Integrand size = 25, antiderivative size = 531 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx=-\frac {\left (A e \left (10 c^3 d^3-b^3 e^3+3 b c e^2 (3 b d-2 a e)-9 c^2 d e (2 b d-a e)\right )-3 B \left (5 c^3 d^4-2 c^2 d^2 e (5 b d-3 a e)-b^2 e^3 (b d-a e)+c e^2 \left (6 b^2 d^2-6 a b d e+a^2 e^2\right )\right )\right ) x}{e^7}-\frac {\left (B \left (10 c^3 d^3-b^3 e^3+3 b c e^2 (3 b d-2 a e)-9 c^2 d e (2 b d-a e)\right )-3 A c e \left (2 c^2 d^2+b^2 e^2-c e (3 b d-a e)\right )\right ) x^2}{2 e^6}-\frac {c \left (A c e (c d-b e)-B \left (2 c^2 d^2+b^2 e^2-c e (3 b d-a e)\right )\right ) x^3}{e^5}-\frac {c^2 (3 B c d-3 b B e-A c e) x^4}{4 e^4}+\frac {B c^3 x^5}{5 e^3}+\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3}{2 e^8 (d+e x)^2}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right )}{e^8 (d+e x)}-\frac {3 \left (c d^2-b d e+a e^2\right ) \left (B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )-A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) \log (d+e x)}{e^8} \]
-(A*e*(10*c^3*d^3-b^3*e^3+3*b*c*e^2*(-2*a*e+3*b*d)-9*c^2*d*e*(-a*e+2*b*d)) -3*B*(5*c^3*d^4-2*c^2*d^2*e*(-3*a*e+5*b*d)-b^2*e^3*(-a*e+b*d)+c*e^2*(a^2*e ^2-6*a*b*d*e+6*b^2*d^2)))*x/e^7-1/2*(B*(10*c^3*d^3-b^3*e^3+3*b*c*e^2*(-2*a *e+3*b*d)-9*c^2*d*e*(-a*e+2*b*d))-3*A*c*e*(2*c^2*d^2+b^2*e^2-c*e*(-a*e+3*b *d)))*x^2/e^6-c*(A*c*e*(-b*e+c*d)-B*(2*c^2*d^2+b^2*e^2-c*e*(-a*e+3*b*d)))* x^3/e^5-1/4*c^2*(-A*c*e-3*B*b*e+3*B*c*d)*x^4/e^4+1/5*B*c^3*x^5/e^3+1/2*(-A *e+B*d)*(a*e^2-b*d*e+c*d^2)^3/e^8/(e*x+d)^2+(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*(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)))*ln(e*x+d)/e^8
Time = 0.17 (sec) , antiderivative size = 503, normalized size of antiderivative = 0.95 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx=\frac {20 e \left (A e \left (-10 c^3 d^3+b^3 e^3+9 c^2 d e (2 b d-a e)+3 b c e^2 (-3 b d+2 a e)\right )+3 B \left (5 c^3 d^4+b^2 e^3 (-b d+a e)+2 c^2 d^2 e (-5 b d+3 a e)+c e^2 \left (6 b^2 d^2-6 a b d e+a^2 e^2\right )\right )\right ) x+10 e^2 \left (3 A c e \left (2 c^2 d^2+b^2 e^2+c e (-3 b d+a e)\right )+B \left (-10 c^3 d^3+b^3 e^3+9 c^2 d e (2 b d-a e)+3 b c e^2 (-3 b d+2 a e)\right )\right ) x^2+20 c e^3 \left (A c e (-c d+b e)+B \left (2 c^2 d^2+b^2 e^2+c e (-3 b d+a e)\right )\right ) x^3+5 c^2 e^4 (-3 B c d+3 b B e+A c e) x^4+4 B c^3 e^5 x^5+\frac {10 (B d-A e) \left (c d^2+e (-b d+a e)\right )^3}{(d+e x)^2}-\frac {20 \left (c d^2+e (-b d+a e)\right )^2 \left (7 B c d^2+B e (-4 b d+a e)+3 A e (-2 c d+b e)\right )}{d+e x}-60 \left (c d^2+e (-b d+a e)\right ) \left (-A e \left (5 c^2 d^2+b^2 e^2+c e (-5 b d+a e)\right )+B \left (7 c^2 d^3+b e^2 (2 b d-a e)+c d e (-8 b d+3 a e)\right )\right ) \log (d+e x)}{20 e^8} \]
(20*e*(A*e*(-10*c^3*d^3 + b^3*e^3 + 9*c^2*d*e*(2*b*d - a*e) + 3*b*c*e^2*(- 3*b*d + 2*a*e)) + 3*B*(5*c^3*d^4 + b^2*e^3*(-(b*d) + a*e) + 2*c^2*d^2*e*(- 5*b*d + 3*a*e) + c*e^2*(6*b^2*d^2 - 6*a*b*d*e + a^2*e^2)))*x + 10*e^2*(3*A *c*e*(2*c^2*d^2 + b^2*e^2 + c*e*(-3*b*d + a*e)) + B*(-10*c^3*d^3 + b^3*e^3 + 9*c^2*d*e*(2*b*d - a*e) + 3*b*c*e^2*(-3*b*d + 2*a*e)))*x^2 + 20*c*e^3*( A*c*e*(-(c*d) + b*e) + B*(2*c^2*d^2 + b^2*e^2 + c*e*(-3*b*d + a*e)))*x^3 + 5*c^2*e^4*(-3*B*c*d + 3*b*B*e + A*c*e)*x^4 + 4*B*c^3*e^5*x^5 + (10*(B*d - A*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x)^2 - (20*(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 ) - 60*(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))) *Log[d + e*x])/(20*e^8)
Time = 1.34 (sec) , antiderivative size = 530, 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.
