Integrand size = 25, antiderivative size = 534 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx=-\frac {c^2 (6 B c d-3 b B e-A c e) x}{e^7}+\frac {B c^3 x^2}{2 e^6}+\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3}{5 e^8 (d+e x)^5}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right )}{4 e^8 (d+e x)^4}+\frac {\left (c d^2-b d e+a e^2\right ) \left (B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )-A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right )}{e^8 (d+e x)^3}+\frac {A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )}{2 e^8 (d+e x)^2}+\frac {B \left (35 c^3 d^3-b^3 e^3+3 b c e^2 (5 b d-2 a e)-15 c^2 d e (3 b d-a e)\right )-3 A c e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^8 (d+e x)}-\frac {3 c \left (A c e (2 c d-b e)-B \left (7 c^2 d^2+b^2 e^2-c e (6 b d-a e)\right )\right ) \log (d+e x)}{e^8} \]
-c^2*(-A*c*e-3*B*b*e+6*B*c*d)*x/e^7+1/2*B*c^3*x^2/e^6+1/5*(-A*e+B*d)*(a*e^ 2-b*d*e+c*d^2)^3/e^8/(e*x+d)^5+1/4*(a*e^2-b*d*e+c*d^2)^2*(3*A*e*(-b*e+2*c* d)-B*(7*c*d^2-e*(-a*e+4*b*d)))/e^8/(e*x+d)^4+(a*e^2-b*d*e+c*d^2)*(B*(7*c^2 *d^3-c*d*e*(-3*a*e+8*b*d)+b*e^2*(-a*e+2*b*d))-A*e*(5*c^2*d^2+b^2*e^2-c*e*( -a*e+5*b*d)))/e^8/(e*x+d)^3+1/2*(A*e*(-b*e+2*c*d)*(10*c^2*d^2+b^2*e^2-2*c* e*(-3*a*e+5*b*d))-B*(35*c^3*d^4-b^2*e^3*(-3*a*e+4*b*d)-30*c^2*d^2*e*(-a*e+ 2*b*d)+3*c*e^2*(a^2*e^2-8*a*b*d*e+10*b^2*d^2)))/e^8/(e*x+d)^2+(B*(35*c^3*d ^3-b^3*e^3+3*b*c*e^2*(-2*a*e+5*b*d)-15*c^2*d*e*(-a*e+3*b*d))-3*A*c*e*(5*c^ 2*d^2+b^2*e^2-c*e*(-a*e+5*b*d)))/e^8/(e*x+d)-3*c*(A*c*e*(-b*e+2*c*d)-B*(7* c^2*d^2+b^2*e^2-c*e*(-a*e+6*b*d)))*ln(e*x+d)/e^8
Time = 0.39 (sec) , antiderivative size = 885, normalized size of antiderivative = 1.66 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx=\frac {A e \left (-2 c^3 \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )-e^3 \left (4 a^3 e^3+3 a^2 b e^2 (d+5 e x)+2 a b^2 e \left (d^2+5 d e x+10 e^2 x^2\right )+b^3 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )-2 c e^2 \left (a^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+3 a b e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+6 b^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )+c^2 e \left (-12 a e \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )+b d \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )\right )\right )+B \left (c^3 \left (459 d^7+1875 d^6 e x+2700 d^5 e^2 x^2+1300 d^4 e^3 x^3-400 d^3 e^4 x^4-500 d^2 e^5 x^5-70 d e^6 x^6+10 e^7 x^7\right )-e^3 \left (a^3 e^3 (d+5 e x)+2 a^2 b e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+3 a b^2 e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+4 b^3 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )+c e^2 \left (-3 a^2 e^2 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )-24 a b e \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )+b^2 d \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )\right )+c^2 e \left (a d e \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )-6 b \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )\right )\right )+60 c \left (A c e (-2 c d+b e)+B \left (7 c^2 d^2+b^2 e^2+c e (-6 b d+a e)\right )\right ) (d+e x)^5 \log (d+e x)}{20 e^8 (d+e x)^5} \]
(A*e*(-2*c^3*(87*d^6 + 375*d^5*e*x + 600*d^4*e^2*x^2 + 400*d^3*e^3*x^3 + 5 0*d^2*e^4*x^4 - 50*d*e^5*x^5 - 10*e^6*x^6) - e^3*(4*a^3*e^3 + 3*a^2*b*e^2* (d + 5*e*x) + 2*a*b^2*e*(d^2 + 5*d*e*x + 10*e^2*x^2) + b^3*(d^3 + 5*d^2*e* x + 10*d*e^2*x^2 + 10*e^3*x^3)) - 2*c*e^2*(a^2*e^2*(d^2 + 5*d*e*x + 10*e^2 *x^2) + 3*a*b*e*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + 6*b^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)) + c^2*e*(-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))) + B *(c^3*(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) - e^3*(a^3*e^3* (d + 5*e*x) + 2*a^2*b*e^2*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*a*b^2*e*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + 4*b^3*(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)) + c*e^2*(-3*a^2*e^2*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) - 24*a*b*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^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)) + c^2*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 - 1 0*e^6*x^6))) + 60*c*(A*c*e*(-2*c*d + b*e) + B*(7*c^2*d^2 + b^2*e^2 + c*e*( -6*b*d + a*e)))*(d + e*x)^5*Log[d + e*x])/(20*e^8*(d + e*x)^5)
Time = 1.29 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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.
