Integrand size = 20, antiderivative size = 213 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^8} \, dx=-\frac {(B d-A e) (a+b x)^7}{7 e (b d-a e) (d+e x)^7}-\frac {B (b d-a e)^6}{6 e^8 (d+e x)^6}+\frac {6 b B (b d-a e)^5}{5 e^8 (d+e x)^5}-\frac {15 b^2 B (b d-a e)^4}{4 e^8 (d+e x)^4}+\frac {20 b^3 B (b d-a e)^3}{3 e^8 (d+e x)^3}-\frac {15 b^4 B (b d-a e)^2}{2 e^8 (d+e x)^2}+\frac {6 b^5 B (b d-a e)}{e^8 (d+e x)}+\frac {b^6 B \log (d+e x)}{e^8} \] Output:
-1/7*(-A*e+B*d)*(b*x+a)^7/e/(-a*e+b*d)/(e*x+d)^7-1/6*B*(-a*e+b*d)^6/e^8/(e *x+d)^6+6/5*b*B*(-a*e+b*d)^5/e^8/(e*x+d)^5-15/4*b^2*B*(-a*e+b*d)^4/e^8/(e* x+d)^4+20/3*b^3*B*(-a*e+b*d)^3/e^8/(e*x+d)^3-15/2*b^4*B*(-a*e+b*d)^2/e^8/( e*x+d)^2+6*b^5*B*(-a*e+b*d)/e^8/(e*x+d)+b^6*B*ln(e*x+d)/e^8
Leaf count is larger than twice the leaf count of optimal. \(615\) vs. \(2(213)=426\).
Time = 0.28 (sec) , antiderivative size = 615, normalized size of antiderivative = 2.89 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^8} \, dx=-\frac {10 a^6 e^6 (6 A e+B (d+7 e x))+12 a^5 b e^5 \left (5 A e (d+7 e x)+2 B \left (d^2+7 d e x+21 e^2 x^2\right )\right )+15 a^4 b^2 e^4 \left (4 A e \left (d^2+7 d e x+21 e^2 x^2\right )+3 B \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )+20 a^3 b^3 e^3 \left (3 A e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+4 B \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )\right )+30 a^2 b^4 e^2 \left (2 A e \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+5 B \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )\right )+60 a b^5 e \left (A e \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )+6 B \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )\right )+b^6 \left (60 A e \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )-B d \left (1089 d^6+7203 d^5 e x+20139 d^4 e^2 x^2+30625 d^3 e^3 x^3+26950 d^2 e^4 x^4+13230 d e^5 x^5+2940 e^6 x^6\right )\right )-420 b^6 B (d+e x)^7 \log (d+e x)}{420 e^8 (d+e x)^7} \] Input:
Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^8,x]
Output:
-1/420*(10*a^6*e^6*(6*A*e + B*(d + 7*e*x)) + 12*a^5*b*e^5*(5*A*e*(d + 7*e* x) + 2*B*(d^2 + 7*d*e*x + 21*e^2*x^2)) + 15*a^4*b^2*e^4*(4*A*e*(d^2 + 7*d* e*x + 21*e^2*x^2) + 3*B*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3)) + 2 0*a^3*b^3*e^3*(3*A*e*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 4*B*( d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4)) + 30*a^2*b^ 4*e^2*(2*A*e*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4 ) + 5*B*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5)) + 60*a*b^5*e*(A*e*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^ 2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5) + 6*B*(d^6 + 7*d^5*e*x + 21*d^4*e^2 *x^2 + 35*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6)) + b^6* (60*A*e*(d^6 + 7*d^5*e*x + 21*d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 35*d^2*e^4*x^ 4 + 21*d*e^5*x^5 + 7*e^6*x^6) - B*d*(1089*d^6 + 7203*d^5*e*x + 20139*d^4*e ^2*x^2 + 30625*d^3*e^3*x^3 + 26950*d^2*e^4*x^4 + 13230*d*e^5*x^5 + 2940*e^ 6*x^6)) - 420*b^6*B*(d + e*x)^7*Log[d + e*x])/(e^8*(d + e*x)^7)
Time = 0.57 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {87, 49, 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.
