Integrand size = 20, antiderivative size = 272 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^6} \, dx=-\frac {b^5 (6 b B d-A b e-6 a B e) x}{e^7}+\frac {b^6 B x^2}{2 e^6}+\frac {(b d-a e)^6 (B d-A e)}{5 e^8 (d+e x)^5}-\frac {(b d-a e)^5 (7 b B d-6 A b e-a B e)}{4 e^8 (d+e x)^4}+\frac {b (b d-a e)^4 (7 b B d-5 A b e-2 a B e)}{e^8 (d+e x)^3}-\frac {5 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e)}{2 e^8 (d+e x)^2}+\frac {5 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e)}{e^8 (d+e x)}+\frac {3 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e) \log (d+e x)}{e^8} \] Output:
-b^5*(-A*b*e-6*B*a*e+6*B*b*d)*x/e^7+1/2*b^6*B*x^2/e^6+1/5*(-a*e+b*d)^6*(-A *e+B*d)/e^8/(e*x+d)^5-1/4*(-a*e+b*d)^5*(-6*A*b*e-B*a*e+7*B*b*d)/e^8/(e*x+d )^4+b*(-a*e+b*d)^4*(-5*A*b*e-2*B*a*e+7*B*b*d)/e^8/(e*x+d)^3-5/2*b^2*(-a*e+ b*d)^3*(-4*A*b*e-3*B*a*e+7*B*b*d)/e^8/(e*x+d)^2+5*b^3*(-a*e+b*d)^2*(-3*A*b *e-4*B*a*e+7*B*b*d)/e^8/(e*x+d)+3*b^4*(-a*e+b*d)*(-2*A*b*e-5*B*a*e+7*B*b*d )*ln(e*x+d)/e^8
Leaf count is larger than twice the leaf count of optimal. \(633\) vs. \(2(272)=544\).
Time = 0.20 (sec) , antiderivative size = 633, normalized size of antiderivative = 2.33 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^6} \, dx=\frac {-a^6 e^6 (4 A e+B (d+5 e x))-2 a^5 b e^5 \left (3 A e (d+5 e x)+2 B \left (d^2+5 d e x+10 e^2 x^2\right )\right )-5 a^4 b^2 e^4 \left (2 A e \left (d^2+5 d e x+10 e^2 x^2\right )+3 B \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )-20 a^3 b^3 e^3 \left (A e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+4 B \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )+5 a^2 b^4 e^2 \left (-12 A e \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )+B d \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )\right )+2 a b^5 e \left (A d e \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )-6 B \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )\right )+b^6 \left (-2 A e \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )+B \left (459 d^7+1875 d^6 e x+2700 d^5 e^2 x^2+1300 d^4 e^3 x^3-400 d^3 e^4 x^4-500 d^2 e^5 x^5-70 d e^6 x^6+10 e^7 x^7\right )\right )+60 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e) (d+e x)^5 \log (d+e x)}{20 e^8 (d+e x)^5} \] Input:
Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^6,x]
Output:
(-(a^6*e^6*(4*A*e + B*(d + 5*e*x))) - 2*a^5*b*e^5*(3*A*e*(d + 5*e*x) + 2*B *(d^2 + 5*d*e*x + 10*e^2*x^2)) - 5*a^4*b^2*e^4*(2*A*e*(d^2 + 5*d*e*x + 10* e^2*x^2) + 3*B*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3)) - 20*a^3*b^3 *e^3*(A*e*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + 4*B*(d^4 + 5*d^3 *e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4)) + 5*a^2*b^4*e^2*(-12*A* e*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4) + B*d*(137 *d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d*e^3*x^3 + 300*e^4*x^4)) + 2* a*b^5*e*(A*d*e*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d*e^3*x^3 + 300*e^4*x^4) - 6*B*(87*d^6 + 375*d^5*e*x + 600*d^4*e^2*x^2 + 400*d^3*e^3* x^3 + 50*d^2*e^4*x^4 - 50*d*e^5*x^5 - 10*e^6*x^6)) + b^6*(-2*A*e*(87*d^6 + 375*d^5*e*x + 600*d^4*e^2*x^2 + 400*d^3*e^3*x^3 + 50*d^2*e^4*x^4 - 50*d*e ^5*x^5 - 10*e^6*x^6) + B*(459*d^7 + 1875*d^6*e*x + 2700*d^5*e^2*x^2 + 1300 *d^4*e^3*x^3 - 400*d^3*e^4*x^4 - 500*d^2*e^5*x^5 - 70*d*e^6*x^6 + 10*e^7*x ^7)) + 60*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^5*Log[d + e*x])/(20*e^8*(d + e*x)^5)
Time = 0.67 (sec) , antiderivative size = 272, 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.
