Integrand size = 25, antiderivative size = 533 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^5} \, dx=-\frac {c \left (A c e (5 c d-3 b e)-3 B \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) x}{e^7}-\frac {c^2 (5 B c d-3 b B e-A c e) x^2}{2 e^6}+\frac {B c^3 x^3}{3 e^5}+\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3}{4 e^8 (d+e x)^4}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right )}{3 e^8 (d+e x)^3}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )-A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right )}{2 e^8 (d+e x)^2}+\frac {A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )}{e^8 (d+e x)}-\frac {\left (B \left (35 c^3 d^3-b^3 e^3+3 b c e^2 (5 b d-2 a e)-15 c^2 d e (3 b d-a e)\right )-3 A c e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) \log (d+e x)}{e^8} \]
-c*(A*c*e*(-3*b*e+5*c*d)-3*B*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d)))*x/e^7-1 /2*c^2*(-A*c*e-3*B*b*e+5*B*c*d)*x^2/e^6+1/3*B*c^3*x^3/e^5+1/4*(-A*e+B*d)*( a*e^2-b*d*e+c*d^2)^3/e^8/(e*x+d)^4+1/3*(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)^3+3/2*(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)^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)-(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)))*ln(e*x+d)/e^8
Time = 0.18 (sec) , antiderivative size = 496, normalized size of antiderivative = 0.93 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^5} \, dx=\frac {12 c e \left (A c e (-5 c d+3 b e)+3 B \left (5 c^2 d^2+b^2 e^2+c e (-5 b d+a e)\right )\right ) x+6 c^2 e^2 (-5 B c d+3 b B e+A c e) x^2+4 B c^3 e^3 x^3+\frac {3 (B d-A e) \left (c d^2+e (-b d+a e)\right )^3}{(d+e x)^4}-\frac {4 \left (c d^2+e (-b d+a e)\right )^2 \left (7 B c d^2+B e (-4 b d+a e)+3 A e (-2 c d+b e)\right )}{(d+e x)^3}+\frac {18 \left (c d^2+e (-b d+a e)\right ) \left (-A e \left (5 c^2 d^2+b^2 e^2+c e (-5 b d+a e)\right )+B \left (7 c^2 d^3+b e^2 (2 b d-a e)+c d e (-8 b d+3 a e)\right )\right )}{(d+e x)^2}-\frac {12 \left (A e (-2 c d+b e) \left (10 c^2 d^2+b^2 e^2+2 c e (-5 b d+3 a e)\right )+B \left (35 c^3 d^4+30 c^2 d^2 e (-2 b d+a e)+b^2 e^3 (-4 b d+3 a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )\right )}{d+e x}+12 \left (3 A c e \left (5 c^2 d^2+b^2 e^2+c e (-5 b d+a e)\right )+B \left (-35 c^3 d^3+b^3 e^3+15 c^2 d e (3 b d-a e)+3 b c e^2 (-5 b d+2 a e)\right )\right ) \log (d+e x)}{12 e^8} \]
(12*c*e*(A*c*e*(-5*c*d + 3*b*e) + 3*B*(5*c^2*d^2 + b^2*e^2 + c*e*(-5*b*d + a*e)))*x + 6*c^2*e^2*(-5*B*c*d + 3*b*B*e + A*c*e)*x^2 + 4*B*c^3*e^3*x^3 + (3*(B*d - A*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x)^4 - (4*(c*d^2 + e* (-(b*d) + a*e))^2*(7*B*c*d^2 + B*e*(-4*b*d + a*e) + 3*A*e*(-2*c*d + b*e))) /(d + e*x)^3 + (18*(c*d^2 + e*(-(b*d) + a*e))*(-(A*e*(5*c^2*d^2 + b^2*e^2 + c*e*(-5*b*d + a*e))) + B*(7*c^2*d^3 + b*e^2*(2*b*d - a*e) + c*d*e*(-8*b* d + 3*a*e))))/(d + e*x)^2 - (12*(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 + 30*c^2*d^2*e*(-2*b*d + a*e) + b^2*e^3*(-4*b*d + 3*a*e) + 3*c*e^2*(10*b^2*d^2 - 8*a*b*d*e + a^2*e^2))))/( d + e*x) + 12*(3*A*c*e*(5*c^2*d^2 + b^2*e^2 + c*e*(-5*b*d + a*e)) + B*(-35 *c^3*d^3 + b^3*e^3 + 15*c^2*d*e*(3*b*d - a*e) + 3*b*c*e^2*(-5*b*d + 2*a*e) ))*Log[d + e*x])/(12*e^8)
Time = 1.28 (sec) , antiderivative size = 531, 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.