| \(\displaystyle \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^8} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {B \int \frac {(a+b x)^6}{(d+e x)^7}dx}{e}-\frac {(a+b x)^7 (B d-A e)}{7 e (d+e x)^7 (b d-a e)}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {B \int \left (\frac {b^6}{e^6 (d+e x)}-\frac {6 (b d-a e) b^5}{e^6 (d+e x)^2}+\frac {15 (b d-a e)^2 b^4}{e^6 (d+e x)^3}-\frac {20 (b d-a e)^3 b^3}{e^6 (d+e x)^4}+\frac {15 (b d-a e)^4 b^2}{e^6 (d+e x)^5}-\frac {6 (b d-a e)^5 b}{e^6 (d+e x)^6}+\frac {(a e-b d)^6}{e^6 (d+e x)^7}\right )dx}{e}-\frac {(a+b x)^7 (B d-A e)}{7 e (d+e x)^7 (b d-a e)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {B \left (\frac {6 b^5 (b d-a e)}{e^7 (d+e x)}-\frac {15 b^4 (b d-a e)^2}{2 e^7 (d+e x)^2}+\frac {20 b^3 (b d-a e)^3}{3 e^7 (d+e x)^3}-\frac {15 b^2 (b d-a e)^4}{4 e^7 (d+e x)^4}+\frac {6 b (b d-a e)^5}{5 e^7 (d+e x)^5}-\frac {(b d-a e)^6}{6 e^7 (d+e x)^6}+\frac {b^6 \log (d+e x)}{e^7}\right )}{e}-\frac {(a+b x)^7 (B d-A e)}{7 e (d+e x)^7 (b d-a e)}\) |
Input:
Int[((a + b*x)^6*(A + B*x))/(d + e*x)^8,x]
Output:
-1/7*((B*d - A*e)*(a + b*x)^7)/(e*(b*d - a*e)*(d + e*x)^7) + (B*(-1/6*(b*d - a*e)^6/(e^7*(d + e*x)^6) + (6*b*(b*d - a*e)^5)/(5*e^7*(d + e*x)^5) - (1 5*b^2*(b*d - a*e)^4)/(4*e^7*(d + e*x)^4) + (20*b^3*(b*d - a*e)^3)/(3*e^7*( d + e*x)^3) - (15*b^4*(b*d - a*e)^2)/(2*e^7*(d + e*x)^2) + (6*b^5*(b*d - a *e))/(e^7*(d + e*x)) + (b^6*Log[d + e*x])/e^7))/e
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Leaf count of result is larger than twice the leaf count of optimal. \(790\) vs. \(2(201)=402\).
Time = 0.23 (sec) , antiderivative size = 791, normalized size of antiderivative = 3.71
| method | result | size |
| risch | \(\frac {-\frac {b^{5} \left (A b e +6 B a e -7 B b d \right ) x^{6}}{e^{2}}-\frac {3 b^{4} \left (2 A a b \,e^{2}+2 A \,b^{2} d e +5 B \,a^{2} e^{2}+12 B a b d e -21 b^{2} B \,d^{2}\right ) x^{5}}{2 e^{3}}-\frac {5 b^{3} \left (6 A \,a^{2} b \,e^{3}+6 A a \,b^{2} d \,e^{2}+6 A \,b^{3} d^{2} e +8 B \,a^{3} e^{3}+15 B \,a^{2} b d \,e^{2}+36 B a \,b^{2} d^{2} e -77 b^{3} B \,d^{3}\right ) x^{4}}{6 e^{4}}-\frac {5 b^{2} \left (12 A \,a^{3} b \,e^{4}+12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}+12 A \,b^{4} d^{3} e +9 B \,a^{4} e^{4}+16 B \,a^{3} b d \,e^{3}+30 B \,a^{2} b^{2} d^{2} e^{2}+72 B a \,b^{3} d^{3} e -175 B \,b^{4} d^{4}\right ) x^{3}}{12 e^{5}}-\frac {b \left (60 A \,a^{4} b \,e^{5}+60 A \,a^{3} b^{2} d \,e^{4}+60 A \,a^{2} b^{3} d^{2} e^{3}+60 A a \,b^{4} d^{3} e^{2}+60 A \,b^{5} d^{4} e +24 B \,a^{5} e^{5}+45 B \,a^{4} b d \,e^{4}+80 B \,a^{3} b^{2} d^{2} e^{3}+150 B \,a^{2} b^{3} d^{3} e^{2}+360 B a \,b^{4} d^{4} e -959 B \,b^{5} d^{5}\right ) x^{2}}{20 e^{6}}-\frac {\left (60 A \,a^{5} b \,e^{6}+60 A \,a^{4} b^{2} d \,e^{5}+60 A \,a^{3} b^{3} d^{2} e^{4}+60 A \,a^{2} b^{4} d^{3} e^{3}+60 A a \,b^{5} d^{4} e^{2}+60 A \,b^{6} d^{5} e +10 B \,a^{6} e^{6}+24 B \,a^{5} b d \,e^{5}+45 B \,a^{4} b^{2} d^{2} e^{4}+80 B \,a^{3} b^{3} d^{3} e^{3}+150 B \,a^{2} b^{4} d^{4} e^{2}+360 B a \,b^{5} d^{5} e -1029 b^{6} B \,d^{6}\right ) x}{60 e^{7}}-\frac {60 a^{6} A \,e^{7}+60 A \,a^{5} b d \,e^{6}+60 A \,a^{4} b^{2} d^{2} e^{5}+60 A \,a^{3} b^{3} d^{3} e^{4}+60 A \,a^{2} b^{4} d^{4} e^{3}+60 A a \,b^{5} d^{5} e^{2}+60 A \,b^{6} d^{6} e +10 B \,a^{6} d \,e^{6}+24 B \,a^{5} b \,d^{2} e^{5}+45 B \,a^{4} b^{2} d^{3} e^{4}+80 B \,a^{3} b^{3} d^{4} e^{3}+150 B \,a^{2} b^{4} d^{5} e^{2}+360 B a \,b^{5} d^{6} e -1089 b^{6} B \,d^{7}}{420 e^{8}}}{\left (e x +d \right )^{7}}+\frac {b^{6} B \ln \left (e x +d \right )}{e^{8}}\) | \(791\) |
| norman | \(\frac {-\frac {60 a^{6} A \,e^{7}+60 A \,a^{5} b d \,e^{6}+60 A \,a^{4} b^{2} d^{2} e^{5}+60 A \,a^{3} b^{3} d^{3} e^{4}+60 A \,a^{2} b^{4} d^{4} e^{3}+60 A a \,b^{5} d^{5} e^{2}+60 A \,b^{6} d^{6} e +10 B \,a^{6} d \,e^{6}+24 B \,a^{5} b \,d^{2} e^{5}+45 B \,a^{4} b^{2} d^{3} e^{4}+80 B \,a^{3} b^{3} d^{4} e^{3}+150 B \,a^{2} b^{4} d^{5} e^{2}+360 B a \,b^{5} d^{6} e -1089 b^{6} B \,d^{7}}{420 e^{8}}-\frac {\left (A \,b^{6} e +6 B a \,b^{5} e -7 b^{6} B d \right ) x^{6}}{e^{2}}-\frac {3 \left (2 A a \,b^{5} e^{2}+2 A \,b^{6} d e +5 B \,a^{2} b^{4} e^{2}+12 B a \,b^{5} d e -21 b^{6} B \,d^{2}\right ) x^{5}}{2 e^{3}}-\frac {5 \left (6 A \,a^{2} b^{4} e^{3}+6 A a \,b^{5} d \,e^{2}+6 A \,b^{6} d^{2} e +8 B \,a^{3} b^{3} e^{3}+15 B \,a^{2} b^{4} d \,e^{2}+36 B a \,b^{5} d^{2} e -77 b^{6} B \,d^{3}\right ) x^{4}}{6 e^{4}}-\frac {5 \left (12 A \,a^{3} b^{3} e^{4}+12 A \,a^{2} b^{4} d \,e^{3}+12 A a \,b^{5} d^{2} e^{2}+12 A \,b^{6} d^{3} e +9 B \,a^{4} b^{2} e^{4}+16 B \,a^{3} b^{3} d \,e^{3}+30 B \,a^{2} b^{4} d^{2} e^{2}+72 B a \,b^{5} d^{3} e -175 b^{6} B \,d^{4}\right ) x^{3}}{12 e^{5}}-\frac {\left (60 A \,a^{4} b^{2} e^{5}+60 A \,a^{3} b^{3} d \,e^{4}+60 A \,a^{2} b^{4} d^{2} e^{3}+60 A a \,b^{5} d^{3} e^{2}+60 A \,b^{6} d^{4} e +24 B \,a^{5} b \,e^{5}+45 B \,a^{4} b^{2} d \,e^{4}+80 B \,a^{3} b^{3} d^{2} e^{3}+150 B \,a^{2} b^{4} d^{3} e^{2}+360 B a \,b^{5} d^{4} e -959 b^{6} B \,d^{5}\right ) x^{2}}{20 e^{6}}-\frac {\left (60 A \,a^{5} b \,e^{6}+60 A \,a^{4} b^{2} d \,e^{5}+60 A \,a^{3} b^{3} d^{2} e^{4}+60 A \,a^{2} b^{4} d^{3} e^{3}+60 A a \,b^{5} d^{4} e^{2}+60 A \,b^{6} d^{5} e +10 B \,a^{6} e^{6}+24 B \,a^{5} b d \,e^{5}+45 B \,a^{4} b^{2} d^{2} e^{4}+80 B \,a^{3} b^{3} d^{3} e^{3}+150 B \,a^{2} b^{4} d^{4} e^{2}+360 B a \,b^{5} d^{5} e -1029 b^{6} B \,d^{6}\right ) x}{60 e^{7}}}{\left (e x +d \right )^{7}}+\frac {b^{6} B \ln \left (e x +d \right )}{e^{8}}\) | \(811\) |
