6*A*a^5*b)*e^7)*x)/(e^20*x^12 +
12*d*e^19*x^11 + 66*d^2*e^18*x^10 + 220*d^3*e^17*x^9 + 495*d^4*e^16*x^8 + 792*d^5*e^15*x^7 + 924*d^6*e^14*x^6
+ 792*d^7*e^13*x^5 + 495*d^8*e^12*x^4 + 220*d^9*e^11*x^3 + 66*d^10*e^10*x^2 + 12*d^11*e^9*x + d^12*e^8)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 912 vs. \(2 (276) = 552\).

Time = 0.29 (sec) , antiderivative size = 912, normalized size of antiderivative = 3.12 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{13}} \, dx=-\frac {5544 \, B b^{6} e^{7} x^{7} + 6468 \, B b^{6} d e^{6} x^{6} + 27720 \, B a b^{5} e^{7} x^{6} + 4620 \, A b^{6} e^{7} x^{6} + 5544 \, B b^{6} d^{2} e^{5} x^{5} + 23760 \, B a b^{5} d e^{6} x^{5} + 3960 \, A b^{6} d e^{6} x^{5} + 59400 \, B a^{2} b^{4} e^{7} x^{5} + 23760 \, A a b^{5} e^{7} x^{5} + 3465 \, B b^{6} d^{3} e^{4} x^{4} + 14850 \, B a b^{5} d^{2} e^{5} x^{4} + 2475 \, A b^{6} d^{2} e^{5} x^{4} + 37125 \, B a^{2} b^{4} d e^{6} x^{4} + 14850 \, A a b^{5} d e^{6} x^{4} + 69300 \, B a^{3} b^{3} e^{7} x^{4} + 51975 \, A a^{2} b^{4} e^{7} x^{4} + 1540 \, B b^{6} d^{4} e^{3} x^{3} + 6600 \, B a b^{5} d^{3} e^{4} x^{3} + 1100 \, A b^{6} d^{3} e^{4} x^{3} + 16500 \, B a^{2} b^{4} d^{2} e^{5} x^{3} + 6600 \, A a b^{5} d^{2} e^{5} x^{3} + 30800 \, B a^{3} b^{3} d e^{6} x^{3} + 23100 \, A a^{2} b^{4} d e^{6} x^{3} + 46200 \, B a^{4} b^{2} e^{7} x^{3} + 61600 \, A a^{3} b^{3} e^{7} x^{3} + 462 \, B b^{6} d^{5} e^{2} x^{2} + 1980 \, B a b^{5} d^{4} e^{3} x^{2} + 330 \, A b^{6} d^{4} e^{3} x^{2} + 4950 \, B a^{2} b^{4} d^{3} e^{4} x^{2} + 1980 \, A a b^{5} d^{3} e^{4} x^{2} + 9240 \, B a^{3} b^{3} d^{2} e^{5} x^{2} + 6930 \, A a^{2} b^{4} d^{2} e^{5} x^{2} + 13860 \, B a^{4} b^{2} d e^{6} x^{2} + 18480 \, A a^{3} b^{3} d e^{6} x^{2} + 16632 \, B a^{5} b e^{7} x^{2} + 41580 \, A a^{4} b^{2} e^{7} x^{2} + 84 \, B b^{6} d^{6} e x + 360 \, B a b^{5} d^{5} e^{2} x + 60 \, A b^{6} d^{5} e^{2} x + 900 \, B a^{2} b^{4} d^{4} e^{3} x + 360 \, A a b^{5} d^{4} e^{3} x + 1680 \, B a^{3} b^{3} d^{3} e^{4} x + 1260 \, A a^{2} b^{4} d^{3} e^{4} x + 2520 \, B a^{4} b^{2} d^{2} e^{5} x + 3360 \, A a^{3} b^{3} d^{2} e^{5} x + 3024 \, B a^{5} b d e^{6} x + 7560 \, A a^{4} b^{2} d e^{6} x + 2520 \, B a^{6} e^{7} x + 15120 \, A a^{5} b e^{7} x + 7 \, B b^{6} d^{7} + 30 \, B a b^{5} d^{6} e + 5 \, A b^{6} d^{6} e + 75 \, B a^{2} b^{4} d^{5} e^{2} + 30 \, A a b^{5} d^{5} e^{2} + 140 \, B a^{3} b^{3} d^{4} e^{3} + 105 \, A a^{2} b^{4} d^{4} e^{3} + 210 \, B a^{4} b^{2} d^{3} e^{4} + 280 \, A a^{3} b^{3} d^{3} e^{4} + 252 \, B a^{5} b d^{2} e^{5} + 630 \, A a^{4} b^{2} d^{2} e^{5} + 210 \, B a^{6} d e^{6} + 1260 \, A a^{5} b d e^{6} + 2310 \, A a^{6} e^{7}}{27720 \, {\left (e x + d\right )}^{12} e^{8}} \]