| \(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^5} \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )-A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7 (d+e x)^2}+\frac {c \left (3 B \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )-A c e (5 c d-3 b e)\right )}{e^7}+\frac {3 \left (a e^2-b d e+c d^2\right ) \left (A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )-B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )\right )}{e^7 (d+e x)^3}+\frac {3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )-B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )}{e^7 (d+e x)}+\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{e^7 (d+e x)^4}+\frac {(A e-B d) \left (a e^2-b d e+c d^2\right )^3}{e^7 (d+e x)^5}+\frac {c^2 x (A c e+3 b B e-5 B c d)}{e^6}+\frac {B c^3 x^2}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )}{e^8 (d+e x)}-\frac {c x \left (A c e (5 c d-3 b e)-3 B \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{e^7}+\frac {3 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{2 e^8 (d+e x)^2}-\frac {\log (d+e x) \left (B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{e^8}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{4 e^8 (d+e x)^4}-\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{3 e^8 (d+e x)^3}-\frac {c^2 x^2 (-A c e-3 b B e+5 B c d)}{2 e^6}+\frac {B c^3 x^3}{3 e^5}\) |
-((c*(A*c*e*(5*c*d - 3*b*e) - 3*B*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e) ))*x)/e^7) - (c^2*(5*B*c*d - 3*b*B*e - A*c*e)*x^2)/(2*e^6) + (B*c^3*x^3)/( 3*e^5) + ((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^3)/(4*e^8*(d + e*x)^4) - ((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)))/(3*e^8*(d + e*x)^3) + (3*(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))))/(2*e^8*(d + e*x)^2) + (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))) /(e^8*(d + e*x)) - ((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)))*Log[d + e*x])/e^8
3.24.43.3.1 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 = 0.34 (sec) , antiderivative size = 1042, normalized size of antiderivative = 1.95
| method | result | size |
| norman | \(\text {Expression too large to display}\) | \(1042\) |
| default | \(\text {Expression too large to display}\) | \(1066\) |
| risch | \(\text {Expression too large to display}\) | \(1122\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(2070\) |
(c*(3*A*b*c*e^2-3*A*c^2*d*e+3*B*a*c*e^2+3*B*b^2*e^2-9*B*b*c*d*e+7*B*c^2*d^ 2)/e^3*x^5-1/12*(3*A*a^3*e^7+3*A*a^2*b*d*e^6+3*A*a^2*c*d^2*e^5+3*A*a*b^2*d ^2*e^5+18*A*a*b*c*d^3*e^4-75*A*a*c^2*d^4*e^3+3*A*b^3*d^3*e^4-75*A*b^2*c*d^ 4*e^3+375*A*b*c^2*d^5*e^2-375*A*c^3*d^6*e+B*a^3*d*e^6+3*B*a^2*b*d^2*e^5+9* B*a^2*c*d^3*e^4+9*B*a*b^2*d^3*e^4-150*B*a*b*c*d^4*e^3+375*B*a*c^2*d^5*e^2- 25*B*b^3*d^4*e^3+375*B*b^2*c*d^5*e^2-1125*B*b*c^2*d^6*e+875*B*c^3*d^7)/e^8 -(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+60*A*b*c^2*d^2 *e^2-60*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+60*B*a*c^ 2*d^2*e^2-4*B*b^3*d*e^3+60*B*b^2*c*d^2*e^2-180*B*b*c^2*d^3*e+140*B*c^3*d^4 )/e^5*x^3-3/2*(A*a^2*c*e^5+A*a*b^2*e^5+6*A*a*b*c*d*e^4-18*A*a*c^2*d^2*e^3+ A*b^3*d*e^4-18*A*b^2*c*d^2*e^3+90*A*b*c^2*d^3*e^2-90*A*c^3*d^4*e+B*a^2*b*e ^5+3*B*a^2*c*d*e^4+3*B*a*b^2*d*e^4-36*B*a*b*c*d^2*e^3+90*B*a*c^2*d^3*e^2-6 *B*b^3*d^2*e^3+90*B*b^2*c*d^3*e^2-270*B*b*c^2*d^4*e+210*B*c^3*d^5)/e^6*x^2 -1/3*(3*A*a^2*b*e^6+3*A*a^2*c*d*e^5+3*A*a*b^2*d*e^5+18*A*a*b*c*d^2*e^4-66* A*a*c^2*d^3*e^3+3*A*b^3*d^2*e^4-66*A*b^2*c*d^3*e^3+330*A*b*c^2*d^4*e^2-330 *A*c^3*d^5*e+B*a^3*e^6+3*B*a^2*b*d*e^5+9*B*a^2*c*d^2*e^4+9*B*a*b^2*d^2*e^4 -132*B*a*b*c*d^3*e^3+330*B*a*c^2*d^4*e^2-22*B*b^3*d^3*e^3+330*B*b^2*c*d^4* e^2-990*B*b*c^2*d^5*e+770*B*c^3*d^6)/e^7*x+1/3*B*c^3*x^7/e+1/6*c^2*(3*A*c* e+9*B*b*e-7*B*c*d)/e^2*x^6)/(e*x+d)^4+1/e^8*(3*A*a*c^2*e^3+3*A*b^2*c*e^3-1 5*A*b*c^2*d*e^2+15*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...
Leaf count of result is larger than twice the leaf count of optimal. 1343 vs. \(2 (523) = 1046\).
Time = 0.35 (sec) , antiderivative size = 1343, normalized size of antiderivative = 2.52 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^5} \, dx=\text {Too large to display} \]
1/12*(4*B*c^3*e^7*x^7 - 319*B*c^3*d^7 - 3*A*a^3*e^7 + 171*(3*B*b*c^2 + A*c ^3)*d^6*e - 231*(B*b^2*c + (B*a + A*b)*c^2)*d^5*e^2 + 25*(B*b^3 + 3*A*a*c^ 2 + 3*(2*B*a*b + A*b^2)*c)*d^4*e^3 - 3*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A *a*b)*c)*d^3*e^4 - 3*(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 - 2*(7*B*c^3*d*e^6 - 3*(3*B*b*c^2 + A*c^3)*e^7)*x^6 + 12*(7*B *c^3*d^2*e^5 - 3*(3*B*b*c^2 + A*c^3)*d*e^6 + 3*(B*b^2*c + (B*a + A*b)*c^2) *e^7)*x^5 + 4*(139*B*c^3*d^3*e^4 - 51*(3*B*b*c^2 + A*c^3)*d^2*e^5 + 36*(B* b^2*c + (B*a + A*b)*c^2)*d*e^6)*x^4 + 4*(136*B*c^3*d^4*e^3 - 24*(3*B*b*c^2 + A*c^3)*d^3*e^4 - 36*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^5 + 12*(B*b^3 + 3 *A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d*e^6 - 3*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*e^7)*x^3 - 6*(74*B*c^3*d^5*e^2 - 66*(3*B*b*c^2 + A*c^3)*d^4* e^3 + 126*(B*b^2*c + (B*a + A*b)*c^2)*d^3*e^4 - 18*(B*b^3 + 3*A*a*c^2 + 3* (2*B*a*b + A*b^2)*c)*d^2*e^5 + 3*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)* c)*d*e^6 + 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*e^7)*x^2 - 4*(214*B*c^3*d^6*e - 126*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 186*(B*b^2*c + (B*a + A*b)*c^2)*d^4*e^3 - 22*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^3*e^4 + 3*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^2*e^5 + 3*(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 - 12*(35*B*c^3*d^7 - 15*(3*B*b*c^2 + A* c^3)*d^6*e + 15*(B*b^2*c + (B*a + A*b)*c^2)*d^5*e^2 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^4*e^3 + (35*B*c^3*d^3*e^4 - 15*(3*B*b*c^2 + A...
Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^5} \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 884, normalized size of antiderivative = 1.66 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^5} \, dx=-\frac {319 \, B c^{3} d^{7} + 3 \, A a^{3} e^{7} - 171 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 231 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{5} e^{2} - 25 \, {\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} + 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} + 3 \, {\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} + 12 \, {\left (35 \, B c^{3} d^{4} e^{3} - 20 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{4} + 30 \, {\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} + 18 \, {\left (63 \, B c^{3} d^{5} e^{2} - 35 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} + 50 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} e^{4} - 6 \, {\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} + {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e^{7}\right )} x^{2} + 4 \, {\left (259 \, B c^{3} d^{6} e - 141 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 195 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} e^{3} - 22 \, {\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} + 3 \, {\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} + 3 \, {\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}{12 \, {\left (e^{12} x^{4} + 4 \, d e^{11} x^{3} + 6 \, d^{2} e^{10} x^{2} + 4 \, d^{3} e^{9} x + d^{4} e^{8}\right )}} + \frac {2 \, B c^{3} e^{2} x^{3} - 3 \, {\left (5 \, B c^{3} d e - {\left (3 \, B b c^{2} + A c^{3}\right )} e^{2}\right )} x^{2} + 6 \, {\left (15 \, B c^{3} d^{2} - 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d e + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} e^{2}\right )} x}{6 \, e^{7}} - \frac {{\left (35 \, B c^{3} d^{3} - 15 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e + 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d e^{2} - {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} e^{3}\right )} \log \left (e x + d\right )}{e^{8}} \]
-1/12*(319*B*c^3*d^7 + 3*A*a^3*e^7 - 171*(3*B*b*c^2 + A*c^3)*d^6*e + 231*( B*b^2*c + (B*a + A*b)*c^2)*d^5*e^2 - 25*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^4*e^3 + 3*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^3*e^4 + 3*(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 + 12 *(35*B*c^3*d^4*e^3 - 20*(3*B*b*c^2 + A*c^3)*d^3*e^4 + 30*(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 + 18*(63*B*c^3*d^5*e ^2 - 35*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 50*(B*b^2*c + (B*a + A*b)*c^2)*d^3*e ^4 - 6*(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 + (B*a^2*b + A*a*b^2 + A*a^2*c)*e^7)* x^2 + 4*(259*B*c^3*d^6*e - 141*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 195*(B*b^2*c + (B*a + A*b)*c^2)*d^4*e^3 - 22*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c )*d^3*e^4 + 3*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^2*e^5 + 3*(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^12*x^4 + 4*d*e^11*x^3 + 6*d^2*e^10*x^2 + 4*d^3*e^9*x + d^4*e^8) + 1/6*(2*B*c^3*e^2* x^3 - 3*(5*B*c^3*d*e - (3*B*b*c^2 + A*c^3)*e^2)*x^2 + 6*(15*B*c^3*d^2 - 5* (3*B*b*c^2 + A*c^3)*d*e + 3*(B*b^2*c + (B*a + A*b)*c^2)*e^2)*x)/e^7 - (35* B*c^3*d^3 - 15*(3*B*b*c^2 + A*c^3)*d^2*e + 15*(B*b^2*c + (B*a + A*b)*c^2)* d*e^2 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e^3)*log(e*x + d)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 1581 vs. \(2 (523) = 1046\).
