Integrand size = 25, antiderivative size = 297 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx=\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^2}{5 e^6 (d+e x)^5}+\frac {\left (c d^2-b d e+a e^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{4 e^6 (d+e x)^4}+\frac {B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )-A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )}{3 e^6 (d+e x)^3}+\frac {2 A c e (2 c d-b e)-B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )}{2 e^6 (d+e x)^2}+\frac {c (5 B c d-2 b B e-A c e)}{e^6 (d+e x)}+\frac {B c^2 \log (d+e x)}{e^6} \]
1/5*(-A*e+B*d)*(a*e^2-b*d*e+c*d^2)^2/e^6/(e*x+d)^5+1/4*(a*e^2-b*d*e+c*d^2) *(2*A*e*(-b*e+2*c*d)-B*(5*c*d^2-e*(-a*e+3*b*d)))/e^6/(e*x+d)^4+1/3*(B*(10* c^2*d^3+b*e^2*(-2*a*e+3*b*d)-6*c*d*e*(-a*e+2*b*d))-A*e*(6*c^2*d^2+b^2*e^2- 2*c*e*(-a*e+3*b*d)))/e^6/(e*x+d)^3+1/2*(2*A*c*e*(-b*e+2*c*d)-B*(10*c^2*d^2 +b^2*e^2-2*c*e*(-a*e+4*b*d)))/e^6/(e*x+d)^2+c*(-A*c*e-2*B*b*e+5*B*c*d)/e^6 /(e*x+d)+B*c^2*ln(e*x+d)/e^6
Time = 0.15 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.30 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx=\frac {-2 A e \left (6 c^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )+e^2 \left (6 a^2 e^2+3 a b e (d+5 e x)+b^2 \left (d^2+5 d e x+10 e^2 x^2\right )\right )+c e \left (2 a e \left (d^2+5 d e x+10 e^2 x^2\right )+3 b \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )\right )+B \left (c^2 d \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )-e^2 \left (3 a^2 e^2 (d+5 e x)+4 a b e \left (d^2+5 d e x+10 e^2 x^2\right )+3 b^2 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )-6 c e \left (a e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+4 b \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )\right )+60 B c^2 (d+e x)^5 \log (d+e x)}{60 e^6 (d+e x)^5} \]
(-2*A*e*(6*c^2*(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) + e^2*(6*a^2*e^2 + 3*a*b*e*(d + 5*e*x) + b^2*(d^2 + 5*d*e*x + 10*e^2*x^ 2)) + c*e*(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))) + B*(c^2*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) - e^2*(3*a^2*e^2*(d + 5*e*x) + 4*a*b*e *(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*b^2*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10 *e^3*x^3)) - 6*c*e*(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))) + 60*B*c ^2*(d + e*x)^5*Log[d + e*x])/(60*e^6*(d + e*x)^5)
Time = 0.58 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1195, 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.