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^13,x, algorithm="giac")

[Out]

-1/27720*(5544*B*b^6*e^7*x^7 + 6468*B*b^6*d*e^6*x^6 + 27720*B*a*b^5*e^7*x^6 + 4620*A*b^6*e^7*x^6 + 5544*B*b^6*
d^2*e^5*x^5 + 23760*B*a*b^5*d*e^6*x^5 + 3960*A*b^6*d*e^6*x^5 + 59400*B*a^2*b^4*e^7*x^5 + 23760*A*a*b^5*e^7*x^5
+ 3465*B*b^6*d^3*e^4*x^4 + 14850*B*a*b^5*d^2*e^5*x^4 + 2475*A*b^6*d^2*e^5*x^4 + 37125*B*a^2*b^4*d*e^6*x^4 + 1
4850*A*a*b^5*d*e^6*x^4 + 69300*B*a^3*b^3*e^7*x^4 + 51975*A*a^2*b^4*e^7*x^4 + 1540*B*b^6*d^4*e^3*x^3 + 6600*B*a
*b^5*d^3*e^4*x^3 + 1100*A*b^6*d^3*e^4*x^3 + 16500*B*a^2*b^4*d^2*e^5*x^3 + 6600*A*a*b^5*d^2*e^5*x^3 + 30800*B*a
^3*b^3*d*e^6*x^3 + 23100*A*a^2*b^4*d*e^6*x^3 + 46200*B*a^4*b^2*e^7*x^3 + 61600*A*a^3*b^3*e^7*x^3 + 462*B*b^6*d
^5*e^2*x^2 + 1980*B*a*b^5*d^4*e^3*x^2 + 330*A*b^6*d^4*e^3*x^2 + 4950*B*a^2*b^4*d^3*e^4*x^2 + 1980*A*a*b^5*d^3*
e^4*x^2 + 9240*B*a^3*b^3*d^2*e^5*x^2 + 6930*A*a^2*b^4*d^2*e^5*x^2 + 13860*B*a^4*b^2*d*e^6*x^2 + 18480*A*a^3*b^
3*d*e^6*x^2 + 16632*B*a^5*b*e^7*x^2 + 41580*A*a^4*b^2*e^7*x^2 + 84*B*b^6*d^6*e*x + 360*B*a*b^5*d^5*e^2*x + 60*
A*b^6*d^5*e^2*x + 900*B*a^2*b^4*d^4*e^3*x + 360*A*a*b^5*d^4*e^3*x + 1680*B*a^3*b^3*d^3*e^4*x + 1260*A*a^2*b^4*
d^3*e^4*x + 2520*B*a^4*b^2*d^2*e^5*x + 3360*A*a^3*b^3*d^2*e^5*x + 3024*B*a^5*b*d*e^6*x + 7560*A*a^4*b^2*d*e^6*
x + 2520*B*a^6*e^7*x + 15120*A*a^5*b*e^7*x + 7*B*b^6*d^7 + 30*B*a*b^5*d^6*e + 5*A*b^6*d^6*e + 75*B*a^2*b^4*d^5
*e^2 + 30*A*a*b^5*d^5*e^2 + 140*B*a^3*b^3*d^4*e^3 + 105*A*a^2*b^4*d^4*e^3 + 210*B*a^4*b^2*d^3*e^4 + 280*A*a^3*
b^3*d^3*e^4 + 252*B*a^5*b*d^2*e^5 + 630*A*a^4*b^2*d^2*e^5 + 210*B*a^6*d*e^6 + 1260*A*a^5*b*d*e^6 + 2310*A*a^6*
e^7)/((e*x + d)^12*e^8)