| default | \(-\frac {6 A \,a^{5} b \,e^{6}-30 A \,a^{4} b^{2} d \,e^{5}+60 A \,a^{3} b^{3} d^{2} e^{4}-60 A \,a^{2} b^{4} d^{3} e^{3}+30 A a \,b^{5} d^{4} e^{2}-6 A \,b^{6} d^{5} e +B \,a^{6} e^{6}-12 B \,a^{5} b d \,e^{5}+45 B \,a^{4} b^{2} d^{2} e^{4}-80 B \,a^{3} b^{3} d^{3} e^{3}+75 B \,a^{2} b^{4} d^{4} e^{2}-36 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}}{6 e^{8} \left (e x +d \right )^{6}}-\frac {a^{6} A \,e^{7}-6 A \,a^{5} b d \,e^{6}+15 A \,a^{4} b^{2} d^{2} e^{5}-20 A \,a^{3} b^{3} d^{3} e^{4}+15 A \,a^{2} b^{4} d^{4} e^{3}-6 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e -B \,a^{6} d \,e^{6}+6 B \,a^{5} b \,d^{2} e^{5}-15 B \,a^{4} b^{2} d^{3} e^{4}+20 B \,a^{3} b^{3} d^{4} e^{3}-15 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e -b^{6} B \,d^{7}}{7 e^{8} \left (e x +d \right )^{7}}-\frac {b^{5} \left (A b e +6 B a e -7 B b d \right )}{e^{8} \left (e x +d \right )}-\frac {3 b^{4} \left (2 A a b \,e^{2}-2 A \,b^{2} d e +5 B \,a^{2} e^{2}-12 B a b d e +7 b^{2} B \,d^{2}\right )}{2 e^{8} \left (e x +d \right )^{2}}-\frac {3 b \left (5 A \,a^{4} b \,e^{5}-20 A \,a^{3} b^{2} d \,e^{4}+30 A \,a^{2} b^{3} d^{2} e^{3}-20 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e +2 B \,a^{5} e^{5}-15 B \,a^{4} b d \,e^{4}+40 B \,a^{3} b^{2} d^{2} e^{3}-50 B \,a^{2} b^{3} d^{3} e^{2}+30 B a \,b^{4} d^{4} e -7 B \,b^{5} d^{5}\right )}{5 e^{8} \left (e x +d \right )^{5}}-\frac {5 b^{2} \left (4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e +3 B \,a^{4} e^{4}-16 B \,a^{3} b d \,e^{3}+30 B \,a^{2} b^{2} d^{2} e^{2}-24 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right )}{4 e^{8} \left (e x +d \right )^{4}}+\frac {b^{6} B \ln \left (e x +d \right )}{e^{8}}-\frac {5 b^{3} \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +4 B \,a^{3} e^{3}-15 B \,a^{2} b d \,e^{2}+18 B a \,b^{2} d^{2} e -7 b^{3} B \,d^{3}\right )}{3 e^{8} \left (e x +d \right )^{3}}\) | \(812\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1054\) |
Input:
int((b*x+a)^6*(B*x+A)/(e*x+d)^8,x,method=_RETURNVERBOSE)
Output:
(-b^5*(A*b*e+6*B*a*e-7*B*b*d)/e^2*x^6-3/2*b^4*(2*A*a*b*e^2+2*A*b^2*d*e+5*B *a^2*e^2+12*B*a*b*d*e-21*B*b^2*d^2)/e^3*x^5-5/6*b^3*(6*A*a^2*b*e^3+6*A*a*b ^2*d*e^2+6*A*b^3*d^2*e+8*B*a^3*e^3+15*B*a^2*b*d*e^2+36*B*a*b^2*d^2*e-77*B* b^3*d^3)/e^4*x^4-5/12*b^2*(12*A*a^3*b*e^4+12*A*a^2*b^2*d*e^3+12*A*a*b^3*d^ 2*e^2+12*A*b^4*d^3*e+9*B*a^4*e^4+16*B*a^3*b*d*e^3+30*B*a^2*b^2*d^2*e^2+72* B*a*b^3*d^3*e-175*B*b^4*d^4)/e^5*x^3-1/20*b*(60*A*a^4*b*e^5+60*A*a^3*b^2*d *e^4+60*A*a^2*b^3*d^2*e^3+60*A*a*b^4*d^3*e^2+60*A*b^5*d^4*e+24*B*a^5*e^5+4 