| \(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {3 B \left (c e^2 \left (a^2 e^2-6 a b d e+6 b^2 d^2\right )-b^2 e^3 (b d-a e)-2 c^2 d^2 e (5 b d-3 a e)+5 c^3 d^4\right )-A e \left (-9 c^2 d e (2 b d-a e)+3 b c e^2 (3 b d-2 a e)-b^3 e^3+10 c^3 d^3\right )}{e^7}+\frac {3 c x^2 \left (B \left (-c e (3 b d-a e)+b^2 e^2+2 c^2 d^2\right )-A c e (c d-b e)\right )}{e^5}+\frac {3 \left (a e^2-b d e+c d^2\right ) \left (A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )-B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )\right )}{e^7 (d+e x)}+\frac {x \left (3 A c e \left (-c e (3 b d-a e)+b^2 e^2+2 c^2 d^2\right )-B \left (-9 c^2 d e (2 b d-a e)+3 b c e^2 (3 b d-2 a e)-b^3 e^3+10 c^3 d^3\right )\right )}{e^6}+\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{e^7 (d+e x)^2}+\frac {(A e-B d) \left (a e^2-b d e+c d^2\right )^3}{e^7 (d+e x)^3}+\frac {c^2 x^3 (A c e+3 b B e-3 B c d)}{e^4}+\frac {B c^3 x^4}{e^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {x \left (A e \left (-9 c^2 d e (2 b d-a e)+3 b c e^2 (3 b d-2 a e)-b^3 e^3+10 c^3 d^3\right )-3 B \left (c e^2 \left (a^2 e^2-6 a b d e+6 b^2 d^2\right )-b^2 e^3 (b d-a e)-2 c^2 d^2 e (5 b d-3 a e)+5 c^3 d^4\right )\right )}{e^7}-\frac {c x^3 \left (A c e (c d-b e)-B \left (-c e (3 b d-a e)+b^2 e^2+2 c^2 d^2\right )\right )}{e^5}-\frac {3 \log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{e^8}-\frac {x^2 \left (B \left (-9 c^2 d e (2 b d-a e)+3 b c e^2 (3 b d-2 a e)-b^3 e^3+10 c^3 d^3\right )-3 A c e \left (-c e (3 b d-a e)+b^2 e^2+2 c^2 d^2\right )\right )}{2 e^6}-\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{e^8 (d+e x)}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{2 e^8 (d+e x)^2}-\frac {c^2 x^4 (-A c e-3 b B e+3 B c d)}{4 e^4}+\frac {B c^3 x^5}{5 e^3}\) |
-(((A*e*(10*c^3*d^3 - b^3*e^3 + 3*b*c*e^2*(3*b*d - 2*a*e) - 9*c^2*d*e*(2*b *d - a*e)) - 3*B*(5*c^3*d^4 - 2*c^2*d^2*e*(5*b*d - 3*a*e) - b^2*e^3*(b*d - a*e) + c*e^2*(6*b^2*d^2 - 6*a*b*d*e + a^2*e^2)))*x)/e^7) - ((B*(10*c^3*d^ 3 - b^3*e^3 + 3*b*c*e^2*(3*b*d - 2*a*e) - 9*c^2*d*e*(2*b*d - a*e)) - 3*A*c *e*(2*c^2*d^2 + b^2*e^2 - c*e*(3*b*d - a*e)))*x^2)/(2*e^6) - (c*(A*c*e*(c* d - b*e) - B*(2*c^2*d^2 + b^2*e^2 - c*e*(3*b*d - a*e)))*x^3)/e^5 - (c^2*(3 *B*c*d - 3*b*B*e - A*c*e)*x^4)/(4*e^4) + (B*c^3*x^5)/(5*e^3) + ((B*d - A*e )*(c*d^2 - b*d*e + a*e^2)^3)/(2*e^8*(d + e*x)^2) - ((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)))/(e^8*(d + e*x)) - (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)))*Log[d + e*x])/e^8
3.24.41.3.1 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 = 1044, normalized size of antiderivative = 1.97
| method | result | size |
| norman | \(\text {Expression too large to display}\) | \(1044\) |
| default | \(\text {Expression too large to display}\) | \(1116\) |
| risch | \(\text {Expression too large to display}\) | \(1215\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1951\) |