Mupad [B] (verification not implemented)

Time = 1.92 (sec) , antiderivative size = 910, normalized size of antiderivative = 3.12 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{13}} \, dx=-\frac {\frac {210\,B\,a^6\,d\,e^6+2310\,A\,a^6\,e^7+252\,B\,a^5\,b\,d^2\,e^5+1260\,A\,a^5\,b\,d\,e^6+210\,B\,a^4\,b^2\,d^3\,e^4+630\,A\,a^4\,b^2\,d^2\,e^5+140\,B\,a^3\,b^3\,d^4\,e^3+280\,A\,a^3\,b^3\,d^3\,e^4+75\,B\,a^2\,b^4\,d^5\,e^2+105\,A\,a^2\,b^4\,d^4\,e^3+30\,B\,a\,b^5\,d^6\,e+30\,A\,a\,b^5\,d^5\,e^2+7\,B\,b^6\,d^7+5\,A\,b^6\,d^6\,e}{27720\,e^8}+\frac {x\,\left (210\,B\,a^6\,e^6+252\,B\,a^5\,b\,d\,e^5+1260\,A\,a^5\,b\,e^6+210\,B\,a^4\,b^2\,d^2\,e^4+630\,A\,a^4\,b^2\,d\,e^5+140\,B\,a^3\,b^3\,d^3\,e^3+280\,A\,a^3\,b^3\,d^2\,e^4+75\,B\,a^2\,b^4\,d^4\,e^2+105\,A\,a^2\,b^4\,d^3\,e^3+30\,B\,a\,b^5\,d^5\,e+30\,A\,a\,b^5\,d^4\,e^2+7\,B\,b^6\,d^6+5\,A\,b^6\,d^5\,e\right )}{2310\,e^7}+\frac {b^3\,x^4\,\left (140\,B\,a^3\,e^3+75\,B\,a^2\,b\,d\,e^2+105\,A\,a^2\,b\,e^3+30\,B\,a\,b^2\,d^2\,e+30\,A\,a\,b^2\,d\,e^2+7\,B\,b^3\,d^3+5\,A\,b^3\,d^2\,e\right )}{56\,e^4}+\frac {b^5\,x^6\,\left (5\,A\,b\,e+30\,B\,a\,e+7\,B\,b\,d\right )}{30\,e^2}+\frac {b\,x^2\,\left (252\,B\,a^5\,e^5+210\,B\,a^4\,b\,d\,e^4+630\,A\,a^4\,b\,e^5+140\,B\,a^3\,b^2\,d^2\,e^3+280\,A\,a^3\,b^2\,d\,e^4+75\,B\,a^2\,b^3\,d^3\,e^2+105\,A\,a^2\,b^3\,d^2\,e^3+30\,B\,a\,b^4\,d^4\,e+30\,A\,a\,b^4\,d^3\,e^2+7\,B\,b^5\,d^5+5\,A\,b^5\,d^4\,e\right )}{420\,e^6}+\frac {b^2\,x^3\,\left (210\,B\,a^4\,e^4+140\,B\,a^3\,b\,d\,e^3+280\,A\,a^3\,b\,e^4+75\,B\,a^2\,b^2\,d^2\,e^2+105\,A\,a^2\,b^2\,d\,e^3+30\,B\,a\,b^3\,d^3\,e+30\,A\,a\,b^3\,d^2\,e^2+7\,B\,b^4\,d^4+5\,A\,b^4\,d^3\,e\right )}{126\,e^5}+\frac {b^4\,x^5\,\left (75\,B\,a^2\,e^2+30\,B\,a\,b\,d\,e+30\,A\,a\,b\,e^2+7\,B\,b^2\,d^2+5\,A\,b^2\,d\,e\right )}{35\,e^3}+\frac {B\,b^6\,x^7}{5\,e}}{d^{12}+12\,d^{11}\,e\,x+66\,d^{10}\,e^2\,x^2+220\,d^9\,e^3\,x^3+495\,d^8\,e^4\,x^4+792\,d^7\,e^5\,x^5+924\,d^6\,e^6\,x^6+792\,d^5\,e^7\,x^7+495\,d^4\,e^8\,x^8+220\,d^3\,e^9\,x^9+66\,d^2\,e^{10}\,x^{10}+12\,d\,e^{11}\,x^{11}+e^{12}\,x^{12}} \]