| \(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )-2 A c e (2 c d-b e)}{e^5 (d+e x)^3}+\frac {A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )-B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )}{e^5 (d+e x)^4}+\frac {\left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{e^5 (d+e x)^5}+\frac {(A e-B d) \left (a e^2-b d e+c d^2\right )^2}{e^5 (d+e x)^6}+\frac {c (A c e+2 b B e-5 B c d)}{e^5 (d+e x)^2}+\frac {B c^2}{e^5 (d+e x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^6 (d+e x)^2}+\frac {B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{3 e^6 (d+e x)^3}-\frac {\left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{4 e^6 (d+e x)^4}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{5 e^6 (d+e x)^5}+\frac {c (-A c e-2 b B e+5 B c d)}{e^6 (d+e x)}+\frac {B c^2 \log (d+e x)}{e^6}\) |
((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^2)/(5*e^6*(d + e*x)^5) - ((c*d^2 - b* d*e + a*e^2)*(5*B*c*d^2 - B*e*(3*b*d - a*e) - 2*A*e*(2*c*d - b*e)))/(4*e^6 *(d + e*x)^4) + (B*(10*c^2*d^3 + b*e^2*(3*b*d - 2*a*e) - 6*c*d*e*(2*b*d - a*e)) - A*e*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e)))/(3*e^6*(d + e*x)^ 3) + (2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(4*b*d - a*e )))/(2*e^6*(d + e*x)^2) + (c*(5*B*c*d - 2*b*B*e - A*c*e))/(e^6*(d + e*x)) + (B*c^2*Log[d + e*x])/e^6
3.24.29.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 0.36 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.48
| method | result | size |
| risch | \(\frac {-\frac {c \left (A c e +2 B b e -5 B c d \right ) x^{4}}{e^{2}}-\frac {\left (2 A b c \,e^{2}+4 A \,c^{2} d e +2 B \,e^{2} a c +B \,b^{2} e^{2}+8 B b c d e -30 B \,c^{2} d^{2}\right ) x^{3}}{2 e^{3}}-\frac {\left (4 A a c \,e^{3}+2 A \,b^{2} e^{3}+6 A b c d \,e^{2}+12 A \,c^{2} d^{2} e +4 B a b \,e^{3}+6 B a c d \,e^{2}+3 B \,b^{2} d \,e^{2}+24 B b c \,d^{2} e -110 B \,c^{2} d^{3}\right ) x^{2}}{6 e^{4}}-\frac {\left (6 A a b \,e^{4}+4 A a c d \,e^{3}+2 A \,b^{2} d \,e^{3}+6 A b c \,d^{2} e^{2}+12 A \,c^{2} d^{3} e +3 B \,e^{4} a^{2}+4 B a b d \,e^{3}+6 B a c \,d^{2} e^{2}+3 B \,b^{2} d^{2} e^{2}+24 B b c \,d^{3} e -125 B \,c^{2} d^{4}\right ) x}{12 e^{5}}-\frac {12 A \,a^{2} e^{5}+6 A a b d \,e^{4}+4 A a c \,d^{2} e^{3}+2 A \,b^{2} d^{2} e^{3}+6 A b c \,d^{3} e^{2}+12 A \,c^{2} d^{4} e +3 B \,a^{2} d \,e^{4}+4 B a b \,d^{2} e^{3}+6 B a c \,d^{3} e^{2}+3 B \,b^{2} d^{3} e^{2}+24 B b c \,d^{4} e -137 B \,c^{2} d^{5}}{60 e^{6}}}{\left (e x +d \right )^{5}}+\frac {B \,c^{2} \ln \left (e x +d \right )}{e^{6}}\) | \(440\) |