| \(\displaystyle \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^6} \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {b^5 (6 a B e+A b e-6 b B d)}{e^7}-\frac {3 b^4 (b d-a e) (5 a B e+2 A b e-7 b B d)}{e^7 (d+e x)}+\frac {5 b^3 (b d-a e)^2 (4 a B e+3 A b e-7 b B d)}{e^7 (d+e x)^2}-\frac {5 b^2 (b d-a e)^3 (3 a B e+4 A b e-7 b B d)}{e^7 (d+e x)^3}+\frac {3 b (b d-a e)^4 (2 a B e+5 A b e-7 b B d)}{e^7 (d+e x)^4}+\frac {(a e-b d)^5 (a B e+6 A b e-7 b B d)}{e^7 (d+e x)^5}+\frac {(a e-b d)^6 (A e-B d)}{e^7 (d+e x)^6}+\frac {b^6 B x}{e^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b^5 x (-6 a B e-A b e+6 b B d)}{e^7}+\frac {3 b^4 (b d-a e) \log (d+e x) (-5 a B e-2 A b e+7 b B d)}{e^8}+\frac {5 b^3 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{e^8 (d+e x)}-\frac {5 b^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{2 e^8 (d+e x)^2}+\frac {b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^8 (d+e x)^3}-\frac {(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{4 e^8 (d+e x)^4}+\frac {(b d-a e)^6 (B d-A e)}{5 e^8 (d+e x)^5}+\frac {b^6 B x^2}{2 e^6}\) |
Input:
Int[((a + b*x)^6*(A + B*x))/(d + e*x)^6,x]
Output:
-((b^5*(6*b*B*d - A*b*e - 6*a*B*e)*x)/e^7) + (b^6*B*x^2)/(2*e^6) + ((b*d - a*e)^6*(B*d - A*e))/(5*e^8*(d + e*x)^5) - ((b*d - a*e)^5*(7*b*B*d - 6*A*b *e - a*B*e))/(4*e^8*(d + e*x)^4) + (b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2 *a*B*e))/(e^8*(d + e*x)^3) - (5*b^2*(b*d - a*e)^3*(7*b*B*d - 4*A*b*e - 3*a *B*e))/(2*e^8*(d + e*x)^2) + (5*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*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)* Log[d + e*x])/e^8
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]))))
Leaf count of result is larger than twice the leaf count of optimal. \(799\) vs. \(2(264)=528\).
Time = 0.23 (sec) , antiderivative size = 800, normalized size of antiderivative = 2.94
| method | result | size |
| default | \(\frac {b^{5} \left (\frac {1}{2} B b e \,x^{2}+A b e x +6 B a e x -6 B b d x \right )}{e^{7}}-\frac {5 b^{3} \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +4 B \,a^{3} e^{3}-15 B \,a^{2} b d \,e^{2}+18 B a \,b^{2} d^{2} e -7 b^{3} B \,d^{3}\right )}{e^{8} \left (e x +d \right )}-\frac {5 b^{2} \left (4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e +3 B \,a^{4} e^{4}-16 B \,a^{3} b d \,e^{3}+30 B \,a^{2} b^{2} d^{2} e^{2}-24 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right )}{2 e^{8} \left (e x +d \right )^{2}}-\frac {a^{6} A \,e^{7}-6 A \,a^{5} b d \,e^{6}+15 A \,a^{4} b^{2} d^{2} e^{5}-20 A \,a^{3} b^{3} d^{3} e^{4}+15 A \,a^{2} b^{4} d^{4} e^{3}-6 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e -B \,a^{6} d \,e^{6}+6 B \,a^{5} b \,d^{2} e^{5}-15 B \,a^{4} b^{2} d^{3} e^{4}+20 B \,a^{3} b^{3} d^{4} e^{3}-15 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e -b^{6} B \,d^{7}}{5 e^{8} \left (e x +d \right )^{5}}-\frac {6 A \,a^{5} b \,e^{6}-30 A \,a^{4} b^{2} d \,e^{5}+60 A \,a^{3} b^{3} d^{2} e^{4}-60 A \,a^{2} b^{4} d^{3} e^{3}+30 A a \,b^{5} d^{4} e^{2}-6 A \,b^{6} d^{5} e +B \,a^{6} e^{6}-12 B \,a^{5} b d \,e^{5}+45 B \,a^{4} b^{2} d^{2} e^{4}-80 B \,a^{3} b^{3} d^{3} e^{3}+75 B \,a^{2} b^{4} d^{4} e^{2}-36 