[In]

int(((A + B*x)*(a + b*x)^6)/(d + e*x)^13,x)

[Out]

-((2310*A*a^6*e^7 + 7*B*b^6*d^7 + 5*A*b^6*d^6*e + 210*B*a^6*d*e^6 + 30*A*a*b^5*d^5*e^2 + 252*B*a^5*b*d^2*e^5 +
105*A*a^2*b^4*d^4*e^3 + 280*A*a^3*b^3*d^3*e^4 + 630*A*a^4*b^2*d^2*e^5 + 75*B*a^2*b^4*d^5*e^2 + 140*B*a^3*b^3*
d^4*e^3 + 210*B*a^4*b^2*d^3*e^4 + 1260*A*a^5*b*d*e^6 + 30*B*a*b^5*d^6*e)/(27720*e^8) + (x*(210*B*a^6*e^6 + 7*B
*b^6*d^6 + 1260*A*a^5*b*e^6 + 5*A*b^6*d^5*e + 30*A*a*b^5*d^4*e^2 + 630*A*a^4*b^2*d*e^5 + 105*A*a^2*b^4*d^3*e^3
+ 280*A*a^3*b^3*d^2*e^4 + 75*B*a^2*b^4*d^4*e^2 + 140*B*a^3*b^3*d^3*e^3 + 210*B*a^4*b^2*d^2*e^4 + 30*B*a*b^5*d
^5*e + 252*B*a^5*b*d*e^5))/(2310*e^7) + (b^3*x^4*(140*B*a^3*e^3 + 7*B*b^3*d^3 + 105*A*a^2*b*e^3 + 5*A*b^3*d^2*
e + 30*A*a*b^2*d*e^2 + 30*B*a*b^2*d^2*e + 75*B*a^2*b*d*e^2))/(56*e^4) + (b^5*x^6*(5*A*b*e + 30*B*a*e + 7*B*b*d
))/(30*e^2) + (b*x^2*(252*B*a^5*e^5 + 7*B*b^5*d^5 + 630*A*a^4*b*e^5 + 5*A*b^5*d^4*e + 30*A*a*b^4*d^3*e^2 + 280
*A*a^3*b^2*d*e^4 + 105*A*a^2*b^3*d^2*e^3 + 75*B*a^2*b^3*d^3*e^2 + 140*B*a^3*b^2*d^2*e^3 + 30*B*a*b^4*d^4*e + 2
10*B*a^4*b*d*e^4))/(420*e^6) + (b^2*x^3*(210*B*a^4*e^4 + 7*B*b^4*d^4 + 280*A*a^3*b*e^4 + 5*A*b^4*d^3*e + 30*A*
a*b^3*d^2*e^2 + 105*A*a^2*b^2*d*e^3 + 75*B*a^2*b^2*d^2*e^2 + 30*B*a*b^3*d^3*e + 140*B*a^3*b*d*e^3))/(126*e^5)
+ (b^4*x^5*(75*B*a^2*e^2 + 7*B*b^2*d^2 + 30*A*a*b*e^2 + 5*A*b^2*d*e + 30*B*a*b*d*e))/(35*e^3) + (B*b^6*x^7)/(5
*e))/(d^12 + e^12*x^12 + 12*d*e^11*x^11 + 66*d^10*e^2*x^2 + 220*d^9*e^3*x^3 + 495*d^8*e^4*x^4 + 792*d^7*e^5*x^
5 + 924*d^6*e^6*x^6 + 792*d^5*e^7*x^7 + 495*d^4*e^8*x^8 + 220*d^3*e^9*x^9 + 66*d^2*e^10*x^10 + 12*d^11*e*x)