| norman | \(\frac {-\frac {12 A \,a^{2} e^{5}+6 A a b d \,e^{4}+4 A a c \,d^{2} e^{3}+2 A \,b^{2} d^{2} e^{3}+6 A b c \,d^{3} e^{2}+12 A \,c^{2} d^{4} e +3 B \,a^{2} d \,e^{4}+4 B a b \,d^{2} e^{3}+6 B a c \,d^{3} e^{2}+3 B \,b^{2} d^{3} e^{2}+24 B b c \,d^{4} e -137 B \,c^{2} d^{5}}{60 e^{6}}-\frac {\left (A \,c^{2} e +2 B e b c -5 B \,c^{2} d \right ) x^{4}}{e^{2}}-\frac {\left (2 A b c \,e^{2}+4 A \,c^{2} d e +2 B \,e^{2} a c +B \,b^{2} e^{2}+8 B b c d e -30 B \,c^{2} d^{2}\right ) x^{3}}{2 e^{3}}-\frac {\left (4 A a c \,e^{3}+2 A \,b^{2} e^{3}+6 A b c d \,e^{2}+12 A \,c^{2} d^{2} e +4 B a b \,e^{3}+6 B a c d \,e^{2}+3 B \,b^{2} d \,e^{2}+24 B b c \,d^{2} e -110 B \,c^{2} d^{3}\right ) x^{2}}{6 e^{4}}-\frac {\left (6 A a b \,e^{4}+4 A a c d \,e^{3}+2 A \,b^{2} d \,e^{3}+6 A b c \,d^{2} e^{2}+12 A \,c^{2} d^{3} e +3 B \,e^{4} a^{2}+4 B a b d \,e^{3}+6 B a c \,d^{2} e^{2}+3 B \,b^{2} d^{2} e^{2}+24 B b c \,d^{3} e -125 B \,c^{2} d^{4}\right ) x}{12 e^{5}}}{\left (e x +d \right )^{5}}+\frac {B \,c^{2} \ln \left (e x +d \right )}{e^{6}}\) | \(444\) |
| default | \(-\frac {A \,a^{2} e^{5}-2 A a b d \,e^{4}+2 A a c \,d^{2} e^{3}+A \,b^{2} d^{2} e^{3}-2 A b c \,d^{3} e^{2}+A \,c^{2} d^{4} e -B \,a^{2} d \,e^{4}+2 B a b \,d^{2} e^{3}-2 B a c \,d^{3} e^{2}-B \,b^{2} d^{3} e^{2}+2 B b c \,d^{4} e -B \,c^{2} d^{5}}{5 e^{6} \left (e x +d \right )^{5}}-\frac {2 A a c \,e^{3}+A \,b^{2} e^{3}-6 A b c d \,e^{2}+6 A \,c^{2} d^{2} e +2 B a b \,e^{3}-6 B a c d \,e^{2}-3 B \,b^{2} d \,e^{2}+12 B b c \,d^{2} e -10 B \,c^{2} d^{3}}{3 e^{6} \left (e x +d \right )^{3}}-\frac {c \left (A c e +2 B b e -5 B c d \right )}{e^{6} \left (e x +d \right )}+\frac {B \,c^{2} \ln \left (e x +d \right )}{e^{6}}-\frac {2 A a b \,e^{4}-4 A a c d \,e^{3}-2 A \,b^{2} d \,e^{3}+6 A b c \,d^{2} e^{2}-4 A \,c^{2} d^{3} e +B \,e^{4} a^{2}-4 B a b d \,e^{3}+6 B a c \,d^{2} e^{2}+3 B \,b^{2} d^{2} e^{2}-8 B b c \,d^{3} e +5 B \,c^{2} d^{4}}{4 e^{6} \left (e x +d \right )^{4}}-\frac {2 A b c \,e^{2}-4 A \,c^{2} d e +2 B \,e^{2} a c +B \,b^{2} e^{2}-8 B b c d e +10 B \,c^{2} d^{2}}{2 e^{6} \left (e x +d \right )^{2}}\) | \(451\) |