| \(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )-A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7 (d+e x)^3}+\frac {3 c \left (B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )-A c e (2 c d-b e)\right )}{e^7 (d+e x)}+\frac {3 \left (a e^2-b d e+c d^2\right ) \left (A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )-B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )\right )}{e^7 (d+e x)^4}+\frac {3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )-B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )}{e^7 (d+e x)^2}+\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{e^7 (d+e x)^5}+\frac {(A e-B d) \left (a e^2-b d e+c d^2\right )^3}{e^7 (d+e x)^6}+\frac {c^2 (A c e+3 b B e-6 B c d)}{e^7}+\frac {B c^3 x}{e^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )}{2 e^8 (d+e x)^2}-\frac {3 c \log (d+e x) \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{e^8}+\frac {\left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{e^8 (d+e x)^3}+\frac {B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^8 (d+e x)}-\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{4 e^8 (d+e x)^4}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{5 e^8 (d+e x)^5}-\frac {c^2 x (-A c e-3 b B e+6 B c d)}{e^7}+\frac {B c^3 x^2}{2 e^6}\) |
-((c^2*(6*B*c*d - 3*b*B*e - A*c*e)*x)/e^7) + (B*c^3*x^2)/(2*e^6) + ((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^3)/(5*e^8*(d + e*x)^5) - ((c*d^2 - b*d*e + a *e^2)^2*(7*B*c*d^2 - B*e*(4*b*d - a*e) - 3*A*e*(2*c*d - b*e)))/(4*e^8*(d + e*x)^4) + ((c*d^2 - b*d*e + a*e^2)*(B*(7*c^2*d^3 - c*d*e*(8*b*d - 3*a*e) + b*e^2*(2*b*d - a*e)) - A*e*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))))/( e^8*(d + e*x)^3) + (A*e*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e)) - B*(35*c^3*d^4 - b^2*e^3*(4*b*d - 3*a*e) - 30*c^2*d^2*e*(2*b*d - a*e) + 3*c*e^2*(10*b^2*d^2 - 8*a*b*d*e + a^2*e^2)))/(2*e^8*(d + e*x)^2) + (B*(35*c^3*d^3 - b^3*e^3 + 3*b*c*e^2*(5*b*d - 2*a*e) - 15*c^2*d*e*(3*b* d - a*e)) - 3*A*c*e*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))/(e^8*(d + e *x)) - (3*c*(A*c*e*(2*c*d - b*e) - B*(7*c^2*d^2 + b^2*e^2 - c*e*(6*b*d - a *e)))*Log[d + e*x])/e^8