5*B*a^4*b*d*e^4+80*B*a^3*b^2*d^2*e^3+150*B*a^2*b^3*d^3*e^2+360*B*a*b^4*d^4 *e-959*B*b^5*d^5)/e^6*x^2-1/60*(60*A*a^5*b*e^6+60*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+60*A*a*b^5*d^4*e^2+60*A*b^6*d^5*e+10*B*a ^6*e^6+24*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+150*B*a^ 2*b^4*d^4*e^2+360*B*a*b^5*d^5*e-1029*B*b^6*d^6)/e^7*x-1/420*(60*A*a^6*e^7+ 60*A*a^5*b*d*e^6+60*A*a^4*b^2*d^2*e^5+60*A*a^3*b^3*d^3*e^4+60*A*a^2*b^4*d^ 4*e^3+60*A*a*b^5*d^5*e^2+60*A*b^6*d^6*e+10*B*a^6*d*e^6+24*B*a^5*b*d^2*e^5+ 45*B*a^4*b^2*d^3*e^4+80*B*a^3*b^3*d^4*e^3+150*B*a^2*b^4*d^5*e^2+360*B*a*b^ 5*d^6*e-1089*B*b^6*d^7)/e^8)/(e*x+d)^7+b^6*B*ln(e*x+d)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 939 vs. \(2 (201) = 402\).
Time = 0.11 (sec) , antiderivative size = 939, normalized size of antiderivative = 4.41 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^8} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)^6*(B*x+A)/(e*x+d)^8,x, algorithm="fricas")
Output:
1/420*(1089*B*b^6*d^7 - 60*A*a^6*e^7 - 60*(6*B*a*b^5 + A*b^6)*d^6*e - 30*( 5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 - 15*(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 - 10*(B*a^6 + 6*A*a^5*b)*d*e^6 + 420*(7*B*b^6*d*e^6 - (6*B*a*b^5 + A *b^6)*e^7)*x^6 + 630*(21*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 + 350*(77*B*b^6*d^3*e^4 - 6*(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 - 2*(4*B*a^3*b^3 + 3*A*a ^2*b^4)*e^7)*x^4 + 175*(175*B*b^6*d^4*e^3 - 12*(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*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 21*(959*B*b^6*d^5*e^2 - 60*( 6*B*a*b^5 + A*b^6)*d^4*e^3 - 30*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 - 20*(4* B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 - 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 - 12*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 7*(1029*B*b^6*d^6*e - 60*(6*B*a*b^ 5 + A*b^6)*d^5*e^2 - 30*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - 20*(4*B*a^3*b^ 3 + 3*A*a^2*b^4)*d^3*e^4 - 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 - 12*(2* B*a^5*b + 5*A*a^4*b^2)*d*e^6 - 10*(B*a^6 + 6*A*a^5*b)*e^7)*x + 420*(B*b^6* e^7*x^7 + 7*B*b^6*d*e^6*x^6 + 21*B*b^6*d^2*e^5*x^5 + 35*B*b^6*d^3*e^4*x^4 + 35*B*b^6*d^4*e^3*x^3 + 21*B*b^6*d^5*e^2*x^2 + 7*B*b^6*d^6*e*x + B*b^6*d^ 7)*log(e*x + d))/(e^15*x^7 + 7*d*e^14*x^6 + 21*d^2*e^13*x^5 + 35*d^3*e^12* x^4 + 35*d^4*e^11*x^3 + 21*d^5*e^10*x^2 + 7*d^6*e^9*x + d^7*e^8)
Timed out. \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^8} \, dx=\text {Timed out} \] Input:
integrate((b*x+a)**6*(B*x+A)/(e*x+d)**8,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 842 vs. \(2 (201) = 402\).