((6*A*a*b*c*e^4-6*A*a*c^2*d*e^3+A*b^3*e^4-6*A*b^2*c*d*e^3+10*A*b*c^2*d^2*e ^2-5*A*c^3*d^3*e+3*B*a^2*c*e^4+3*B*a*b^2*e^4-12*B*a*b*c*d*e^3+10*B*a*c^2*d ^2*e^2-2*B*b^3*d*e^3+10*B*b^2*c*d^2*e^2-15*B*b*c^2*d^3*e+7*B*c^3*d^4)/e^5* x^3-1/2*(A*a^3*e^7+3*A*a^2*b*d*e^6-9*A*a^2*c*d^2*e^5-9*A*a*b^2*d^2*e^5+54* A*a*b*c*d^3*e^4-54*A*a*c^2*d^4*e^3+9*A*b^3*d^3*e^4-54*A*b^2*c*d^4*e^3+90*A *b*c^2*d^5*e^2-45*A*c^3*d^6*e+B*a^3*d*e^6-9*B*a^2*b*d^2*e^5+27*B*a^2*c*d^3 *e^4+27*B*a*b^2*d^3*e^4-108*B*a*b*c*d^4*e^3+90*B*a*c^2*d^5*e^2-18*B*b^3*d^ 4*e^3+90*B*b^2*c*d^5*e^2-135*B*b*c^2*d^6*e+63*B*c^3*d^7)/e^8+1/4*(6*A*a*c^ 2*e^3+6*A*b^2*c*e^3-10*A*b*c^2*d*e^2+5*A*c^3*d^2*e+12*B*a*b*c*e^3-10*B*a*c ^2*d*e^2+2*B*b^3*e^3-10*B*b^2*c*d*e^2+15*B*b*c^2*d^2*e-7*B*c^3*d^3)/e^4*x^ 4-(3*A*a^2*b*e^6-6*A*a^2*c*d*e^5-6*A*a*b^2*d*e^5+36*A*a*b*c*d^2*e^4-36*A*a *c^2*d^3*e^3+6*A*b^3*d^2*e^4-36*A*b^2*c*d^3*e^3+60*A*b*c^2*d^4*e^2-30*A*c^ 3*d^5*e+B*a^3*e^6-6*B*a^2*b*d*e^5+18*B*a^2*c*d^2*e^4+18*B*a*b^2*d^2*e^4-72 *B*a*b*c*d^3*e^3+60*B*a*c^2*d^4*e^2-12*B*b^3*d^3*e^3+60*B*b^2*c*d^4*e^2-90 *B*b*c^2*d^5*e+42*B*c^3*d^6)/e^7*x+1/5*B*c^3*x^7/e+1/10*c*(10*A*b*c*e^2-5* A*c^2*d*e+10*B*a*c*e^2+10*B*b^2*e^2-15*B*b*c*d*e+7*B*c^2*d^2)/e^3*x^5+1/20 *c^2*(5*A*c*e+15*B*b*e-7*B*c*d)/e^2*x^6)/(e*x+d)^2+3*(A*a^2*c*e^5+A*a*b^2* e^5-6*A*a*b*c*d*e^4+6*A*a*c^2*d^2*e^3-A*b^3*d*e^4+6*A*b^2*c*d^2*e^3-10*A*b *c^2*d^3*e^2+5*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+12* B*a*b*c*d^2*e^3-10*B*a*c^2*d^3*e^2+2*B*b^3*d^2*e^3-10*B*b^2*c*d^3*e^2+1...
Leaf count of result is larger than twice the leaf count of optimal. 1311 vs. \(2 (524) = 1048\).
Time = 0.32 (sec) , antiderivative size = 1311, normalized size of antiderivative = 2.47 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx=\text {Too large to display} \]
1/20*(4*B*c^3*e^7*x^7 - 130*B*c^3*d^7 - 10*A*a^3*e^7 + 110*(3*B*b*c^2 + A* c^3)*d^6*e - 270*(B*b^2*c + (B*a + A*b)*c^2)*d^5*e^2 + 70*(B*b^3 + 3*A*a*c ^2 + 3*(2*B*a*b + A*b^2)*c)*d^4*e^3 - 50*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2 *A*a*b)*c)*d^3*e^4 + 90*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^2*e^5 - 10*(B*a^3 + 3*A*a^2*b)*d*e^6 - (7*B*c^3*d*e^6 - 5*(3*B*b*c^2 + A*c^3)*e^7)*x^6 + 2*( 7*B*c^3*d^2*e^5 - 5*(3*B*b*c^2 + A*c^3)*d*e^6 + 10*(B*b^2*c + (B*a + A*b)* c^2)*e^7)*x^5 - 5*(7*B*c^3*d^3*e^4 - 5*(3*B*b*c^2 + A*c^3)*d^2*e^5 + 10*(B *b^2*c + (B*a + A*b)*c^2)*d*e^6 - 2*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^ 2)*c)*e^7)*x^4 + 20*(7*B*c^3*d^4*e^3 - 5*(3*B*b*c^2 + A*c^3)*d^3*e^4 + 10* (B*b^2*c + (B*a + A*b)*c^2)*d^2*e^5 - 2*(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 + 1 0*(50*B*c^3*d^5*e^2 - 34*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 63*(B*b^2*c + (B*a + A*b)*c^2)*d^3*e^4 - 11*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e ^5 + 4*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*e^6)*x^2 + 20*(8*B*c^ 3*d^6*e - 4*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 3*(B*b^2*c + (B*a + A*b)*c^2)*d^ 4*e^3 + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^3*e^4 - 2*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^2*e^5 + 6*(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 - 5*(3*B*b*c^2 + A* c^3)*d^6*e + 10*(B*b^2*c + (B*a + A*b)*c^2)*d^5*e^2 - 2*(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...
Leaf count of result is larger than twice the leaf count of optimal. 1149 vs. \(2 (518) = 1036\).