3.24.44.3.1 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 = 0.37 (sec) , antiderivative size = 1042, normalized size of antiderivative = 1.95
| method | result | size |
| norman | \(\text {Expression too large to display}\) | \(1042\) |
| default | \(\text {Expression too large to display}\) | \(1053\) |
| risch | \(\text {Expression too large to display}\) | \(1089\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1854\) |
(-1/20*(4*A*a^3*e^7+3*A*a^2*b*d*e^6+2*A*a^2*c*d^2*e^5+2*A*a*b^2*d^2*e^5+6* A*a*b*c*d^3*e^4+12*A*a*c^2*d^4*e^3+A*b^3*d^3*e^4+12*A*b^2*c*d^4*e^3-137*A* b*c^2*d^5*e^2+274*A*c^3*d^6*e+B*a^3*d*e^6+2*B*a^2*b*d^2*e^5+3*B*a^2*c*d^3* e^4+3*B*a*b^2*d^3*e^4+24*B*a*b*c*d^4*e^3-137*B*a*c^2*d^5*e^2+4*B*b^3*d^4*e ^3-137*B*b^2*c*d^5*e^2+822*B*b*c^2*d^6*e-959*B*c^3*d^7)/e^8-(3*A*a*c^2*e^3 +3*A*b^2*c*e^3-15*A*b*c^2*d*e^2+30*A*c^3*d^2*e+6*B*a*b*c*e^3-15*B*a*c^2*d* e^2+B*b^3*e^3-15*B*b^2*c*d*e^2+90*B*b*c^2*d^2*e-105*B*c^3*d^3)/e^4*x^4-1/2 *(6*A*a*b*c*e^4+12*A*a*c^2*d*e^3+A*b^3*e^4+12*A*b^2*c*d*e^3-90*A*b*c^2*d^2 *e^2+180*A*c^3*d^3*e+3*B*a^2*c*e^4+3*B*a*b^2*e^4+24*B*a*b*c*d*e^3-90*B*a*c ^2*d^2*e^2+4*B*b^3*d*e^3-90*B*b^2*c*d^2*e^2+540*B*b*c^2*d^3*e-630*B*c^3*d^ 4)/e^5*x^3-1/2*(2*A*a^2*c*e^5+2*A*a*b^2*e^5+6*A*a*b*c*d*e^4+12*A*a*c^2*d^2 *e^3+A*b^3*d*e^4+12*A*b^2*c*d^2*e^3-110*A*b*c^2*d^3*e^2+220*A*c^3*d^4*e+2* B*a^2*b*e^5+3*B*a^2*c*d*e^4+3*B*a*b^2*d*e^4+24*B*a*b*c*d^2*e^3-110*B*a*c^2 *d^3*e^2+4*B*b^3*d^2*e^3-110*B*b^2*c*d^3*e^2+660*B*b*c^2*d^4*e-770*B*c^3*d ^5)/e^6*x^2-1/4*(3*A*a^2*b*e^6+2*A*a^2*c*d*e^5+2*A*a*b^2*d*e^5+6*A*a*b*c*d ^2*e^4+12*A*a*c^2*d^3*e^3+A*b^3*d^2*e^4+12*A*b^2*c*d^3*e^3-125*A*b*c^2*d^4 *e^2+250*A*c^3*d^5*e+B*a^3*e^6+2*B*a^2*b*d*e^5+3*B*a^2*c*d^2*e^4+3*B*a*b^2 *d^2*e^4+24*B*a*b*c*d^3*e^3-125*B*a*c^2*d^4*e^2+4*B*b^3*d^3*e^3-125*B*b^2* c*d^4*e^2+750*B*b*c^2*d^5*e-875*B*c^3*d^6)/e^7*x+1/2*B*c^3*x^7/e+1/2*c^2*( 2*A*c*e+6*B*b*e-7*B*c*d)/e^2*x^6)/(e*x+d)^5+3*c/e^8*(A*b*c*e^2-2*A*c^2*...
Leaf count of result is larger than twice the leaf count of optimal. 1247 vs. \(2 (529) = 1058\).