Time = 0.06 (sec) , antiderivative size = 842, normalized size of antiderivative = 3.95 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^8} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)^6*(B*x+A)/(e*x+d)^8,x, algorithm="maxima")
Output:
1/420*(1089*B*b^6*d^7 - 60*A*a^6*e^7 - 60*(6*B*a*b^5 + A*b^6)*d^6*e - 30*( 5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 - 15*(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 - 10*(B*a^6 + 6*A*a^5*b)*d*e^6 + 420*(7*B*b^6*d*e^6 - (6*B*a*b^5 + A *b^6)*e^7)*x^6 + 630*(21*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 + 350*(77*B*b^6*d^3*e^4 - 6*(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 - 2*(4*B*a^3*b^3 + 3*A*a ^2*b^4)*e^7)*x^4 + 175*(175*B*b^6*d^4*e^3 - 12*(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*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 21*(959*B*b^6*d^5*e^2 - 60*( 6*B*a*b^5 + A*b^6)*d^4*e^3 - 30*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 - 20*(4* B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 - 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 - 12*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 7*(1029*B*b^6*d^6*e - 60*(6*B*a*b^ 5 + A*b^6)*d^5*e^2 - 30*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - 20*(4*B*a^3*b^ 3 + 3*A*a^2*b^4)*d^3*e^4 - 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 - 12*(2* B*a^5*b + 5*A*a^4*b^2)*d*e^6 - 10*(B*a^6 + 6*A*a^5*b)*e^7)*x)/(e^15*x^7 + 7*d*e^14*x^6 + 21*d^2*e^13*x^5 + 35*d^3*e^12*x^4 + 35*d^4*e^11*x^3 + 21*d^ 5*e^10*x^2 + 7*d^6*e^9*x + d^7*e^8) + B*b^6*log(e*x + d)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 830 vs. \(2 (201) = 402\).
Time = 0.12 (sec) , antiderivative size = 830, normalized size of antiderivative = 3.90 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^8} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)^6*(B*x+A)/(e*x+d)^8,x, algorithm="giac")
Output:
B*b^6*log(abs(e*x + d))/e^8 + 1/420*(420*(7*B*b^6*d*e^5 - 6*B*a*b^5*e^6 - A*b^6*e^6)*x^6 + 630*(21*B*b^6*d^2*e^4 - 12*B*a*b^5*d*e^5 - 2*A*b^6*d*e^5 - 5*B*a^2*b^4*e^6 - 2*A*a*b^5*e^6)*x^5 + 350*(77*B*b^6*d^3*e^3 - 36*B*a*b^ 5*d^2*e^4 - 6*A*b^6*d^2*e^4 - 15*B*a^2*b^4*d*e^5 - 6*A*a*b^5*d*e^5 - 8*B*a ^3*b^3*e^6 - 6*A*a^2*b^4*e^6)*x^4 + 175*(175*B*b^6*d^4*e^2 - 72*B*a*b^5*d^ 3*e^3 - 12*A*b^6*d^3*e^3 - 30*B*a^2*b^4*d^2*e^4 - 12*A*a*b^5*d^2*e^4 - 16* B*a^3*b^3*d*e^5 - 12*A*a^2*b^4*d*e^5 - 9*B*a^4*b^2*e^6 - 12*A*a^3*b^3*e^6) *x^3 + 21*(959*B*b^6*d^5*e - 360*B*a*b^5*d^4*e^2 - 60*A*b^6*d^4*e^2 - 