| parallelrisch | \(-\frac {12 A \,a^{2} e^{5}+20 B x a b d \,e^{4}+40 B \,x^{2} a b \,e^{5}+30 A x a b \,e^{5}-137 B \,c^{2} d^{5}+30 B x a c \,d^{2} e^{3}+20 A x a c d \,e^{4}+60 B \,x^{2} a c d \,e^{4}+15 B x \,a^{2} e^{5}-300 B \ln \left (e x +d \right ) x \,c^{2} d^{4} e -600 B \ln \left (e x +d \right ) x^{3} c^{2} d^{2} e^{3}-300 B \ln \left (e x +d \right ) x^{4} c^{2} d \,e^{4}-600 B \ln \left (e x +d \right ) x^{2} c^{2} d^{3} e^{2}-60 B \ln \left (e x +d \right ) c^{2} d^{5}+60 A \,x^{4} c^{2} e^{5}+30 B \,x^{3} b^{2} e^{5}+20 A \,x^{2} b^{2} e^{5}+120 B x b c \,d^{3} e^{2}+30 A x b c \,d^{2} e^{3}+240 B \,x^{2} b c \,d^{2} e^{3}+60 A \,x^{2} b c d \,e^{4}+240 B \,x^{3} b c d \,e^{4}+3 B \,a^{2} d \,e^{4}+2 A \,b^{2} d^{2} e^{3}+12 A \,c^{2} d^{4} e +3 B \,b^{2} d^{3} e^{2}+6 A b c \,d^{3} e^{2}+24 B b c \,d^{4} e -60 B \ln \left (e x +d \right ) x^{5} c^{2} e^{5}+60 B \,x^{3} a c \,e^{5}+40 A \,x^{2} a c \,e^{5}+4 A a c \,d^{2} e^{3}+6 B a c \,d^{3} e^{2}+4 B a b \,d^{2} e^{3}+60 A \,x^{3} b c \,e^{5}+120 A \,x^{3} c^{2} d \,e^{4}-900 B \,x^{3} c^{2} d^{2} e^{3}+120 A \,x^{2} c^{2} d^{2} e^{3}+30 B \,x^{2} b^{2} d \,e^{4}-1100 B \,x^{2} c^{2} d^{3} e^{2}+10 A x \,b^{2} d \,e^{4}+60 A x \,c^{2} d^{3} e^{2}+15 B x \,b^{2} d^{2} e^{3}-625 B x \,c^{2} d^{4} e +120 B \,x^{4} b c \,e^{5}-300 B \,x^{4} c^{2} d \,e^{4}+6 A a b d \,e^{4}}{60 e^{6} \left (e x +d \right )^{5}}\) | \(597\) |
(-c*(A*c*e+2*B*b*e-5*B*c*d)/e^2*x^4-1/2*(2*A*b*c*e^2+4*A*c^2*d*e+2*B*a*c*e ^2+B*b^2*e^2+8*B*b*c*d*e-30*B*c^2*d^2)/e^3*x^3-1/6*(4*A*a*c*e^3+2*A*b^2*e^ 3+6*A*b*c*d*e^2+12*A*c^2*d^2*e+4*B*a*b*e^3+6*B*a*c*d*e^2+3*B*b^2*d*e^2+24* B*b*c*d^2*e-110*B*c^2*d^3)/e^4*x^2-1/12*(6*A*a*b*e^4+4*A*a*c*d*e^3+2*A*b^2 *d*e^3+6*A*b*c*d^2*e^2+12*A*c^2*d^3*e+3*B*a^2*e^4+4*B*a*b*d*e^3+6*B*a*c*d^ 2*e^2+3*B*b^2*d^2*e^2+24*B*b*c*d^3*e-125*B*c^2*d^4)/e^5*x-1/60*(12*A*a^2*e ^5+6*A*a*b*d*e^4+4*A*a*c*d^2*e^3+2*A*b^2*d^2*e^3+6*A*b*c*d^3*e^2+12*A*c^2* d^4*e+3*B*a^2*d*e^4+4*B*a*b*d^2*e^3+6*B*a*c*d^3*e^2+3*B*b^2*d^3*e^2+24*B*b *c*d^4*e-137*B*c^2*d^5)/e^6)/(e*x+d)^5+B*c^2*ln(e*x+d)/e^6
Time = 0.64 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.70 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx=\frac {137 \, B c^{2} d^{5} - 12 \, A a^{2} e^{5} - 12 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e - 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} - 2 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} - 3 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{4} + 60 \, {\left (5 \, B c^{2} d e^{4} - {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 30 \, {\left (30 \, B c^{2} d^{2} e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} - {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} + 10 \, {\left (110 \, B c^{2} d^{3} e^{2} - 12 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} - 