Time = 0.32 (sec) , antiderivative size = 1247, normalized size of antiderivative = 2.34 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx=\text {Too large to display} \]
1/20*(10*B*c^3*e^7*x^7 + 459*B*c^3*d^7 - 4*A*a^3*e^7 - 174*(3*B*b*c^2 + A* c^3)*d^6*e + 137*(B*b^2*c + (B*a + A*b)*c^2)*d^5*e^2 - 4*(B*b^3 + 3*A*a*c^ 2 + 3*(2*B*a*b + A*b^2)*c)*d^4*e^3 - (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a *b)*c)*d^3*e^4 - 2*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^2*e^5 - (B*a^3 + 3*A*a^ 2*b)*d*e^6 - 10*(7*B*c^3*d*e^6 - 2*(3*B*b*c^2 + A*c^3)*e^7)*x^6 - 100*(5*B *c^3*d^2*e^5 - (3*B*b*c^2 + A*c^3)*d*e^6)*x^5 - 20*(20*B*c^3*d^3*e^4 + 5*( 3*B*b*c^2 + A*c^3)*d^2*e^5 - 15*(B*b^2*c + (B*a + A*b)*c^2)*d*e^6 + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e^7)*x^4 + 10*(130*B*c^3*d^4*e^3 - 8 0*(3*B*b*c^2 + A*c^3)*d^3*e^4 + 90*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^5 - 4 *(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d*e^6 - (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*e^7)*x^3 + 10*(270*B*c^3*d^5*e^2 - 120*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 110*(B*b^2*c + (B*a + A*b)*c^2)*d^3*e^4 - 4*(B*b^3 + 3*A *a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e^5 - (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*e^6 - 2*(B*a^2*b + A*a*b^2 + A*a^2*c)*e^7)*x^2 + 5*(375*B*c^ 3*d^6*e - 150*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 125*(B*b^2*c + (B*a + A*b)*c^2 )*d^4*e^3 - 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^3*e^4 - (3*B*a *b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^2*e^5 - 2*(B*a^2*b + A*a*b^2 + A*a ^2*c)*d*e^6 - (B*a^3 + 3*A*a^2*b)*e^7)*x + 60*(7*B*c^3*d^7 - 2*(3*B*b*c^2 + A*c^3)*d^6*e + (B*b^2*c + (B*a + A*b)*c^2)*d^5*e^2 + (7*B*c^3*d^2*e^5 - 2*(3*B*b*c^2 + A*c^3)*d*e^6 + (B*b^2*c + (B*a + A*b)*c^2)*e^7)*x^5 + 5*...
Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx=\text {Timed out} \]
Time = 0.22 (sec) , antiderivative size = 898, normalized size of antiderivative = 1.68 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx=\frac {459 \, B c^{3} d^{7} - 4 \, A a^{3} e^{7} - 174 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 137 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{5} e^{2} - 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{4} e^{3} - {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{3} e^{4} - 2 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{2} e^{5} - {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{6} + 20 \, {\left (35 \, B c^{3} d^{3} e^{4} - 15 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{5} + 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d e^{6} - {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} e^{7}\right )} x^{4} + 10 \, {\left (245 \, B c^{3} d^{4} e^{3} - 100 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{4} + 90 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} e^{5} - 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d e^{6} - {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} e^{7}\right )} x^{3} + 10 \, {\left (329 \, B c^{3} d^{5} e^{2} - 130 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} + 110 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} e^{4} - 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{2} e^{5} - {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d e^{6} - 2 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e^{7}\right )} x^{2} + 5 \, {\left (399 \, B c^{3} d^{6} e - 154 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 125 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} e^{3} - 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{3} e^{4} - {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{2} e^{5} - 2 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d e^{6} - {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{7}\right )} x}{20 \, {\left (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}\right )}} + \frac {B c^{3} e x^{2} - 2 \, {\left (6 \, B c^{3} d - {\left (3 \, B b c^{2} + A c^{3}\right )} e\right )} x}{2 \, e^{7}} + \frac {3 \, {\left (7 \, B c^{3} d^{2} - 2 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d e + {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} e^{2}\right )} \log \left (e x + d\right )}{e^{8}} \]
1/20*(459*B*c^3*d^7 - 4*A*a^3*e^7 - 174*(3*B*b*c^2 + A*c^3)*d^6*e + 137*(B *b^2*c + (B*a + A*b)*c^2)*d^5*e^2 - 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A* b^2)*c)*d^4*e^3 - (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^3*e^4 - 2* (B*a^2*b + A*a*b^2 + A*a^2*c)*d^2*e^5 - (B*a^3 + 3*A*a^2*b)*d*e^6 + 20*(35 *B*c^3*d^3*e^4 - 15*(3*B*b*c^2 + A*c^3)*d^2*e^5 + 15*(B*b^2*c + (B*a + A*b )*c^2)*d*e^6 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e^7)*x^4 + 10*( 245*B*c^3*d^4*e^3 - 100*(3*B*b*c^2 + A*c^3)*d^3*e^4 + 90*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^5 - 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d*e^6 - (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*e^7)*x^3 + 10*(329*B*c^3*d^5* e^2 - 130*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 110*(B*b^2*c + (B*a + A*b)*c^2)*d^ 3*e^4 - 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e^5 - (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*e^6 - 2*(B*a^2*b + A*a*b^2 + A*a^2*c)* e^7)*x^2 + 5*(399*B*c^3*d^6*e - 154*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 125*(B*b ^2*c + (B*a + A*b)*c^2)*d^4*e^3 - 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^ 2)*c)*d^3*e^4 - (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^2*e^5 - 2*(B *a^2*b + A*a*b^2 + A*a^2*c)*d*e^6 - (B*a^3 + 3*A*a^2*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*c^3*e*x^2 - 2*(6*B*c^3*d - (3*B*b*c^2 + A*c^3)*e)*x)/e^7 + 3*(7 *B*c^3*d^2 - 2*(3*B*b*c^2 + A*c^3)*d*e + (B*b^2*c + (B*a + A*b)*c^2)*e^2)* log(e*x + d)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 1077 vs. \(2 (529) = 1058\).