Time = 32.65 (sec) , antiderivative size = 1149, normalized size of antiderivative = 2.16 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx=\frac {B c^{3} x^{5}}{5 e^{3}} + x^{4} \left (\frac {A c^{3}}{4 e^{3}} + \frac {3 B b c^{2}}{4 e^{3}} - \frac {3 B c^{3} d}{4 e^{4}}\right ) + x^{3} \left (\frac {A b c^{2}}{e^{3}} - \frac {A c^{3} d}{e^{4}} + \frac {B a c^{2}}{e^{3}} + \frac {B b^{2} c}{e^{3}} - \frac {3 B b c^{2} d}{e^{4}} + \frac {2 B c^{3} d^{2}}{e^{5}}\right ) + x^{2} \cdot \left (\frac {3 A a c^{2}}{2 e^{3}} + \frac {3 A b^{2} c}{2 e^{3}} - \frac {9 A b c^{2} d}{2 e^{4}} + \frac {3 A c^{3} d^{2}}{e^{5}} + \frac {3 B a b c}{e^{3}} - \frac {9 B a c^{2} d}{2 e^{4}} + \frac {B b^{3}}{2 e^{3}} - \frac {9 B b^{2} c d}{2 e^{4}} + \frac {9 B b c^{2} d^{2}}{e^{5}} - \frac {5 B c^{3} d^{3}}{e^{6}}\right ) + x \left (\frac {6 A a b c}{e^{3}} - \frac {9 A a c^{2} d}{e^{4}} + \frac {A b^{3}}{e^{3}} - \frac {9 A b^{2} c d}{e^{4}} + \frac {18 A b c^{2} d^{2}}{e^{5}} - \frac {10 A c^{3} d^{3}}{e^{6}} + \frac {3 B a^{2} c}{e^{3}} + \frac {3 B a b^{2}}{e^{3}} - \frac {18 B a b c d}{e^{4}} + \frac {18 B a c^{2} d^{2}}{e^{5}} - \frac {3 B b^{3} d}{e^{4}} + \frac {18 B b^{2} c d^{2}}{e^{5}} - \frac {30 B b c^{2} d^{3}}{e^{6}} + \frac {15 B c^{3} d^{4}}{e^{7}}\right ) + \frac {- A a^{3} e^{7} - 3 A a^{2} b d e^{6} + 9 A a^{2} c d^{2} e^{5} + 9 A a b^{2} d^{2} e^{5} - 30 A a b c d^{3} e^{4} + 21 A a c^{2} d^{4} e^{3} - 5 A b^{3} d^{3} e^{4} + 21 A b^{2} c d^{4} e^{3} - 27 A b c^{2} d^{5} e^{2} + 11 A c^{3} d^{6} e - B a^{3} d e^{6} + 9 B a^{2} b d^{2} e^{5} - 15 B a^{2} c d^{3} e^{4} - 15 B a b^{2} d^{3} e^{4} + 42 B a b c d^{4} e^{3} - 27 B a c^{2} d^{5} e^{2} + 7 B b^{3} d^{4} e^{3} - 27 B b^{2} c d^{5} e^{2} + 33 B b c^{2} d^{6} e - 13 B c^{3} d^{7} + x \left (- 6 A a^{2} b e^{7} + 12 A a^{2} c d e^{6} + 12 A a b^{2} d e^{6} - 36 A a b c d^{2} e^{5} + 24 A a c^{2} d^{3} e^{4} - 6 A b^{3} d^{2} e^{5} + 24 A b^{2} c d^{3} e^{4} - 30 A b c^{2} d^{4} e^{3} + 12 A c^{3} d^{5} e^{2} - 2 B a^{3} e^{7} + 12 B a^{2} b d e^{6} - 18 B a^{2} c d^{2} e^{5} - 18 B a b^{2} d^{2} e^{5} + 48 B a b c d^{3} e^{4} - 30 B a c^{2} d^{4} e^{3} + 8 B b^{3} d^{3} e^{4} - 30 B b^{2} c d^{4} e^{3} + 36 B b c^{2} d^{5} e^{2} - 14 B c^{3} d^{6} e\right )}{2 d^{2} e^{8} + 4 d e^{9} x + 2 e^{10} x^{2}} + \frac {3 \left (a e^{2} - b d e + c d^{2}\right ) \left (A a c e^{3} + A b^{2} e^{3} - 5 A b c d e^{2} + 5 A c^{2} d^{2} e + B a b e^{3} - 3 B a c d e^{2} - 2 B b^{2} d e^{2} + 8 B b c d^{2} e - 7 B c^{2} d^{3}\right ) \log {\left (d + e x \right )}}{e^{8}} \]
B*c**3*x**5/(5*e**3) + x**4*(A*c**3/(4*e**3) + 3*B*b*c**2/(4*e**3) - 3*B*c **3*d/(4*e**4)) + x**3*(A*b*c**2/e**3 - A*c**3*d/e**4 + B*a*c**2/e**3 + B* b**2*c/e**3 - 3*B*b*c**2*d/e**4 + 2*B*c**3*d**2/e**5) + x**2*(3*A*a*c**2/( 2*e**3) + 3*A*b**2*c/(2*e**3) - 9*A*b*c**2*d/(2*e**4) + 3*A*c**3*d**2/e**5 + 3*B*a*b*c/e**3 - 9*B*a*c**2*d/(2*e**4) + B*b**3/(2*e**3) - 9*B*b**2*c*d /(2*e**4) + 9*B*b*c**2*d**2/e**5 - 5*B*c**3*d**3/e**6) + x*(6*A*a*b*c/e**3 - 9*A*a*c**2*d/e**4 + A*b**3/e**3 - 9*A*b**2*c*d/e**4 + 18*A*b*c**2*d**2/ e**5 - 10*A*c**3*d**3/e**6 + 3*B*a**2*c/e**3 + 3*B*a*b**2/e**3 - 18*B*a*b* c*d/e**4 + 18*B*a*c**2*d**2/e**5 - 3*B*b**3*d/e**4 + 18*B*b**2*c*d**2/e**5 - 30*B*b*c**2*d**3/e**6 + 15*B*c**3*d**4/e**7) + (-A*a**3*e**7 - 3*A*a**2 *b*d*e**6 + 9*A*a**2*c*d**2*e**5 + 9*A*a*b**2*d**2*e**5 - 30*A*a*b*c*d**3* e**4 + 21*A*a*c**2*d**4*e**3 - 5*A*b**3*d**3*e**4 + 21*A*b**2*c*d**4*e**3 - 27*A*b*c**2*d**5*e**2 + 11*A*c**3*d**6*e - B*a**3*d*e**6 + 9*B*a**2*b*d* *2*e**5 - 15*B*a**2*c*d**3*e**4 - 15*B*a*b**2*d**3*e**4 + 42*B*a*b*c*d**4* e**3 - 27*B*a*c**2*d**5*e**2 + 7*B*b**3*d**4*e**3 - 27*B*b**2*c*d**5*e**2 + 33*B*b*c**2*d**6*e - 13*B*c**3*d**7 + x*(-6*A*a**2*b*e**7 + 12*A*a**2*c* d*e**6 + 12*A*a*b**2*d*e**6 - 36*A*a*b*c*d**2*e**5 + 24*A*a*c**2*d**3*e**4 - 6*A*b**3*d**2*e**5 + 24*A*b**2*c*d**3*e**4 - 30*A*b*c**2*d**4*e**3 + 12 *A*c**3*d**5*e**2 - 2*B*a**3*e**7 + 12*B*a**2*b*d*e**6 - 18*B*a**2*c*d**2* e**5 - 18*B*a*b**2*d**2*e**5 + 48*B*a*b*c*d**3*e**4 - 30*B*a*c**2*d**4*...