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}}{4 e^{8} \left (e x +d \right )^{4}}+\frac {3 b^{4} \left (2 A a b \,e^{2}-2 A \,b^{2} d e +5 B \,a^{2} e^{2}-12 B a b d e +7 b^{2} B \,d^{2}\right ) \ln \left (e x +d \right )}{e^{8}}-\frac {b \left (5 A \,a^{4} b \,e^{5}-20 A \,a^{3} b^{2} d \,e^{4}+30 A \,a^{2} b^{3} d^{2} e^{3}-20 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e +2 B \,a^{5} e^{5}-15 B \,a^{4} b d \,e^{4}+40 B \,a^{3} b^{2} d^{2} e^{3}-50 B \,a^{2} b^{3} d^{3} e^{2}+30 B a \,b^{4} d^{4} e -7 B \,b^{5} d^{5}\right )}{e^{8} \left (e x +d \right )^{3}}\) | \(800\) |
| norman | \(\frac {-\frac {4 a^{6} A \,e^{7}+6 A \,a^{5} b d \,e^{6}+10 A \,a^{4} b^{2} d^{2} e^{5}+20 A \,a^{3} b^{3} d^{3} e^{4}+60 A \,a^{2} b^{4} d^{4} e^{3}-274 A a \,b^{5} d^{5} e^{2}+274 A \,b^{6} d^{6} e +B \,a^{6} d \,e^{6}+4 B \,a^{5} b \,d^{2} e^{5}+15 B \,a^{4} b^{2} d^{3} e^{4}+80 B \,a^{3} b^{3} d^{4} e^{3}-685 B \,a^{2} b^{4} d^{5} e^{2}+1644 B a \,b^{5} d^{6} e -959 b^{6} B \,d^{7}}{20 e^{8}}-\frac {5 \left (3 A \,a^{2} b^{4} e^{3}-6 A a \,b^{5} d \,e^{2}+6 A \,b^{6} d^{2} e +4 B \,a^{3} b^{3} e^{3}-15 B \,a^{2} b^{4} d \,e^{2}+36 B a \,b^{5} d^{2} e -21 b^{6} B \,d^{3}\right ) x^{4}}{e^{4}}-\frac {5 \left (4 A \,a^{3} b^{3} e^{4}+12 A \,a^{2} b^{4} d \,e^{3}-36 A a \,b^{5} d^{2} e^{2}+36 A \,b^{6} d^{3} e +3 B \,a^{4} b^{2} e^{4}+16 B \,a^{3} b^{3} d \,e^{3}-90 B \,a^{2} b^{4} d^{2} e^{2}+216 B a \,b^{5} d^{3} e -126 b^{6} B \,d^{4}\right ) x^{3}}{2 e^{5}}-\frac {\left (10 A \,a^{4} b^{2} e^{5}+20 A \,a^{3} b^{3} d \,e^{4}+60 A \,a^{2} b^{4} d^{2} e^{3}-220 A a \,b^{5} d^{3} e^{2}+220 A \,b^{6} d^{4} e +4 B \,a^{5} b \,e^{5}+15 B \,a^{4} b^{2} d \,e^{4}+80 B \,a^{3} b^{3} d^{2} e^{3}-550 B \,a^{2} b^{4} d^{3} e^{2}+1320 B a \,b^{5} d^{4} e -770 b^{6} B \,d^{5}\right ) x^{2}}{2 e^{6}}-\frac {\left (6 A \,a^{5} b \,e^{6}+10 A \,a^{4} b^{2} d \,e^{5}+20 A \,a^{3} b^{3} d^{2} e^{4}+60 A \,a^{2} b^{4} d^{3} e^{3}-250 A a \,b^{5} d^{4} e^{2}+250 A \,b^{6} d^{5} e +B \,a^{6} e^{6}+4 B \,a^{5} b d \,e^{5}+15 B \,a^{4} b^{2} d^{2} e^{4}+80 B \,a^{3} b^{3} d^{3} e^{3}-625 B \,a^{2} b^{4} d^{4} e^{2}+1500 B a \,b^{5} d^{5} e -875 b^{6} B \,d^{6}\right ) x}{4 e^{7}}+\frac {b^{5} \left (2 A b e +12 B a e -7 B b d \right ) x^{6}}{2 e^{2}}+\frac {b^{6} B \,x^{7}}{2 e}}{\left (e x +d \right )^{5}}+\frac {3 b^{4} \left (2 A a b \,e^{2}-2 A \,b^{2} d e +5 B \,a^{2} e^{2}-12 B a b d e +7 b^{2} B \,d^{2}\right ) \ln \left (e x +d \right )}{e^{8}}\) | \(803\) |
| risch | \(\frac {b^{6} B \,x^{2}}{2 e^{6}}+\frac {b^{6} A x}{e^{6}}+\frac {6 b^{5} B a x}{e^{6}}-\frac {6 b^{6} B d x}{e^{7}}+\frac {\left (-15 A \,a^{2} b^{4} e^{6}+30 A a \,b^{5} d \,e^{5}-15 A \,b^{6} d^{2} e^{4}-20 B \,a^{3} b^{3} e^{6}+75 B \,a^{2} b^{4} d \,e^{5}-90 B a \,b^{5} d^{2} e^{4}+35 b^{6} B \,d^{3} e^{3}\right ) x^{4}-\frac {5 b^{2} e^{2} \left (4 A \,a^{3} b \,e^{4}+12 A \,a^{2} b^{2} d \,e^{3}-36 A a \,b^{3} d^{2} e^{2}+20 A \,b^{4} d^{3} e +3 B \,a^{4} e^{4}+16 B \,a^{3} b d \,e^{3}-90 B \,a^{2} b^{2} d^{2} e^{2}+120 B a \,b^{3} d^{3} e -49 B \,b^{4} d^{4}\right ) x^{3}}{2}-\frac {b e \left (10 A \,a^{4} b \,e^{5}+20 A \,a^{3} b^{2} d \,e^{4}+60 A \,a^{2} b^{3} d^{2} e^{3}-220 