Time = 0.29 (sec) , antiderivative size = 1077, normalized size of antiderivative = 2.02 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx=\frac {3 \, {\left (7 \, B c^{3} d^{2} - 6 \, B b c^{2} d e - 2 \, A c^{3} d e + B b^{2} c e^{2} + B a c^{2} e^{2} + A b c^{2} e^{2}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{8}} + \frac {B c^{3} e^{6} x^{2} - 12 \, B c^{3} d e^{5} x + 6 \, B b c^{2} e^{6} x + 2 \, A c^{3} e^{6} x}{2 \, e^{12}} + \frac {459 \, B c^{3} d^{7} - 522 \, B b c^{2} d^{6} e - 174 \, A c^{3} d^{6} e + 137 \, B b^{2} c d^{5} e^{2} + 137 \, B a c^{2} d^{5} e^{2} + 137 \, A b c^{2} d^{5} e^{2} - 4 \, B b^{3} d^{4} e^{3} - 24 \, B a b c d^{4} e^{3} - 12 \, A b^{2} c d^{4} e^{3} - 12 \, A a c^{2} d^{4} e^{3} - 3 \, B a b^{2} d^{3} e^{4} - A b^{3} d^{3} e^{4} - 3 \, B a^{2} c d^{3} e^{4} - 6 \, A a b c d^{3} e^{4} - 2 \, B a^{2} b d^{2} e^{5} - 2 \, A a b^{2} d^{2} e^{5} - 2 \, A a^{2} c d^{2} e^{5} - B a^{3} d e^{6} - 3 \, A a^{2} b d e^{6} - 4 \, A a^{3} e^{7} + 20 \, {\left (35 \, B c^{3} d^{3} e^{4} - 45 \, B b c^{2} d^{2} e^{5} - 15 \, A c^{3} d^{2} e^{5} + 15 \, B b^{2} c d e^{6} + 15 \, B a c^{2} d e^{6} + 15 \, A b c^{2} d e^{6} - B b^{3} e^{7} - 6 \, B a b c e^{7} - 3 \, A b^{2} c e^{7} - 3 \, A a c^{2} e^{7}\right )} x^{4} + 10 \, {\left (245 \, B c^{3} d^{4} e^{3} - 300 \, B b c^{2} d^{3} e^{4} - 100 \, A c^{3} d^{3} e^{4} + 90 \, B b^{2} c d^{2} e^{5} + 90 \, B a c^{2} d^{2} e^{5} + 90 \, A b c^{2} d^{2} e^{5} - 4 \, B b^{3} d e^{6} - 24 \, B a b c d e^{6} - 12 \, A b^{2} c d e^{6} - 12 \, A a c^{2} d e^{6} - 3 \, B a b^{2} e^{7} - A b^{3} e^{7} - 3 \, B a^{2} c e^{7} - 6 \, A a b c e^{7}\right )} x^{3} + 10 \, {\left (329 \, B c^{3} d^{5} e^{2} - 390 \, B b c^{2} d^{4} e^{3} - 130 \, A c^{3} d^{4} e^{3} + 110 \, B b^{2} c d^{3} e^{4} + 110 \, B a c^{2} d^{3} e^{4} + 110 \, A b c^{2} d^{3} e^{4} - 4 \, B b^{3} d^{2} e^{5} - 24 \, B a b c d^{2} e^{5} - 12 \, A b^{2} c d^{2} e^{5} - 12 \, A a c^{2} d^{2} e^{5} - 3 \, B a b^{2} d e^{6} - A b^{3} d e^{6} - 3 \, B a^{2} c d e^{6} - 6 \, A a b c d e^{6} - 2 \, B a^{2} b e^{7} - 2 \, A a b^{2} e^{7} - 2 \, A a^{2} c e^{7}\right )} x^{2} + 5 \, {\left (399 \, B c^{3} d^{6} e - 462 \, B b c^{2} d^{5} e^{2} - 154 \, A c^{3} d^{5} e^{2} + 125 \, B b^{2} c d^{4} e^{3} + 125 \, B a c^{2} d^{4} e^{3} + 125 \, A b c^{2} d^{4} e^{3} - 4 \, B b^{3} d^{3} e^{4} - 24 \, B a b c d^{3} e^{4} - 12 \, A b^{2} c d^{3} e^{4} - 12 \, A a c^{2} d^{3} e^{4} - 3 \, B a b^{2} d^{2} e^{5} - A b^{3} d^{2} e^{5} - 3 \, B a^{2} c d^{2} e^{5} - 6 \, A a b c d^{2} e^{5} - 2 \, B a^{2} b d e^{6} - 2 \, A a b^{2} d e^{6} - 2 \, A a^{2} c d e^{6} - B a^{3} e^{7} - 3 \, A a^{2} b e^{7}\right )} x}{20 \, {\left (e x + d\right )}^{5} e^{8}} \]
3*(7*B*c^3*d^2 - 6*B*b*c^2*d*e - 2*A*c^3*d*e + B*b^2*c*e^2 + B*a*c^2*e^2 + A*b*c^2*e^2)*log(abs(e*x + d))/e^8 + 1/2*(B*c^3*e^6*x^2 - 12*B*c^3*d*e^5* x + 6*B*b*c^2*e^6*x + 2*A*c^3*e^6*x)/e^12 + 1/20*(459*B*c^3*d^7 - 522*B*b* c^2*d^6*e - 174*A*c^3*d^6*e + 137*B*b^2*c*d^5*e^2 + 137*B*a*c^2*d^5*e^2 + 137*A*b*c^2*d^5*e^2 - 4*B*b^3*d^4*e^3 - 24*B*a*b*c*d^4*e^3 - 12*A*b^2*c*d^ 4*e^3 - 12*A*a*c^2*d^4*e^3 - 3*B*a*b^2*d^3*e^4 - A*b^3*d^3*e^4 - 3*B*a^2*c *d^3*e^4 - 6*A*a*b*c*d^3*e^4 - 2*B*a^2*b*d^2*e^5 - 2*A*a*b^2*d^2*e^5 - 2*A *a^2*c*d^2*e^5 - B*a^3*d*e^6 - 3*A*a^2*b*d*e^6 - 4*A*a^3*e^7 + 20*(35*B*c^ 3*d^3*e^4 - 45*B*b*c^2*d^2*e^5 - 15*A*c^3*d^2*e^5 + 15*B*b^2*c*d*e^6 + 15* B*a*c^2*d*e^6 + 15*A*b*c^2*d*e^6 - B*b^3*e^7 - 6*B*a*b*c*e^7 - 3*A*b^2*c*e ^7 - 3*A*a*c^2*e^7)*x^4 + 10*(245*B*c^3*d^4*e^3 - 300*B*b*c^2*d^3*e^4 - 10 0*A*c^3*d^3*e^4 + 90*B*b^2*c*d^2*e^5 + 90*B*a*c^2*d^2*e^5 + 90*A*b*c^2*d^2 *e^5 - 4*B*b^3*d*e^6 - 24*B*a*b*c*d*e^6 - 12*A*b^2*c*d*e^6 - 12*A*a*c^2*d* e^6 - 3*B*a*b^2*e^7 - A*b^3*e^7 - 3*B*a^2*c*e^7 - 6*A*a*b*c*e^7)*x^3 + 10* (329*B*c^3*d^5*e^2 - 390*B*b*c^2*d^4*e^3 - 130*A*c^3*d^4*e^3 + 110*B*b^2*c *d^3*e^4 + 110*B*a*c^2*d^3*e^4 + 110*A*b*c^2*d^3*e^4 - 4*B*b^3*d^2*e^5 - 2 4*B*a*b*c*d^2*e^5 - 12*A*b^2*c*d^2*e^5 - 12*A*a*c^2*d^2*e^5 - 3*B*a*b^2*d* e^6 - A*b^3*d*e^6 - 3*B*a^2*c*d*e^6 - 6*A*a*b*c*d*e^6 - 2*B*a^2*b*e^7 - 2* A*a*b^2*e^7 - 2*A*a^2*c*e^7)*x^2 + 5*(399*B*c^3*d^6*e - 462*B*b*c^2*d^5*e^ 2 - 154*A*c^3*d^5*e^2 + 125*B*b^2*c*d^4*e^3 + 125*B*a*c^2*d^4*e^3 + 125...