Time = 0.20 (sec) , antiderivative size = 861, normalized size of antiderivative = 1.62 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx=-\frac {13 \, B c^{3} d^{7} + A a^{3} e^{7} - 11 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 27 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{5} e^{2} - 7 \, {\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} + 5 \, {\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} - 9 \, {\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} + 2 \, {\left (7 \, B c^{3} d^{6} e - 6 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 15 \, {\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} + 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} - 6 \, {\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}{2 \, {\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} e^{8}\right )}} + \frac {4 \, B c^{3} e^{4} x^{5} - 5 \, {\left (3 \, B c^{3} d e^{3} - {\left (3 \, B b c^{2} + A c^{3}\right )} e^{4}\right )} x^{4} + 20 \, {\left (2 \, B c^{3} d^{2} e^{2} - {\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^{4}\right )} x^{3} - 10 \, {\left (10 \, B c^{3} d^{3} e - 6 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{2} + 9 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d e^{3} - {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} e^{4}\right )} x^{2} + 20 \, {\left (15 \, B c^{3} d^{4} - 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e + 18 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} e^{2} - 3 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d e^{3} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} e^{4}\right )} x}{20 \, e^{7}} - \frac {3 \, {\left (7 \, B c^{3} d^{5} - 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e + 10 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} e^{2} - 2 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{2} e^{3} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d e^{4} - {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e^{5}\right )} \log \left (e x + d\right )}{e^{8}} \]
-1/2*(13*B*c^3*d^7 + A*a^3*e^7 - 11*(3*B*b*c^2 + A*c^3)*d^6*e + 27*(B*b^2* c + (B*a + A*b)*c^2)*d^5*e^2 - 7*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)* c)*d^4*e^3 + 5*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^3*e^4 - 9*(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^6*e - 6*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 15*(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*(3*B*a* b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^2*e^5 - 6*(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^10*x^2 + 2*d*e^9*x + d^2*e^8) + 1/20*(4*B*c^3*e^4*x^5 - 5*(3*B*c^3*d*e^3 - (3*B*b*c^2 + A*c^3)*e^4)*x^4 + 20*(2*B*c^3*d^2*e^2 - (3*B*b*c^2 + A*c^3)*d*e^3 + (B*b^2*c + (B*a + A*b) *c^2)*e^4)*x^3 - 10*(10*B*c^3*d^3*e - 6*(3*B*b*c^2 + A*c^3)*d^2*e^2 + 9*(B *b^2*c + (B*a + A*b)*c^2)*d*e^3 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2) *c)*e^4)*x^2 + 20*(15*B*c^3*d^4 - 10*(3*B*b*c^2 + A*c^3)*d^3*e + 18*(B*b^2 *c + (B*a + A*b)*c^2)*d^2*e^2 - 3*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2) *c)*d*e^3 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*e^4)*x)/e^7 - 3*(7 *B*c^3*d^5 - 5*(3*B*b*c^2 + A*c^3)*d^4*e + 10*(B*b^2*c + (B*a + A*b)*c^2)* d^3*e^2 - 2*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e^3 + (3*B*a*b ^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*e^4 - (B*a^2*b + A*a*b^2 + A*a^2*c)* e^5)*log(e*x + d)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 1127 vs. \(2 (524) = 1048\).