Time = 0.30 (sec) , antiderivative size = 1581, normalized size of antiderivative = 2.97 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^5} \, dx=\text {Too large to display} \]
1/6*(2*B*c^3 - 3*(7*B*c^3*d*e - 3*B*b*c^2*e^2 - A*c^3*e^2)/((e*x + d)*e) + 18*(7*B*c^3*d^2*e^2 - 6*B*b*c^2*d*e^3 - 2*A*c^3*d*e^3 + B*b^2*c*e^4 + B*a *c^2*e^4 + A*b*c^2*e^4)/((e*x + d)^2*e^2))*(e*x + d)^3/e^8 + (35*B*c^3*d^3 - 45*B*b*c^2*d^2*e - 15*A*c^3*d^2*e + 15*B*b^2*c*d*e^2 + 15*B*a*c^2*d*e^2 + 15*A*b*c^2*d*e^2 - B*b^3*e^3 - 6*B*a*b*c*e^3 - 3*A*b^2*c*e^3 - 3*A*a*c^ 2*e^3)*log(abs(e*x + d)/((e*x + d)^2*abs(e)))/e^8 - 1/12*(420*B*c^3*d^4*e^ 36/(e*x + d) - 126*B*c^3*d^5*e^36/(e*x + d)^2 + 28*B*c^3*d^6*e^36/(e*x + d )^3 - 3*B*c^3*d^7*e^36/(e*x + d)^4 - 720*B*b*c^2*d^3*e^37/(e*x + d) - 240* A*c^3*d^3*e^37/(e*x + d) + 270*B*b*c^2*d^4*e^37/(e*x + d)^2 + 90*A*c^3*d^4 *e^37/(e*x + d)^2 - 72*B*b*c^2*d^5*e^37/(e*x + d)^3 - 24*A*c^3*d^5*e^37/(e *x + d)^3 + 9*B*b*c^2*d^6*e^37/(e*x + d)^4 + 3*A*c^3*d^6*e^37/(e*x + d)^4 + 360*B*b^2*c*d^2*e^38/(e*x + d) + 360*B*a*c^2*d^2*e^38/(e*x + d) + 360*A* b*c^2*d^2*e^38/(e*x + d) - 180*B*b^2*c*d^3*e^38/(e*x + d)^2 - 180*B*a*c^2* d^3*e^38/(e*x + d)^2 - 180*A*b*c^2*d^3*e^38/(e*x + d)^2 + 60*B*b^2*c*d^4*e ^38/(e*x + d)^3 + 60*B*a*c^2*d^4*e^38/(e*x + d)^3 + 60*A*b*c^2*d^4*e^38/(e *x + d)^3 - 9*B*b^2*c*d^5*e^38/(e*x + d)^4 - 9*B*a*c^2*d^5*e^38/(e*x + d)^ 4 - 9*A*b*c^2*d^5*e^38/(e*x + d)^4 - 48*B*b^3*d*e^39/(e*x + d) - 288*B*a*b *c*d*e^39/(e*x + d) - 144*A*b^2*c*d*e^39/(e*x + d) - 144*A*a*c^2*d*e^39/(e *x + d) + 36*B*b^3*d^2*e^39/(e*x + d)^2 + 216*B*a*b*c*d^2*e^39/(e*x + d)^2 + 108*A*b^2*c*d^2*e^39/(e*x + d)^2 + 108*A*a*c^2*d^2*e^39/(e*x + d)^2 ...