Time = 11.31 (sec) , antiderivative size = 1106, normalized size of antiderivative = 2.07 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx=x\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^6}-\frac {6\,B\,c^3\,d}{e^7}\right )-\frac {\frac {B\,a^3\,d\,e^6+4\,A\,a^3\,e^7+2\,B\,a^2\,b\,d^2\,e^5+3\,A\,a^2\,b\,d\,e^6+3\,B\,a^2\,c\,d^3\,e^4+2\,A\,a^2\,c\,d^2\,e^5+3\,B\,a\,b^2\,d^3\,e^4+2\,A\,a\,b^2\,d^2\,e^5+24\,B\,a\,b\,c\,d^4\,e^3+6\,A\,a\,b\,c\,d^3\,e^4-137\,B\,a\,c^2\,d^5\,e^2+12\,A\,a\,c^2\,d^4\,e^3+4\,B\,b^3\,d^4\,e^3+A\,b^3\,d^3\,e^4-137\,B\,b^2\,c\,d^5\,e^2+12\,A\,b^2\,c\,d^4\,e^3+522\,B\,b\,c^2\,d^6\,e-137\,A\,b\,c^2\,d^5\,e^2-459\,B\,c^3\,d^7+174\,A\,c^3\,d^6\,e}{20\,e}+x^4\,\left (B\,b^3\,e^6-15\,B\,b^2\,c\,d\,e^5+3\,A\,b^2\,c\,e^6+45\,B\,b\,c^2\,d^2\,e^4-15\,A\,b\,c^2\,d\,e^5+6\,B\,a\,b\,c\,e^6-35\,B\,c^3\,d^3\,e^3+15\,A\,c^3\,d^2\,e^4-15\,B\,a\,c^2\,d\,e^5+3\,A\,a\,c^2\,e^6\right )+x^3\,\left (\frac {3\,B\,a^2\,c\,e^6}{2}+\frac {3\,B\,a\,b^2\,e^6}{2}+12\,B\,a\,b\,c\,d\,e^5+3\,A\,a\,b\,c\,e^6-45\,B\,a\,c^2\,d^2\,e^4+6\,A\,a\,c^2\,d\,e^5+2\,B\,b^3\,d\,e^5+\frac {A\,b^3\,e^6}{2}-45\,B\,b^2\,c\,d^2\,e^4+6\,A\,b^2\,c\,d\,e^5+150\,B\,b\,c^2\,d^3\,e^3-45\,A\,b\,c^2\,d^2\,e^4-\frac {245\,B\,c^3\,d^4\,e^2}{2}+50\,A\,c^3\,d^3\,e^3\right )+x\,\left (\frac {B\,a^3\,e^6}{4}+\frac {B\,a^2\,b\,d\,e^5}{2}+\frac {3\,A\,a^2\,b\,e^6}{4}+\frac {3\,B\,a^2\,c\,d^2\,e^4}{4}+\frac {A\,a^2\,c\,d\,e^5}{2}+\frac {3\,B\,a\,b^2\,d^2\,e^4}{4}+\frac {A\,a\,b^2\,d\,e^5}{2}+6\,B\,a\,b\,c\,d^3\,e^3+\frac {3\,A\,a\,b\,c\,d^2\,e^4}{2}-\frac {125\,B\,a\,c^2\,d^4\,e^2}{4}+3\,A\,a\,c^2\,d^3\,e^3+B\,b^3\,d^3\,e^3+\frac {A\,b^3\,d^2\,e^4}{4}-\frac {125\,B\,b^2\,c\,d^4\,e^2}{4}+3\,A\,b^2\,c\,d^3\,e^3+\frac {231\,B\,b\,c^2\,d^5\,e}{2}-\frac {125\,A\,b\,c^2\,d^4\,e^2}{4}-\frac {399\,B\,c^3\,d^6}{4}+\frac {77\,A\,c^3\,d^5\,e}{2}\right )+x^2\,\left (B\,a^2\,b\,e^6+\frac {3\,B\,a^2\,c\,d\,e^5}{2}+A\,a^2\,c\,e^6+\frac {3\,B\,a\,b^2\,d\,e^5}{2}+A\,a\,b^2\,e^6+12\,B\,a\,b\,c\,d^2\,e^4+3\,A\,a\,b\,c\,d\,e^5-55\,B\,a\,c^2\,d^3\,e^3+6\,A\,a\,c^2\,d^2\,e^4+2\,B\,b^3\,d^2\,e^4+\frac {A\,b^3\,d\,e^5}{2}-55\,B\,b^2\,c\,d^3\,e^3+6\,A\,b^2\,c\,d^2\,e^4+195\,B\,b\,c^2\,d^4\,e^2-55\,A\,b\,c^2\,d^3\,e^3-\frac {329\,B\,c^3\,d^5\,e}{2}+65\,A\,c^3\,d^4\,e^2\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 (3\,B\,b^2\,c\,e^2-18\,B\,b\,c^2\,d\,e+3\,A\,b\,c^2\,e^2+21\,B\,c^3\,d^2-6\,A\,c^3\,d\,e+3\,B\,a\,c^2\,e^2\right )}{e^8}+\frac {B\,c^3\,x^2}{2\,e^6} \]
x*((A*c^3 + 3*B*b*c^2)/e^6 - (6*B*c^3*d)/e^7) - ((4*A*a^3*e^7 - 459*B*c^3* d^7 + B*a^3*d*e^6 + 174*A*c^3*d^6*e + A*b^3*d^3*e^4 + 4*B*b^3*d^4*e^3 + 2* A*a*b^2*d^2*e^5 + 12*A*a*c^2*d^4*e^3 + 2*A*a^2*c*d^2*e^5 + 3*B*a*b^2*d^3*e ^4 + 2*B*a^2*b*d^2*e^5 - 137*A*b*c^2*d^5*e^2 + 12*A*b^2*c*d^4*e^3 - 137*B* a*c^2*d^5*e^2 + 3*B*a^2*c*d^3*e^4 - 137*B*b^2*c*d^5*e^2 + 3*A*a^2*b*d*e^6 + 522*B*b*c^2*d^6*e + 6*A*a*b*c*d^3*e^4 + 24*B*a*b*c*d^4*e^3)/(20*e) + x^4 *(B*b^3*e^6 + 3*A*a*c^2*e^6 + 3*A*b^2*c*e^6 + 15*A*c^3*d^2*e^4 - 35*B*c^3* d^3*e^3 + 45*B*b*c^2*d^2*e^4 + 6*B*a*b*c*e^6 - 15*A*b*c^2*d*e^5 - 15*B*a*c ^2*d*e^5 - 15*B*b^2*c*d*e^5) + x^3*((A*b^3*e^6)/2 + (3*B*a*b^2*e^6)/2 + (3 *B*a^2*c*e^6)/2 + 2*B*b^3*d*e^5 + 50*A*c^3*d^3*e^3 - (245*B*c^3*d^4*e^2)/2 - 45*A*b*c^2*d^2*e^4 - 45*B*a*c^2*d^2*e^4 + 150*B*b*c^2*d^3*e^3 - 45*B*b^ 2*c*d^2*e^4 + 3*A*a*b*c*e^6 + 6*A*a*c^2*d*e^5 + 6*A*b^2*c*d*e^5 + 12*B*a*b *c*d*e^5) + x*((B*a^3*e^6)/4 - (399*B*c^3*d^6)/4 + (3*A*a^2*b*e^6)/4 + (77 *A*c^3*d^5*e)/2 + (A*b^3*d^2*e^4)/4 + B*b^3*d^3*e^3 + 3*A*a*c^2*d^3*e^3 + (3*B*a*b^2*d^2*e^4)/4 - (125*A*b*c^2*d^4*e^2)/4 + 3*A*b^2*c*d^3*e^3 - (125 *B*a*c^2*d^4*e^2)/4 + (3*B*a^2*c*d^2*e^4)/4 - (125*B*b^2*c*d^4*e^2)/4 + (A *a*b^2*d*e^5)/2 + (A*a^2*c*d*e^5)/2 + (B*a^2*b*d*e^5)/2 + (231*B*b*c^2*d^5 *e)/2 + (3*A*a*b*c*d^2*e^4)/2 + 6*B*a*b*c*d^3*e^3) + x^2*(A*a*b^2*e^6 + A* a^2*c*e^6 + B*a^2*b*e^6 + (A*b^3*d*e^5)/2 - (329*B*c^3*d^5*e)/2 + 65*A*c^3 *d^4*e^2 + 2*B*b^3*d^2*e^4 + 6*A*a*c^2*d^2*e^4 - 55*A*b*c^2*d^3*e^3 + 6...