Time = 0.28 (sec) , antiderivative size = 1127, normalized size of antiderivative = 2.12 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx=-\frac {3 \, {\left (7 \, B c^{3} d^{5} - 15 \, B b c^{2} d^{4} e - 5 \, A c^{3} d^{4} e + 10 \, B b^{2} c d^{3} e^{2} + 10 \, B a c^{2} d^{3} e^{2} + 10 \, A b c^{2} d^{3} e^{2} - 2 \, B b^{3} d^{2} e^{3} - 12 \, B a b c d^{2} e^{3} - 6 \, A b^{2} c d^{2} e^{3} - 6 \, A a c^{2} d^{2} e^{3} + 3 \, B a b^{2} d e^{4} + A b^{3} d e^{4} + 3 \, B a^{2} c d e^{4} + 6 \, A a b c d e^{4} - B a^{2} b e^{5} - A a b^{2} e^{5} - A a^{2} c e^{5}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{8}} - \frac {13 \, B c^{3} d^{7} - 33 \, B b c^{2} d^{6} e - 11 \, A c^{3} d^{6} e + 27 \, B b^{2} c d^{5} e^{2} + 27 \, B a c^{2} d^{5} e^{2} + 27 \, A b c^{2} d^{5} e^{2} - 7 \, B b^{3} d^{4} e^{3} - 42 \, B a b c d^{4} e^{3} - 21 \, A b^{2} c d^{4} e^{3} - 21 \, A a c^{2} d^{4} e^{3} + 15 \, B a b^{2} d^{3} e^{4} + 5 \, A b^{3} d^{3} e^{4} + 15 \, B a^{2} c d^{3} e^{4} + 30 \, A a b c d^{3} e^{4} - 9 \, B a^{2} b d^{2} e^{5} - 9 \, A a b^{2} d^{2} e^{5} - 9 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} + 3 \, A a^{2} b d e^{6} + A a^{3} e^{7} + 2 \, {\left (7 \, B c^{3} d^{6} e - 18 \, B b c^{2} d^{5} e^{2} - 6 \, A c^{3} d^{5} e^{2} + 15 \, B b^{2} c d^{4} e^{3} + 15 \, B a c^{2} d^{4} e^{3} + 15 \, 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} + 9 \, B a b^{2} d^{2} e^{5} + 3 \, A b^{3} d^{2} e^{5} + 9 \, B a^{2} c d^{2} e^{5} + 18 \, A a b c d^{2} e^{5} - 6 \, B a^{2} b d e^{6} - 6 \, A a b^{2} d e^{6} - 6 \, A a^{2} c d e^{6} + B a^{3} e^{7} + 3 \, A a^{2} b e^{7}\right )} x}{2 \, {\left (e x + d\right )}^{2} e^{8}} + \frac {4 \, B c^{3} e^{12} x^{5} - 15 \, B c^{3} d e^{11} x^{4} + 15 \, B b c^{2} e^{12} x^{4} + 5 \, A c^{3} e^{12} x^{4} + 40 \, B c^{3} d^{2} e^{10} x^{3} - 60 \, B b c^{2} d e^{11} x^{3} - 20 \, A c^{3} d e^{11} x^{3} + 20 \, B b^{2} c e^{12} x^{3} + 20 \, B a c^{2} e^{12} x^{3} + 20 \, A b c^{2} e^{12} x^{3} - 100 \, B c^{3} d^{3} e^{9} x^{2} + 180 \, B b c^{2} d^{2} e^{10} x^{2} + 60 \, A c^{3} d^{2} e^{10} x^{2} - 90 \, B b^{2} c d e^{11} x^{2} - 90 \, B a c^{2} d e^{11} x^{2} - 90 \, A b c^{2} d e^{11} x^{2} + 10 \, B b^{3} e^{12} x^{2} + 60 \, B a b c e^{12} x^{2} + 30 \, A b^{2} c e^{12} x^{2} + 30 \, A a c^{2} e^{12} x^{2} + 300 \, B c^{3} d^{4} e^{8} x - 600 \, B b c^{2} d^{3} e^{9} x - 200 \, A c^{3} d^{3} e^{9} x + 360 \, B b^{2} c d^{2} e^{10} x + 360 \, B a c^{2} d^{2} e^{10} x + 360 \, A b c^{2} d^{2} e^{10} x - 60 \, B b^{3} d e^{11} x - 360 \, B a b c d e^{11} x - 180 \, A b^{2} c d e^{11} x - 180 \, A a c^{2} d e^{11} x + 60 \, B a b^{2} e^{12} x + 20 \, A b^{3} e^{12} x + 60 \, B a^{2} c e^{12} x + 120 \, A a b c e^{12} x}{20 \, e^{15}} \]