Time = 11.34 (sec) , antiderivative size = 1106, normalized size of antiderivative = 2.08 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^5} \, dx=x^2\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{2\,e^5}-\frac {5\,B\,c^3\,d}{2\,e^6}\right )-\frac {\frac {B\,a^3\,d\,e^6+3\,A\,a^3\,e^7+3\,B\,a^2\,b\,d^2\,e^5+3\,A\,a^2\,b\,d\,e^6+9\,B\,a^2\,c\,d^3\,e^4+3\,A\,a^2\,c\,d^2\,e^5+9\,B\,a\,b^2\,d^3\,e^4+3\,A\,a\,b^2\,d^2\,e^5-150\,B\,a\,b\,c\,d^4\,e^3+18\,A\,a\,b\,c\,d^3\,e^4+231\,B\,a\,c^2\,d^5\,e^2-75\,A\,a\,c^2\,d^4\,e^3-25\,B\,b^3\,d^4\,e^3+3\,A\,b^3\,d^3\,e^4+231\,B\,b^2\,c\,d^5\,e^2-75\,A\,b^2\,c\,d^4\,e^3-513\,B\,b\,c^2\,d^6\,e+231\,A\,b\,c^2\,d^5\,e^2+319\,B\,c^3\,d^7-171\,A\,c^3\,d^6\,e}{12\,e}+x^3\,\left (3\,B\,a^2\,c\,e^6+3\,B\,a\,b^2\,e^6-24\,B\,a\,b\,c\,d\,e^5+6\,A\,a\,b\,c\,e^6+30\,B\,a\,c^2\,d^2\,e^4-12\,A\,a\,c^2\,d\,e^5-4\,B\,b^3\,d\,e^5+A\,b^3\,e^6+30\,B\,b^2\,c\,d^2\,e^4-12\,A\,b^2\,c\,d\,e^5-60\,B\,b\,c^2\,d^3\,e^3+30\,A\,b\,c^2\,d^2\,e^4+35\,B\,c^3\,d^4\,e^2-20\,A\,c^3\,d^3\,e^3\right )+x\,\left (\frac {B\,a^3\,e^6}{3}+B\,a^2\,b\,d\,e^5+A\,a^2\,b\,e^6+3\,B\,a^2\,c\,d^2\,e^4+A\,a^2\,c\,d\,e^5+3\,B\,a\,b^2\,d^2\,e^4+A\,a\,b^2\,d\,e^5-44\,B\,a\,b\,c\,d^3\,e^3+6\,A\,a\,b\,c\,d^2\,e^4+65\,B\,a\,c^2\,d^4\,e^2-22\,A\,a\,c^2\,d^3\,e^3-\frac {22\,B\,b^3\,d^3\,e^3}{3}+A\,b^3\,d^2\,e^4+65\,B\,b^2\,c\,d^4\,e^2-22\,A\,b^2\,c\,d^3\,e^3-141\,B\,b\,c^2\,d^5\,e+65\,A\,b\,c^2\,d^4\,e^2+\frac {259\,B\,c^3\,d^6}{3}-47\,A\,c^3\,d^5\,e\right )+x^2\,\left (\frac {3\,B\,a^2\,b\,e^6}{2}+\frac {9\,B\,a^2\,c\,d\,e^5}{2}+\frac {3\,A\,a^2\,c\,e^6}{2}+\frac {9\,B\,a\,b^2\,d\,e^5}{2}+\frac {3\,A\,a\,b^2\,e^6}{2}-54\,B\,a\,b\,c\,d^2\,e^4+9\,A\,a\,b\,c\,d\,e^5+75\,B\,a\,c^2\,d^3\,e^3-27\,A\,a\,c^2\,d^2\,e^4-9\,B\,b^3\,d^2\,e^4+\frac {3\,A\,b^3\,d\,e^5}{2}+75\,B\,b^2\,c\,d^3\,e^3-27\,A\,b^2\,c\,d^2\,e^4-\frac {315\,B\,b\,c^2\,d^4\,e^2}{2}+75\,A\,b\,c^2\,d^3\,e^3+\frac {189\,B\,c^3\,d^5\,e}{2}-\frac {105\,A\,c^3\,d^4\,e^2}{2}\right )}{d^4\,e^7+4\,d^3\,e^8\,x+6\,d^2\,e^9\,x^2+4\,d\,e^{10}\,x^3+e^{11}\,x^4}-x\,\left (\frac {5\,d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^5}-\frac {5\,B\,c^3\,d}{e^6}\right )}{e}-\frac {3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2}{e^5}+\frac {10\,B\,c^3\,d^2}{e^7}\right )+\frac {\ln \left (d+e\,x\right )\,\left (B\,b^3\,e^3-15\,B\,b^2\,c\,d\,e^2+3\,A\,b^2\,c\,e^3+45\,B\,b\,c^2\,d^2\,e-15\,A\,b\,c^2\,d\,e^2+6\,B\,a\,b\,c\,e^3-35\,B\,c^3\,d^3+15\,A\,c^3\,d^2\,e-15\,B\,a\,c^2\,d\,e^2+3\,A\,a\,c^2\,e^3\right )}{e^8}+\frac {B\,c^3\,x^3}{3\,e^5} \]
x^2*((A*c^3 + 3*B*b*c^2)/(2*e^5) - (5*B*c^3*d)/(2*e^6)) - ((3*A*a^3*e^7 + 319*B*c^3*d^7 + B*a^3*d*e^6 - 171*A*c^3*d^6*e + 3*A*b^3*d^3*e^4 - 25*B*b^3 *d^4*e^3 + 3*A*a*b^2*d^2*e^5 - 75*A*a*c^2*d^4*e^3 + 3*A*a^2*c*d^2*e^5 + 9* B*a*b^2*d^3*e^4 + 3*B*a^2*b*d^2*e^5 + 231*A*b*c^2*d^5*e^2 - 75*A*b^2*c*d^4 *e^3 + 231*B*a*c^2*d^5*e^2 + 9*B*a^2*c*d^3*e^4 + 231*B*b^2*c*d^5*e^2 + 3*A *a^2*b*d*e^6 - 513*B*b*c^2*d^6*e + 18*A*a*b*c*d^3*e^4 - 150*B*a*b*c*d^4*e^ 3)/(12*e) + x^3*(A*b^3*e^6 + 3*B*a*b^2*e^6 + 3*B*a^2*c*e^6 - 4*B*b^3*d*e^5 - 20*A*c^3*d^3*e^3 + 35*B*c^3*d^4*e^2 + 30*A*b*c^2*d^2*e^4 + 30*B*a*c^2*d ^2*e^4 - 60*B*b*c^2*d^3*e^3 + 30*B*b^2*c*d^2*e^4 + 6*A*a*b*c*e^6 - 12*A*a* c^2*d*e^5 - 12*A*b^2*c*d*e^5 - 24*B*a*b*c*d*e^5) + x*((B*a^3*e^6)/3 + (259 *B*c^3*d^6)/3 + A*a^2*b*e^6 - 47*A*c^3*d^5*e + A*b^3*d^2*e^4 - (22*B*b^3*d ^3*e^3)/3 - 22*A*a*c^2*d^3*e^3 + 3*B*a*b^2*d^2*e^4 + 65*A*b*c^2*d^4*e^2 - 22*A*b^2*c*d^3*e^3 + 65*B*a*c^2*d^4*e^2 + 3*B*a^2*c*d^2*e^4 + 65*B*b^2*c*d ^4*e^2 + A*a*b^2*d*e^5 + A*a^2*c*d*e^5 + B*a^2*b*d*e^5 - 141*B*b*c^2*d^5*e + 6*A*a*b*c*d^2*e^4 - 44*B*a*b*c*d^3*e^3) + x^2*((3*A*a*b^2*e^6)/2 + (3*A *a^2*c*e^6)/2 + (3*B*a^2*b*e^6)/2 + (3*A*b^3*d*e^5)/2 + (189*B*c^3*d^5*e)/ 2 - (105*A*c^3*d^4*e^2)/2 - 9*B*b^3*d^2*e^4 - 27*A*a*c^2*d^2*e^4 + 75*A*b* c^2*d^3*e^3 - 27*A*b^2*c*d^2*e^4 + 75*B*a*c^2*d^3*e^3 - (315*B*b*c^2*d^4*e ^2)/2 + 75*B*b^2*c*d^3*e^3 + (9*B*a*b^2*d*e^5)/2 + (9*B*a^2*c*d*e^5)/2 - 5 4*B*a*b*c*d^2*e^4 + 9*A*a*b*c*d*e^5))/(d^4*e^7 + e^11*x^4 + 4*d^3*e^8*x...