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{4} - 2 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 5 \, {\left (125 \, B c^{2} d^{4} e - 12 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} - 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{3} - 2 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} - 3 \, {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x + 60 \, {\left (B c^{2} e^{5} x^{5} + 5 \, B c^{2} d e^{4} x^{4} + 10 \, B c^{2} d^{2} e^{3} x^{3} + 10 \, B c^{2} d^{3} e^{2} x^{2} + 5 \, B c^{2} d^{4} e x + B c^{2} d^{5}\right )} \log \left (e x + d\right )}{60 \, {\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}\right )}} \]
1/60*(137*B*c^2*d^5 - 12*A*a^2*e^5 - 12*(2*B*b*c + A*c^2)*d^4*e - 3*(B*b^2 + 2*(B*a + A*b)*c)*d^3*e^2 - 2*(2*B*a*b + A*b^2 + 2*A*a*c)*d^2*e^3 - 3*(B *a^2 + 2*A*a*b)*d*e^4 + 60*(5*B*c^2*d*e^4 - (2*B*b*c + A*c^2)*e^5)*x^4 + 3 0*(30*B*c^2*d^2*e^3 - 4*(2*B*b*c + A*c^2)*d*e^4 - (B*b^2 + 2*(B*a + A*b)*c )*e^5)*x^3 + 10*(110*B*c^2*d^3*e^2 - 12*(2*B*b*c + A*c^2)*d^2*e^3 - 3*(B*b ^2 + 2*(B*a + A*b)*c)*d*e^4 - 2*(2*B*a*b + A*b^2 + 2*A*a*c)*e^5)*x^2 + 5*( 125*B*c^2*d^4*e - 12*(2*B*b*c + A*c^2)*d^3*e^2 - 3*(B*b^2 + 2*(B*a + A*b)* c)*d^2*e^3 - 2*(2*B*a*b + A*b^2 + 2*A*a*c)*d*e^4 - 3*(B*a^2 + 2*A*a*b)*e^5 )*x + 60*(B*c^2*e^5*x^5 + 5*B*c^2*d*e^4*x^4 + 10*B*c^2*d^2*e^3*x^3 + 10*B* c^2*d^3*e^2*x^2 + 5*B*c^2*d^4*e*x + B*c^2*d^5)*log(e*x + d))/(e^11*x^5 + 5 *d*e^10*x^4 + 10*d^2*e^9*x^3 + 10*d^3*e^8*x^2 + 5*d^4*e^7*x + d^5*e^6)
Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.47 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx=\frac {137 \, B c^{2} d^{5} - 12 \, A a^{2} e^{5} - 12 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e - 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} - 2 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} - 3 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{4} + 60 \, {\left (5 \, B c^{2} d e^{4} - {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 30 \, {\left (30 \, B c^{2} d^{2} e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} - {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} + 10 \, {\left (110 \, B c^{2} d^{3} e^{2} - 12 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} - 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{4} - 2 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 5 \, {\left (125 \, B c^{2} d^{4} e - 12 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} - 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{3} - 2 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} - 3 \, {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x}{60 \, {\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}\right )}} + \frac {B c^{2} \log \left (e x + d\right )}{e^{6}} \]