150* B*a^2*b^4*d^3*e^3 - 60*A*a*b^5*d^3*e^3 - 80*B*a^3*b^3*d^2*e^4 - 60*A*a^2*b ^4*d^2*e^4 - 45*B*a^4*b^2*d*e^5 - 60*A*a^3*b^3*d*e^5 - 24*B*a^5*b*e^6 - 60 *A*a^4*b^2*e^6)*x^2 + 7*(1029*B*b^6*d^6 - 360*B*a*b^5*d^5*e - 60*A*b^6*d^5 *e - 150*B*a^2*b^4*d^4*e^2 - 60*A*a*b^5*d^4*e^2 - 80*B*a^3*b^3*d^3*e^3 - 6 0*A*a^2*b^4*d^3*e^3 - 45*B*a^4*b^2*d^2*e^4 - 60*A*a^3*b^3*d^2*e^4 - 24*B*a ^5*b*d*e^5 - 60*A*a^4*b^2*d*e^5 - 10*B*a^6*e^6 - 60*A*a^5*b*e^6)*x + (1089 *B*b^6*d^7 - 360*B*a*b^5*d^6*e - 60*A*b^6*d^6*e - 150*B*a^2*b^4*d^5*e^2 - 60*A*a*b^5*d^5*e^2 - 80*B*a^3*b^3*d^4*e^3 - 60*A*a^2*b^4*d^4*e^3 - 45*B*a^ 4*b^2*d^3*e^4 - 60*A*a^3*b^3*d^3*e^4 - 24*B*a^5*b*d^2*e^5 - 60*A*a^4*b^2*d ^2*e^5 - 10*B*a^6*d*e^6 - 60*A*a^5*b*d*e^6 - 60*A*a^6*e^7)/e)/((e*x + d)^7 *e^7)
Time = 1.07 (sec) , antiderivative size = 1046, normalized size of antiderivative = 4.91 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^8} \, dx =\text {Too large to display} \] Input:
int(((A + B*x)*(a + b*x)^6)/(d + e*x)^8,x)
Output:
-((A*a^6*e^7)/7 - (363*B*b^6*d^7)/140 + (A*b^6*d^6*e)/7 + (B*a^6*d*e^6)/42 - B*b^6*d^7*log(d + e*x) + (B*a^6*e^7*x)/6 + A*b^6*e^7*x^6 - (343*B*b^6*d ^6*e*x)/20 + (A*a*b^5*d^5*e^2)/7 + (2*B*a^5*b*d^2*e^5)/35 + 3*A*a*b^5*e^7* x^5 + (6*B*a^5*b*e^7*x^2)/5 + 6*B*a*b^5*e^7*x^6 + A*b^6*d^5*e^2*x + 3*A*b^ 6*d*e^6*x^5 - 7*B*b^6*d*e^6*x^6 - B*b^6*e^7*x^7*log(d + e*x) + (A*a^2*b^4* d^4*e^3)/7 + (A*a^3*b^3*d^3*e^4)/7 + (A*a^4*b^2*d^2*e^5)/7 + (5*B*a^2*b^4* d^5*e^2)/14 + (4*B*a^3*b^3*d^4*e^3)/21 + (3*B*a^4*b^2*d^3*e^4)/28 + 3*A*a^ 4*b^2*e^7*x^2 + 5*A*a^3*b^3*e^7*x^3 + 5*A*a^2*b^4*e^7*x^4 + (15*B*a^4*b^2* e^7*x^3)/4 + (20*B*a^3*b^3*e^7*x^4)/3 + (15*B*a^2*b^4*e^7*x^5)/2 + 3*A*b^6 *d^4*e^3*x^2 + 5*A*b^6*d^3*e^4*x^3 + 5*A*b^6*d^2*e^5*x^4 - (959*B*b^6*d^5* e^2*x^2)/20 - (875*B*b^6*d^4*e^3*x^3)/12 - (385*B*b^6*d^3*e^4*x^4)/6 - (63 *B*b^6*d^2*e^5*x^5)/2 + (A*a^5*b*d*e^6)/7 + (6*B*a*b^5*d^6*e)/7 + A*a^5*b* e^7*x + A*a^2*b^4*d^3*e^4*x + A*a^3*b^3*d^2*e^5*x + 3*A*a*b^5*d^3*e^4*x^2 + 3*A*a^3*b^3*d*e^6*x^2 + 5*A*a*b^5*d^2*e^5*x^3 + 5*A*a^2*b^4*d*e^6*x^3 + (5*B*a^2*b^4*d^4*e^3*x)/2 + (4*B*a^3*b^3*d^3*e^4*x)/3 + (3*B*a^4*b^2*d^2*e ^5*x)/4 + 18*B*a*b^5*d^4*e^3*x^2 + (9*B*a^4*b^2*d*e^6*x^2)/4 + 30*B*a*b^5* d^3*e^4*x^3 + (20*B*a^3*b^3*d*e^6*x^3)/3 + 30*B*a*b^5*d^2*e^5*x^4 + (25*B* a^2*b^4*d*e^6*x^4)/2 - 21*B*b^6*d^5*e^2*x^2*log(d + e*x) - 35*B*b^6*d^4*e^ 3*x^3*log(d + e*x) - 35*B*b^6*d^3*e^4*x^4*log(d + e*x) - 21*B*b^6*d^2*e^5* x^5*log(d + e*x) + (2*B*a^5*b*d*e^6*x)/5 - 7*B*b^6*d^6*e*x*log(d + e*x)...