A a \,b^{4} d^{3} e^{2}+130 A \,b^{5} d^{4} e +4 B \,a^{5} e^{5}+15 B \,a^{4} b d \,e^{4}+80 B \,a^{3} b^{2} d^{2} e^{3}-550 B \,a^{2} b^{3} d^{3} e^{2}+780 B a \,b^{4} d^{4} e -329 B \,b^{5} d^{5}\right ) x^{2}}{2}+\left (-\frac {3}{2} A \,a^{5} b \,e^{6}-\frac {5}{2} A \,a^{4} b^{2} d \,e^{5}-5 A \,a^{3} b^{3} d^{2} e^{4}-15 A \,a^{2} b^{4} d^{3} e^{3}+\frac {125}{2} A a \,b^{5} d^{4} e^{2}-\frac {77}{2} A \,b^{6} d^{5} e -\frac {1}{4} B \,a^{6} e^{6}-B \,a^{5} b d \,e^{5}-\frac {15}{4} B \,a^{4} b^{2} d^{2} e^{4}-20 B \,a^{3} b^{3} d^{3} e^{3}+\frac {625}{4} B \,a^{2} b^{4} d^{4} e^{2}-231 B a \,b^{5} d^{5} e +\frac {399}{4} b^{6} B \,d^{6}\right ) x -\frac {4 a^{6} A \,e^{7}+6 A \,a^{5} b d \,e^{6}+10 A \,a^{4} b^{2} d^{2} e^{5}+20 A \,a^{3} b^{3} d^{3} e^{4}+60 A \,a^{2} b^{4} d^{4} e^{3}-274 A a \,b^{5} d^{5} e^{2}+174 A \,b^{6} d^{6} e +B \,a^{6} d \,e^{6}+4 B \,a^{5} b \,d^{2} e^{5}+15 B \,a^{4} b^{2} d^{3} e^{4}+80 B \,a^{3} b^{3} d^{4} e^{3}-685 B \,a^{2} b^{4} d^{5} e^{2}+1044 B a \,b^{5} d^{6} e -459 b^{6} B \,d^{7}}{20 e}}{e^{7} \left (e x +d \right )^{5}}+\frac {6 b^{5} \ln \left (e x +d \right ) A a}{e^{6}}-\frac {6 b^{6} \ln \left (e x +d \right ) A d}{e^{7}}+\frac {15 b^{4} \ln \left (e x +d \right ) B \,a^{2}}{e^{6}}-\frac {36 b^{5} \ln \left (e x +d \right ) B a d}{e^{7}}+\frac {21 b^{6} \ln \left (e x +d \right ) B \,d^{2}}{e^{8}}\) | \(829\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1455\) |
Input:
int((b*x+a)^6*(B*x+A)/(e*x+d)^6,x,method=_RETURNVERBOSE)
Output:
b^5/e^7*(1/2*B*b*e*x^2+A*b*e*x+6*B*a*e*x-6*B*b*d*x)-5/e^8*b^3*(3*A*a^2*b*e ^3-6*A*a*b^2*d*e^2+3*A*b^3*d^2*e+4*B*a^3*e^3-15*B*a^2*b*d*e^2+18*B*a*b^2*d ^2*e-7*B*b^3*d^3)/(e*x+d)-5/2*b^2/e^8*(4*A*a^3*b*e^4-12*A*a^2*b^2*d*e^3+12 *A*a*b^3*d^2*e^2-4*A*b^4*d^3*e+3*B*a^4*e^4-16*B*a^3*b*d*e^3+30*B*a^2*b^2*d ^2*e^2-24*B*a*b^3*d^3*e+7*B*b^4*d^4)/(e*x+d)^2-1/5*(A*a^6*e^7-6*A*a^5*b*d* e^6+15*A*a^4*b^2*d^2*e^5-20*A*a^3*b^3*d^3*e^4+15*A*a^2*b^4*d^4*e^3-6*A*a*b ^5*d^5*e^2+A*b^6*d^6*e-B*a^6*d*e^6+6*B*a^5*b*d^2*e^5-15*B*a^4*b^2*d^3*e^4+ 20*B*a^3*b^3*d^4*e^3-15*B*a^2*b^4*d^5*e^2+6*B*a*b^5*d^6*e-B*b^6*d^7)/e^8/( e*x+d)^5-1/4/e^8*(6*A*a^5*b*e^6-30*A*a^4*b^2*d*e^5+60*A*a^3*b^3*d^2*e^4-60 *A*a^2*b^4*d^3*e^3+30*A*a*b^5*d^4*e^2-6*A*b^6*d^5*e+B*a^6*e^6-12*B*a^5*b*d *e^5+45*B*a^4*b^2*d^2*e^4-80*B*a^3*b^3*d^3*e^3+75*B*a^2*b^4*d^4*e^2-36*B*a *b^5*d^5*e+7*B*b^6*d^6)/(e*x+d)^4+3*b^4/e^8*(2*A*a*b*e^2-2*A*b^2*d*e+5*B*a ^2*e^2-12*B*a*b*d*e+7*B*b^2*d^2)*ln(e*x+d)-b/e^8*(5*A*a^4*b*e^5-20*A*a^3*b ^2*d*e^4+30*A*a^2*b^3*d^2*e^3-20*A*a*b^4*d^3*e^2+5*A*b^5*d^4*e+2*B*a^5*e^5 -15*B*a^4*b*d*e^4+40*B*a^3*b^2*d^2*e^3-50*B*a^2*b^3*d^3*e^2+30*B*a*b^4*d^4 *e-7*B*b^5*d^5)/(e*x+d)^3
Leaf count of result is larger than twice the leaf count of optimal. 1157 vs. \(2 (264) = 528\).
Time = 0.13 (sec) , antiderivative size = 1157, normalized size of antiderivative = 4.25 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^6} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^6*(B*x+A)/(e*x+d)^6,x, algorithm="fricas")
Output:
1/20*(10*B*b^6*e^7*x^7 + 459*B*b^6*d^7 - 4*A*a^6*e^7 - 174*(6*B*a*b^5 + A* b^6)*d^6*e + 137*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 20*(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 - 2*(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 - 10*(7*B*b^6*d*e^6 - 2* (6*B*a*b^5 + A*b^6)*e^7)*x^6 - 100*(5*B*b^6*d^2*e^5 - (6*B*a*b^5 + A*b^6)* d*e^6)*x^5 - 100*(4*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 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 50*(26* B*b^6*d^4*e^3 - 16*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 18*(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 + 10*(270*B*b^6*d^5*e^2 - 120*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 110*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 - 20*(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*(2*B*a^5*b + 5*A*a^4*b^2)* e^7)*x^2 + 5*(375*B*b^6*d^6*e - 150*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 125*(5*B *a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - 20*(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 - 2*(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 - 2*(6*B*a*b^5 + A*b^6)*d^6* e + (5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + (7*B*b^6*d^2*e^5 - 2*(6*B*a*b^5 + A*b^6)*d*e^6 + (5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 5*(7*B*b^6*d^3*e^4 - 2 *(6*B*a*b^5 + A*b^6)*d^2*e^5 + (5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6)*x^4 + 10*( 7*B*b^6*d^4*e^3 - 2*(6*B*a*b^5 + A*b^6)*d^3*e^4 + (5*B*a^2*b^4 + 2*A*a*...
Timed out. \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^6} \, dx=\text {Timed out} \] Input:
integrate((b*x+a)**6*(B*x+A)/(e*x+d)**6,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 814 vs. \(2 (264) = 528\).
Time = 0.07 (sec) , antiderivative size = 814, normalized size of antiderivative = 2.99 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^6} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)^6*(B*x+A)/(e*x+d)^6,x, algorithm="maxima")
Output:
1/20*(459*B*b^6*d^7 - 4*A*a^6*e^7 - 174*(6*B*a*b^5 + A*b^6)*d^6*e + 137*(5 *B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 20*(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 - 2*(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 + 100*(7*B*b^6*d^3*e^4 - 3*(6*B*a*b^5 + A*b ^6)*d^2*e^5 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 - (4*B*a^3*b^3 + 3*A*a^2*b ^4)*e^7)*x^4 + 50*(49*B*b^6*d^4*e^3 - 20*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 18* (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 + 10*(329*B*b^6*d^5*e^2 - 130*(6*B*a* b^5 + A*b^6)*d^4*e^3 + 110*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 - 20*(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*(2*B *a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 5*(399*B*b^6*d^6*e - 154*(6*B*a*b^5 + A*b ^6)*d^5*e^2 + 125*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - 20*(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 - 2*(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^13*x^5 + 5*d*e^12*x^ 4 + 10*d^2*e^11*x^3 + 10*d^3*e^10*x^2 + 5*d^4*e^9*x + d^5*e^8) + 1/2*(B*b^ 6*e*x^2 - 2*(6*B*b^6*d - (6*B*a*b^5 + A*b^6)*e)*x)/e^7 + 3*(7*B*b^6*d^2 - 2*(6*B*a*b^5 + A*b^6)*d*e + (5*B*a^2*b^4 + 2*A*a*b^5)*e^2)*log(e*x + d)/e^ 8
Leaf count of result is larger than twice the leaf count of optimal. 831 vs. \(2 (264) = 528\).