1/60*(137*B*c^2*d^5 - 12*A*a^2*e^5 - 12*(2*B*b*c + A*c^2)*d^4*e - 3*(B*b^2 + 2*(B*a + A*b)*c)*d^3*e^2 - 2*(2*B*a*b + A*b^2 + 2*A*a*c)*d^2*e^3 - 3*(B *a^2 + 2*A*a*b)*d*e^4 + 60*(5*B*c^2*d*e^4 - (2*B*b*c + A*c^2)*e^5)*x^4 + 3 0*(30*B*c^2*d^2*e^3 - 4*(2*B*b*c + A*c^2)*d*e^4 - (B*b^2 + 2*(B*a + A*b)*c )*e^5)*x^3 + 10*(110*B*c^2*d^3*e^2 - 12*(2*B*b*c + A*c^2)*d^2*e^3 - 3*(B*b ^2 + 2*(B*a + A*b)*c)*d*e^4 - 2*(2*B*a*b + A*b^2 + 2*A*a*c)*e^5)*x^2 + 5*( 125*B*c^2*d^4*e - 12*(2*B*b*c + A*c^2)*d^3*e^2 - 3*(B*b^2 + 2*(B*a + A*b)* c)*d^2*e^3 - 2*(2*B*a*b + A*b^2 + 2*A*a*c)*d*e^4 - 3*(B*a^2 + 2*A*a*b)*e^5 )*x)/(e^11*x^5 + 5*d*e^10*x^4 + 10*d^2*e^9*x^3 + 10*d^3*e^8*x^2 + 5*d^4*e^ 7*x + d^5*e^6) + B*c^2*log(e*x + d)/e^6
Time = 0.27 (sec) , antiderivative size = 456, normalized size of antiderivative = 1.54 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx=\frac {B c^{2} \log \left ({\left | e x + d \right |}\right )}{e^{6}} + \frac {60 \, {\left (5 \, B c^{2} d e^{3} - 2 \, B b c e^{4} - A c^{2} e^{4}\right )} x^{4} + 30 \, {\left (30 \, B c^{2} d^{2} e^{2} - 8 \, B b c d e^{3} - 4 \, A c^{2} d e^{3} - B b^{2} e^{4} - 2 \, B a c e^{4} - 2 \, A b c e^{4}\right )} x^{3} + 10 \, {\left (110 \, B c^{2} d^{3} e - 24 \, B b c d^{2} e^{2} - 12 \, A c^{2} d^{2} e^{2} - 3 \, B b^{2} d e^{3} - 6 \, B a c d e^{3} - 6 \, A b c d e^{3} - 4 \, B a b e^{4} - 2 \, A b^{2} e^{4} - 4 \, A a c e^{4}\right )} x^{2} + 5 \, {\left (125 \, B c^{2} d^{4} - 24 \, B b c d^{3} e - 12 \, A c^{2} d^{3} e - 3 \, B b^{2} d^{2} e^{2} - 6 \, B a c d^{2} e^{2} - 6 \, A b c d^{2} e^{2} - 4 \, B a b d e^{3} - 2 \, A b^{2} d e^{3} - 4 \, A a c d e^{3} - 3 \, B a^{2} e^{4} - 6 \, A a b e^{4}\right )} x + \frac {137 \, B c^{2} d^{5} - 24 \, B b c d^{4} e - 12 \, A c^{2} d^{4} e - 3 \, B b^{2} d^{3} e^{2} - 6 \, B a c d^{3} e^{2} - 6 \, A b c d^{3} e^{2} - 4 \, B a b d^{2} e^{3} - 2 \, A b^{2} d^{2} e^{3} - 4 \, A a c d^{2} e^{3} - 3 \, B a^{2} d e^{4} - 6 \, A a b d e^{4} - 12 \, A a^{2} e^{5}}{e}}{60 \, {\left (e x + d\right )}^{5} e^{5}} \]