Time = 0.16 (sec) , antiderivative size = 638, normalized size of antiderivative = 3.00 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^8} \, dx=\frac {420 \,\mathrm {log}\left (e x +d \right ) b^{7} d \,e^{7} x^{7}+8820 \,\mathrm {log}\left (e x +d \right ) b^{7} d^{3} e^{5} x^{5}+8820 \,\mathrm {log}\left (e x +d \right ) b^{7} d^{6} e^{2} x^{2}-60 a^{7} d \,e^{7}+2940 \,\mathrm {log}\left (e x +d \right ) b^{7} d^{2} e^{6} x^{6}+420 a \,b^{6} e^{8} x^{7}-70 a^{6} b \,d^{2} e^{6}+14700 \,\mathrm {log}\left (e x +d \right ) b^{7} d^{4} e^{4} x^{4}-84 a^{5} b^{2} d^{3} e^{5}-105 a^{4} b^{3} d^{4} e^{4}-140 a^{3} b^{4} d^{5} e^{3}-210 a^{2} b^{5} d^{6} e^{2}+420 \,\mathrm {log}\left (e x +d \right ) b^{7} d^{8}+4263 b^{7} d^{7} e x +11319 b^{7} d^{6} e^{2} x^{2}+15925 b^{7} d^{5} e^{3} x^{3}+12250 b^{7} d^{4} e^{4} x^{4}+4410 b^{7} d^{3} e^{5} x^{5}-420 b^{7} d \,e^{7} x^{7}+2940 \,\mathrm {log}\left (e x +d \right ) b^{7} d^{7} e x -490 a^{6} b d \,e^{7} x -588 a^{5} b^{2} d^{2} e^{6} x -1764 a^{5} b^{2} d \,e^{7} x^{2}-735 a^{4} b^{3} d^{3} e^{5} x -2205 a^{4} b^{3} d^{2} e^{6} x^{2}-3675 a^{4} b^{3} d \,e^{7} x^{3}-980 a^{3} b^{4} d^{4} e^{4} x -2940 a^{3} b^{4} d^{3} e^{5} x^{2}-4900 a^{3} b^{4} d^{2} e^{6} x^{3}-4900 a^{3} b^{4} d \,e^{7} x^{4}-1470 a^{2} b^{5} d^{5} e^{3} x -4410 a^{2} b^{5} d^{4} e^{4} x^{2}-7350 a^{2} b^{5} d^{3} e^{5} x^{3}-7350 a^{2} b^{5} d^{2} e^{6} x^{4}-4410 a^{2} b^{5} d \,e^{7} x^{5}+669 b^{7} d^{8}+14700 \,\mathrm {log}\left (e x +d \right ) b^{7} d^{5} e^{3} x^{3}}{420 d \,e^{8} \left (e^{7} x^{7}+7 d \,e^{6} x^{6}+21 d^{2} e^{5} x^{5}+35 d^{3} e^{4} x^{4}+35 d^{4} e^{3} x^{3}+21 d^{5} e^{2} x^{2}+7 d^{6} e x +d^{7}\right )} \] Input:
int((b*x+a)^6*(B*x+A)/(e*x+d)^8,x)
Output:
(420*log(d + e*x)*b**7*d**8 + 2940*log(d + e*x)*b**7*d**7*e*x + 8820*log(d + e*x)*b**7*d**6*e**2*x**2 + 14700*log(d + e*x)*b**7*d**5*e**3*x**3 + 147 00*log(d + e*x)*b**7*d**4*e**4*x**4 + 8820*log(d + e*x)*b**7*d**3*e**5*x** 5 + 2940*log(d + e*x)*b**7*d**2*e**6*x**6 + 420*log(d + e*x)*b**7*d*e**7*x **7 - 60*a**7*d*e**7 - 70*a**6*b*d**2*e**6 - 490*a**6*b*d*e**7*x - 84*a**5 *b**2*d**3*e**5 - 588*a**5*b**2*d**2*e**6*x - 1764*a**5*b**2*d*e**7*x**2 - 105*a**4*b**3*d**4*e**4 - 735*a**4*b**3*d**3*e**5*x - 2205*a**4*b**3*d**2 *e**6*x**2 - 3675*a**4*b**3*d*e**7*x**3 - 140*a**3*b**4*d**5*e**3 - 980*a* *3*b**4*d**4*e**4*x - 2940*a**3*b**4*d**3*e**5*x**2 - 4900*a**3*b**4*d**2* e**6*x**3 - 4900*a**3*b**4*d*e**7*x**4 - 210*a**2*b**5*d**6*e**2 - 1470*a* *2*b**5*d**5*e**3*x - 4410*a**2*b**5*d**4*e**4*x**2 - 7350*a**2*b**5*d**3* e**5*x**3 - 7350*a**2*b**5*d**2*e**6*x**4 - 4410*a**2*b**5*d*e**7*x**5 + 4 20*a*b**6*e**8*x**7 + 669*b**7*d**8 + 4263*b**7*d**7*e*x + 11319*b**7*d**6 *e**2*x**2 + 15925*b**7*d**5*e**3*x**3 + 12250*b**7*d**4*e**4*x**4 + 4410* b**7*d**3*e**5*x**5 - 420*b**7*d*e**7*x**7)/(420*d*e**8*(d**7 + 7*d**6*e*x + 21*d**5*e**2*x**2 + 35*d**4*e**3*x**3 + 35*d**3*e**4*x**4 + 21*d**2*e** 5*x**5 + 7*d*e**6*x**6 + e**7*x**7))