Time = 0.13 (sec) , antiderivative size = 831, normalized size of antiderivative = 3.06 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^6} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)^6*(B*x+A)/(e*x+d)^6,x, algorithm="giac")
Output:
3*(7*B*b^6*d^2 - 12*B*a*b^5*d*e - 2*A*b^6*d*e + 5*B*a^2*b^4*e^2 + 2*A*a*b^ 5*e^2)*log(abs(e*x + d))/e^8 + 1/2*(B*b^6*e^6*x^2 - 12*B*b^6*d*e^5*x + 12* B*a*b^5*e^6*x + 2*A*b^6*e^6*x)/e^12 + 1/20*(459*B*b^6*d^7 - 1044*B*a*b^5*d ^6*e - 174*A*b^6*d^6*e + 685*B*a^2*b^4*d^5*e^2 + 274*A*a*b^5*d^5*e^2 - 80* B*a^3*b^3*d^4*e^3 - 60*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 - 4*B*a^5*b*d^2*e^5 - 10*A*a^4*b^2*d^2*e^5 - B*a^6*d*e^6 - 6* A*a^5*b*d*e^6 - 4*A*a^6*e^7 + 100*(7*B*b^6*d^3*e^4 - 18*B*a*b^5*d^2*e^5 - 3*A*b^6*d^2*e^5 + 15*B*a^2*b^4*d*e^6 + 6*A*a*b^5*d*e^6 - 4*B*a^3*b^3*e^7 - 3*A*a^2*b^4*e^7)*x^4 + 50*(49*B*b^6*d^4*e^3 - 120*B*a*b^5*d^3*e^4 - 20*A* b^6*d^3*e^4 + 90*B*a^2*b^4*d^2*e^5 + 36*A*a*b^5*d^2*e^5 - 16*B*a^3*b^3*d*e ^6 - 12*A*a^2*b^4*d*e^6 - 3*B*a^4*b^2*e^7 - 4*A*a^3*b^3*e^7)*x^3 + 10*(329 *B*b^6*d^5*e^2 - 780*B*a*b^5*d^4*e^3 - 130*A*b^6*d^4*e^3 + 550*B*a^2*b^4*d ^3*e^4 + 220*A*a*b^5*d^3*e^4 - 80*B*a^3*b^3*d^2*e^5 - 60*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 - 4*B*a^5*b*e^7 - 10*A*a^4*b^2* e^7)*x^2 + 5*(399*B*b^6*d^6*e - 924*B*a*b^5*d^5*e^2 - 154*A*b^6*d^5*e^2 + 625*B*a^2*b^4*d^4*e^3 + 250*A*a*b^5*d^4*e^3 - 80*B*a^3*b^3*d^3*e^4 - 60*A* a^2*b^4*d^3*e^4 - 15*B*a^4*b^2*d^2*e^5 - 20*A*a^3*b^3*d^2*e^5 - 4*B*a^5*b* d*e^6 - 10*A*a^4*b^2*d*e^6 - B*a^6*e^7 - 6*A*a^5*b*e^7)*x)/((e*x + d)^5*e^ 8)
Time = 0.14 (sec) , antiderivative size = 862, normalized size of antiderivative = 3.17 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^6} \, dx=x\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e^6}-\frac {6\,B\,b^6\,d}{e^7}\right )-\frac {x^3\,\left (\frac {15\,B\,a^4\,b^2\,e^6}{2}+40\,B\,a^3\,b^3\,d\,e^5+10\,A\,a^3\,b^3\,e^6-225\,B\,a^2\,b^4\,d^2\,e^4+30\,A\,a^2\,b^4\,d\,e^5+300\,B\,a\,b^5\,d^3\,e^3-90\,A\,a\,b^5\,d^2\,e^4-\frac {245\,B\,b^6\,d^4\,e^2}{2}+50\,A\,b^6\,d^3\,e^3\right )+\frac {B\,a^6\,d\,e^6+4\,A\,a^6\,e^7+4\,B\,a^5\,b\,d^2\,e^5+6\,A\,a^5\,b\,d\,e^6+15\,B\,a^4\,b^2\,d^3\,e^4+10\,A\,a^4\,b^2\,d^2\,e^5+80\,B\,a^3\,b^3\,d^4\,e^3+20\,A\,a^3\,b^3\,d^3\,e^4-685\,B\,a^2\,b^4\,d^5\,e^2+60\,A\,a^2\,b^4\,d^4\,e^3+1044\,B\,a\,b^5\,d^6\,e-274\,A\,a\,b^5\,d^5\,e^2-459\,B\,b^6\,d^7+174\,A\,b^6\,d^6\,e}{20\,e}+x\,\left (\frac {B\,a^6\,e^6}{4}+B\,a^5\,b\,d\,e^5+\frac {3\,A\,a^5\,b\,e^6}{2}+\frac {15\,B\,a^4\,b^2\,d^2\,e^4}{4}+\frac {5\,A\,a^4\,b^2\,d\,e^5}{2}+20\,B\,a^3\,b^3\,d^3\,e^3+5\,A\,a^3\,b^3\,d^2\,e^4-\frac {625\,B\,a^2\,b^4\,d^4\,e^2}{4}+15\,A\,a^2\,b^4\,d^3\,e^3+231\,B\,a\,b^5\,d^5\,e-\frac {125\,A\,a\,b^5\,d^4\,e^2}{2}-\frac {399\,B\,b^6\,d^6}{4}+\frac {77\,A\,b^6\,d^5\,e}{2}\right )+x^2\,\left (2\,B\,a^5\,b\,e^6+\frac {15\,B\,a^4\,b^2\,d\,e^5}{2}+5\,A\,a^4\,b^2\,e^6+40\,B\,a^3\,b^3\,d^2\,e^4+10\,A\,a^3\,b^3\,d\,e^5-275\,B\,a^2\,b^4\,d^3\,e^3+30\,A\,a^2\,b^4\,d^2\,e^4+390\,B\,a\,b^5\,d^4\,e^2-110\,A\,a\,b^5\,d^3\,e^3-\frac {329\,B\,b^6\,d^5\,e}{2}+65\,A\,b^6\,d^4\,e^2\right )+x^4\,\left (20\,B\,a^3\,b^3\,e^6-75\,B\,a^2\,b^4\,d\,e^5+15\,A\,a^2\,b^4\,e^6+90\,B\,a\,b^5\,d^2\,e^4-30\,A\,a\,b^5\,d\,e^5-35\,B\,b^6\,d^3\,e^3+15\,A\,b^6\,d^2\,e^4\right )}{d^5\,e^7+5\,d^4\,e^8\,x+10\,d^3\,e^9\,x^2+10\,d^2\,e^{10}\,x^3+5\,d\,e^{11}\,x^4+e^{12}\,x^5}+\frac {\ln \left (d+e\,x\right )\,\left (15\,B\,a^2\,b^4\,e^2-36\,B\,a\,b^5\,d\,e+6\,A\,a\,b^5\,e^2+21\,B\,b^6\,d^2-6\,A\,b^6\,d\,e\right )}{e^8}+\frac {B\,b^6\,x^2}{2\,e^6} \] Input:
int(((A + B*x)*(a + b*x)^6)/(d + e*x)^6,x)
Output:
x*((A*b^6 + 6*B*a*b^5)/e^6 - (6*B*b^6*d)/e^7) - (x^3*(10*A*a^3*b^3*e^6 + ( 15*B*a^4*b^2*e^6)/2 + 50*A*b^6*d^3*e^3 - (245*B*b^6*d^4*e^2)/2 - 90*A*a*b^ 5*d^2*e^4 + 30*A*a^2*b^4*d*e^5 + 300*B*a*b^5*d^3*e^3 + 40*B*a^3*b^3*d*e^5 - 225*B*a^2*b^4*d^2*e^4) + (4*A*a^6*e^7 - 459*B*b^6*d^7 + 174*A*b^6*d^6*e + B*a^6*d*e^6 - 274*A*a*b^5*d^5*e^2 + 4*B*a^5*b*d^2*e^5 + 60*A*a^2*b^4*d^4 *e^3 + 20*A*a^3*b^3*d^3*e^4 + 10*A*a^4*b^2*d^2*e^5 - 685*B*a^2*b^4*d^5*e^2 + 80*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 + 1044*B* a*b^5*d^6*e)/(20*e) + x*((B*a^6*e^6)/4 - (399*B*b^6*d^6)/4 + (3*A*a^5*b*e^ 6)/2 + (77*A*b^6*d^5*e)/2 - (125*A*a*b^5*d^4*e^2)/2 + (5*A*a^4*b^2*d*e^5)/ 2 + 15*A*a^2*b^4*d^3*e^3 + 5*A*a^3*b^3*d^2*e^4 - (625*B*a^2*b^4*d^4*e^2)/4 + 20*B*a^3*b^3*d^3*e^3 + (15*B*a^4*b^2*d^2*e^4)/4 + 231*B*a*b^5*d^5*e + B *a^5*b*d*e^5) + x^2*(2*B*a^5*b*e^6 - (329*B*b^6*d^5*e)/2 + 5*A*a^4*b^2*e^6 + 65*A*b^6*d^4*e^2 - 110*A*a*b^5*d^3*e^3 + 10*A*a^3*b^3*d*e^5 + 390*B*a*b ^5*d^4*e^2 + (15*B*a^4*b^2*d*e^5)/2 + 30*A*a^2*b^4*d^2*e^4 - 275*B*a^2*b^4 *d^3*e^3 + 40*B*a^3*b^3*d^2*e^4) + x^4*(15*A*a^2*b^4*e^6 + 20*B*a^3*b^3*e^ 6 + 15*A*b^6*d^2*e^4 - 35*B*b^6*d^3*e^3 + 90*B*a*b^5*d^2*e^4 - 75*B*a^2*b^ 4*d*e^5 - 30*A*a*b^5*d*e^5))/(d^5*e^7 + e^12*x^5 + 5*d^4*e^8*x + 5*d*e^11* x^4 + 10*d^3*e^9*x^2 + 10*d^2*e^10*x^3) + (log(d + e*x)*(21*B*b^6*d^2 - 6* A*b^6*d*e + 6*A*a*b^5*e^2 + 15*B*a^2*b^4*e^2 - 36*B*a*b^5*d*e))/e^8 + (B*b ^6*x^2)/(2*e^6)
Time = 0.16 (sec) , antiderivative size = 816, normalized size of antiderivative = 3.00 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^6} \, dx =\text {Too large to display} \] Input:
int((b*x+a)^6*(B*x+A)/(e*x+d)^6,x)
Output:
(420*log(d + e*x)*a**2*b**5*d**6*e**2 + 2100*log(d + e*x)*a**2*b**5*d**5*e **3*x + 4200*log(d + e*x)*a**2*b**5*d**4*e**4*x**2 + 4200*log(d + e*x)*a** 2*b**5*d**3*e**5*x**3 + 2100*log(d + e*x)*a**2*b**5*d**2*e**6*x**4 + 420*l og(d + e*x)*a**2*b**5*d*e**7*x**5 - 840*log(d + e*x)*a*b**6*d**7*e - 4200* log(d + e*x)*a*b**6*d**6*e**2*x - 8400*log(d + e*x)*a*b**6*d**5*e**3*x**2 - 8400*log(d + e*x)*a*b**6*d**4*e**4*x**3 - 4200*log(d + e*x)*a*b**6*d**3* e**5*x**4 - 840*log(d + e*x)*a*b**6*d**2*e**6*x**5 + 420*log(d + e*x)*b**7 *d**8 + 2100*log(d + e*x)*b**7*d**7*e*x + 4200*log(d + e*x)*b**7*d**6*e**2 *x**2 + 4200*log(d + e*x)*b**7*d**5*e**3*x**3 + 2100*log(d + e*x)*b**7*d** 4*e**4*x**4 + 420*log(d + e*x)*b**7*d**3*e**5*x**5 - 4*a**7*d*e**7 - 7*a** 6*b*d**2*e**6 - 35*a**6*b*d*e**7*x - 14*a**5*b**2*d**3*e**5 - 70*a**5*b**2 *d**2*e**6*x - 140*a**5*b**2*d*e**7*x**2 - 35*a**4*b**3*d**4*e**4 - 175*a* *4*b**3*d**3*e**5*x - 350*a**4*b**3*d**2*e**6*x**2 - 350*a**4*b**3*d*e**7* x**3 + 140*a**3*b**4*e**8*x**5 + 539*a**2*b**5*d**6*e**2 + 2275*a**2*b**5* d**5*e**3*x + 3500*a**2*b**5*d**4*e**4*x**2 + 2100*a**2*b**5*d**3*e**5*x** 3 - 420*a**2*b**5*d*e**7*x**5 - 1078*a*b**6*d**7*e - 4550*a*b**6*d**6*e**2 *x - 7000*a*b**6*d**5*e**3*x**2 - 4200*a*b**6*d**4*e**4*x**3 + 840*a*b**6* d**2*e**6*x**5 + 140*a*b**6*d*e**7*x**6 + 539*b**7*d**8 + 2275*b**7*d**7*e *x + 3500*b**7*d**6*e**2*x**2 + 2100*b**7*d**5*e**3*x**3 - 420*b**7*d**3*e **5*x**5 - 70*b**7*d**2*e**6*x**6 + 10*b**7*d*e**7*x**7)/(20*d*e**8*(d*...