B*c^2*log(abs(e*x + d))/e^6 + 1/60*(60*(5*B*c^2*d*e^3 - 2*B*b*c*e^4 - A*c^ 2*e^4)*x^4 + 30*(30*B*c^2*d^2*e^2 - 8*B*b*c*d*e^3 - 4*A*c^2*d*e^3 - B*b^2* e^4 - 2*B*a*c*e^4 - 2*A*b*c*e^4)*x^3 + 10*(110*B*c^2*d^3*e - 24*B*b*c*d^2* e^2 - 12*A*c^2*d^2*e^2 - 3*B*b^2*d*e^3 - 6*B*a*c*d*e^3 - 6*A*b*c*d*e^3 - 4 *B*a*b*e^4 - 2*A*b^2*e^4 - 4*A*a*c*e^4)*x^2 + 5*(125*B*c^2*d^4 - 24*B*b*c* d^3*e - 12*A*c^2*d^3*e - 3*B*b^2*d^2*e^2 - 6*B*a*c*d^2*e^2 - 6*A*b*c*d^2*e ^2 - 4*B*a*b*d*e^3 - 2*A*b^2*d*e^3 - 4*A*a*c*d*e^3 - 3*B*a^2*e^4 - 6*A*a*b *e^4)*x + (137*B*c^2*d^5 - 24*B*b*c*d^4*e - 12*A*c^2*d^4*e - 3*B*b^2*d^3*e ^2 - 6*B*a*c*d^3*e^2 - 6*A*b*c*d^3*e^2 - 4*B*a*b*d^2*e^3 - 2*A*b^2*d^2*e^3 - 4*A*a*c*d^2*e^3 - 3*B*a^2*d*e^4 - 6*A*a*b*d*e^4 - 12*A*a^2*e^5)/e)/((e* x + d)^5*e^5)
Time = 11.38 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.63 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx=\frac {B\,c^2\,\ln \left (d+e\,x\right )}{e^6}-\frac {\frac {3\,B\,a^2\,d\,e^4+12\,A\,a^2\,e^5+4\,B\,a\,b\,d^2\,e^3+6\,A\,a\,b\,d\,e^4+6\,B\,a\,c\,d^3\,e^2+4\,A\,a\,c\,d^2\,e^3+3\,B\,b^2\,d^3\,e^2+2\,A\,b^2\,d^2\,e^3+24\,B\,b\,c\,d^4\,e+6\,A\,b\,c\,d^3\,e^2-137\,B\,c^2\,d^5+12\,A\,c^2\,d^4\,e}{60\,e^6}+\frac {x^3\,\left (B\,b^2\,e^2+8\,B\,b\,c\,d\,e+2\,A\,b\,c\,e^2-30\,B\,c^2\,d^2+4\,A\,c^2\,d\,e+2\,B\,a\,c\,e^2\right )}{2\,e^3}+\frac {x^2\,\left (3\,B\,b^2\,d\,e^2+2\,A\,b^2\,e^3+24\,B\,b\,c\,d^2\,e+6\,A\,b\,c\,d\,e^2+4\,B\,a\,b\,e^3-110\,B\,c^2\,d^3+12\,A\,c^2\,d^2\,e+6\,B\,a\,c\,d\,e^2+4\,A\,a\,c\,e^3\right )}{6\,e^4}+\frac {x\,\left (3\,B\,a^2\,e^4+4\,B\,a\,b\,d\,e^3+6\,A\,a\,b\,e^4+6\,B\,a\,c\,d^2\,e^2+4\,A\,a\,c\,d\,e^3+3\,B\,b^2\,d^2\,e^2+2\,A\,b^2\,d\,e^3+24\,B\,b\,c\,d^3\,e+6\,A\,b\,c\,d^2\,e^2-125\,B\,c^2\,d^4+12\,A\,c^2\,d^3\,e\right )}{12\,e^5}+\frac {c\,x^4\,\left (A\,c\,e+2\,B\,b\,e-5\,B\,c\,d\right )}{e^2}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \]
(B*c^2*log(d + e*x))/e^6 - ((12*A*a^2*e^5 - 137*B*c^2*d^5 + 3*B*a^2*d*e^4 + 12*A*c^2*d^4*e + 2*A*b^2*d^2*e^3 + 3*B*b^2*d^3*e^2 + 6*A*a*b*d*e^4 + 24* B*b*c*d^4*e + 4*A*a*c*d^2*e^3 + 4*B*a*b*d^2*e^3 + 6*A*b*c*d^3*e^2 + 6*B*a* c*d^3*e^2)/(60*e^6) + (x^3*(B*b^2*e^2 - 30*B*c^2*d^2 + 2*A*b*c*e^2 + 2*B*a *c*e^2 + 4*A*c^2*d*e + 8*B*b*c*d*e))/(2*e^3) + (x^2*(2*A*b^2*e^3 - 110*B*c ^2*d^3 + 4*A*a*c*e^3 + 4*B*a*b*e^3 + 12*A*c^2*d^2*e + 3*B*b^2*d*e^2 + 6*A* b*c*d*e^2 + 6*B*a*c*d*e^2 + 24*B*b*c*d^2*e))/(6*e^4) + (x*(3*B*a^2*e^4 - 1 25*B*c^2*d^4 + 6*A*a*b*e^4 + 2*A*b^2*d*e^3 + 12*A*c^2*d^3*e + 3*B*b^2*d^2* e^2 + 4*A*a*c*d*e^3 + 4*B*a*b*d*e^3 + 24*B*b*c*d^3*e + 6*A*b*c*d^2*e^2 + 6 *B*a*c*d^2*e^2))/(12*e^5) + (c*x^4*(A*c*e + 2*B*b*e - 5*B*c*d))/e^2)/(d^5 + e^5*x^5 + 5*d*e^4*x^4 + 10*d^3*e^2*x^2 + 10*d^2*e^3*x^3 + 5*d^4*e*x)