-3*(7*B*c^3*d^5 - 15*B*b*c^2*d^4*e - 5*A*c^3*d^4*e + 10*B*b^2*c*d^3*e^2 + 10*B*a*c^2*d^3*e^2 + 10*A*b*c^2*d^3*e^2 - 2*B*b^3*d^2*e^3 - 12*B*a*b*c*d^2 *e^3 - 6*A*b^2*c*d^2*e^3 - 6*A*a*c^2*d^2*e^3 + 3*B*a*b^2*d*e^4 + A*b^3*d*e ^4 + 3*B*a^2*c*d*e^4 + 6*A*a*b*c*d*e^4 - B*a^2*b*e^5 - A*a*b^2*e^5 - A*a^2 *c*e^5)*log(abs(e*x + d))/e^8 - 1/2*(13*B*c^3*d^7 - 33*B*b*c^2*d^6*e - 11* A*c^3*d^6*e + 27*B*b^2*c*d^5*e^2 + 27*B*a*c^2*d^5*e^2 + 27*A*b*c^2*d^5*e^2 - 7*B*b^3*d^4*e^3 - 42*B*a*b*c*d^4*e^3 - 21*A*b^2*c*d^4*e^3 - 21*A*a*c^2* d^4*e^3 + 15*B*a*b^2*d^3*e^4 + 5*A*b^3*d^3*e^4 + 15*B*a^2*c*d^3*e^4 + 30*A *a*b*c*d^3*e^4 - 9*B*a^2*b*d^2*e^5 - 9*A*a*b^2*d^2*e^5 - 9*A*a^2*c*d^2*e^5 + B*a^3*d*e^6 + 3*A*a^2*b*d*e^6 + A*a^3*e^7 + 2*(7*B*c^3*d^6*e - 18*B*b*c ^2*d^5*e^2 - 6*A*c^3*d^5*e^2 + 15*B*b^2*c*d^4*e^3 + 15*B*a*c^2*d^4*e^3 + 1 5*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 + 9*B*a*b^2*d^2*e^5 + 3*A*b^3*d^2*e^5 + 9*B*a^2*c *d^2*e^5 + 18*A*a*b*c*d^2*e^5 - 6*B*a^2*b*d*e^6 - 6*A*a*b^2*d*e^6 - 6*A*a^ 2*c*d*e^6 + B*a^3*e^7 + 3*A*a^2*b*e^7)*x)/((e*x + d)^2*e^8) + 1/20*(4*B*c^ 3*e^12*x^5 - 15*B*c^3*d*e^11*x^4 + 15*B*b*c^2*e^12*x^4 + 5*A*c^3*e^12*x^4 + 40*B*c^3*d^2*e^10*x^3 - 60*B*b*c^2*d*e^11*x^3 - 20*A*c^3*d*e^11*x^3 + 20 *B*b^2*c*e^12*x^3 + 20*B*a*c^2*e^12*x^3 + 20*A*b*c^2*e^12*x^3 - 100*B*c^3* d^3*e^9*x^2 + 180*B*b*c^2*d^2*e^10*x^2 + 60*A*c^3*d^2*e^10*x^2 - 90*B*b^2* c*d*e^11*x^2 - 90*B*a*c^2*d*e^11*x^2 - 90*A*b*c^2*d*e^11*x^2 + 10*B*b^3...
Time = 11.34 (sec) , antiderivative size = 1297, normalized size of antiderivative = 2.44 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx=x\,\left (\frac {3\,B\,c\,a^2+3\,B\,a\,b^2+6\,A\,c\,a\,b+A\,b^3}{e^3}+\frac {3\,d^2\,\left (\frac {3\,d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e}-\frac {3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2}{e^3}+\frac {3\,B\,c^3\,d^2}{e^5}\right )}{e^2}-\frac {3\,d\,\left (\frac {B\,b^3+3\,A\,b^2\,c+6\,B\,a\,b\,c+3\,A\,a\,c^2}{e^3}-\frac {3\,d^2\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e^2}+\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e}-\frac {3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2}{e^3}+\frac {3\,B\,c^3\,d^2}{e^5}\right )}{e}-\frac {B\,c^3\,d^3}{e^6}\right )}{e}-\frac {d^3\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e^3}\right )-\frac {\frac {B\,a^3\,d\,e^6+A\,a^3\,e^7-9\,B\,a^2\,b\,d^2\,e^5+3\,A\,a^2\,b\,d\,e^6+15\,B\,a^2\,c\,d^3\,e^4-9\,A\,a^2\,c\,d^2\,e^5+15\,B\,a\,b^2\,d^3\,e^4-9\,A\,a\,b^2\,d^2\,e^5-42\,B\,a\,b\,c\,d^4\,e^3+30\,A\,a\,b\,c\,d^3\,e^4+27\,B\,a\,c^2\,d^5\,e^2-21\,A\,a\,c^2\,d^4\,e^3-7\,B\,b^3\,d^4\,e^3+5\,A\,b^3\,d^3\,e^4+27\,B\,b^2\,c\,d^5\,e^2-21\,A\,b^2\,c\,d^4\,e^3-33\,B\,b\,c^2\,d^6\,e+27\,A\,b\,c^2\,d^5\,e^2+13\,B\,c^3\,d^7-11\,A\,c^3\,d^6\,e}{2\,e}+x\,\left (B\,a^3\,e^6-6\,B\,a^2\,b\,d\,e^5+3\,A\,a^2\,b\,e^6+9\,B\,a^2\,c\,d^2\,e^4-6\,A\,a^2\,c\,d\,e^5+9\,B\,a\,b^2\,d^2\,e^4-6\,A\,a\,b^2\,d\,e^5-24\,B\,a\,b\,c\,d^3\,e^3+18\,A\,a\,b\,c\,d^2\,e^4+15\,B\,a\,c^2\,d^4\,e^2-12\,A\,a\,c^2\,d^3\,e^3-4\,B\,b^3\,d^3\,e^3+3\,A\,b^3\,d^2\,e^4+15\,B\,b^2\,c\,d^4\,e^2-12\,A\,b^2\,c\,d^3\,e^3-18\,B\,b\,c^2\,d^5\,e+15\,A\,b\,c^2\,d^4\,e^2+7\,B\,c^3\,d^6-6\,A\,c^3\,d^5\,e\right )}{d^2\,e^7+2\,d\,e^8\,x+e^9\,x^2}-x^3\,\left (\frac {d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e}-\frac {3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2}{3\,e^3}+\frac {B\,c^3\,d^2}{e^5}\right )+x^4\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{4\,e^3}-\frac {3\,B\,c^3\,d}{4\,e^4}\right )+x^2\,\left (\frac {B\,b^3+3\,A\,b^2\,c+6\,B\,a\,b\,c+3\,A\,a\,c^2}{2\,e^3}-\frac {3\,d^2\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{2\,e^2}+\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e}-\frac {3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2}{e^3}+\frac {3\,B\,c^3\,d^2}{e^5}\right )}{2\,e}-\frac {B\,c^3\,d^3}{2\,e^6}\right )+\frac {\ln \left (d+e\,x\right )\,\left (3\,B\,a^2\,b\,e^5-9\,B\,a^2\,c\,d\,e^4+3\,A\,a^2\,c\,e^5-9\,B\,a\,b^2\,d\,e^4+3\,A\,a\,b^2\,e^5+36\,B\,a\,b\,c\,d^2\,e^3-18\,A\,a\,b\,c\,d\,e^4-30\,B\,a\,c^2\,d^3\,e^2+18\,A\,a\,c^2\,d^2\,e^3+6\,B\,b^3\,d^2\,e^3-3\,A\,b^3\,d\,e^4-30\,B\,b^2\,c\,d^3\,e^2+18\,A\,b^2\,c\,d^2\,e^3+45\,B\,b\,c^2\,d^4\,e-30\,A\,b\,c^2\,d^3\,e^2-21\,B\,c^3\,d^5+15\,A\,c^3\,d^4\,e\right )}{e^8}+\frac {B\,c^3\,x^5}{5\,e^3} \]
x*((A*b^3 + 3*B*a*b^2 + 3*B*a^2*c + 6*A*a*b*c)/e^3 + (3*d^2*((3*d*((A*c^3 + 3*B*b*c^2)/e^3 - (3*B*c^3*d)/e^4))/e - (3*A*b*c^2 + 3*B*a*c^2 + 3*B*b^2* c)/e^3 + (3*B*c^3*d^2)/e^5))/e^2 - (3*d*((B*b^3 + 3*A*a*c^2 + 3*A*b^2*c + 6*B*a*b*c)/e^3 - (3*d^2*((A*c^3 + 3*B*b*c^2)/e^3 - (3*B*c^3*d)/e^4))/e^2 + (3*d*((3*d*((A*c^3 + 3*B*b*c^2)/e^3 - (3*B*c^3*d)/e^4))/e - (3*A*b*c^2 + 3*B*a*c^2 + 3*B*b^2*c)/e^3 + (3*B*c^3*d^2)/e^5))/e - (B*c^3*d^3)/e^6))/e - (d^3*((A*c^3 + 3*B*b*c^2)/e^3 - (3*B*c^3*d)/e^4))/e^3) - ((A*a^3*e^7 + 13 *B*c^3*d^7 + B*a^3*d*e^6 - 11*A*c^3*d^6*e + 5*A*b^3*d^3*e^4 - 7*B*b^3*d^4* e^3 - 9*A*a*b^2*d^2*e^5 - 21*A*a*c^2*d^4*e^3 - 9*A*a^2*c*d^2*e^5 + 15*B*a* b^2*d^3*e^4 - 9*B*a^2*b*d^2*e^5 + 27*A*b*c^2*d^5*e^2 - 21*A*b^2*c*d^4*e^3 + 27*B*a*c^2*d^5*e^2 + 15*B*a^2*c*d^3*e^4 + 27*B*b^2*c*d^5*e^2 + 3*A*a^2*b *d*e^6 - 33*B*b*c^2*d^6*e + 30*A*a*b*c*d^3*e^4 - 42*B*a*b*c*d^4*e^3)/(2*e) + x*(B*a^3*e^6 + 7*B*c^3*d^6 + 3*A*a^2*b*e^6 - 6*A*c^3*d^5*e + 3*A*b^3*d^ 2*e^4 - 4*B*b^3*d^3*e^3 - 12*A*a*c^2*d^3*e^3 + 9*B*a*b^2*d^2*e^4 + 15*A*b* c^2*d^4*e^2 - 12*A*b^2*c*d^3*e^3 + 15*B*a*c^2*d^4*e^2 + 9*B*a^2*c*d^2*e^4 + 15*B*b^2*c*d^4*e^2 - 6*A*a*b^2*d*e^5 - 6*A*a^2*c*d*e^5 - 6*B*a^2*b*d*e^5 - 18*B*b*c^2*d^5*e + 18*A*a*b*c*d^2*e^4 - 24*B*a*b*c*d^3*e^3))/(d^2*e^7 + e^9*x^2 + 2*d*e^8*x) - x^3*((d*((A*c^3 + 3*B*b*c^2)/e^3 - (3*B*c^3*d)/e^4 ))/e - (3*A*b*c^2 + 3*B*a*c^2 + 3*B*b^2*c)/(3*e^3) + (B*c^3*d^2)/e^5) + x^ 4*((A*c^3 + 3*B*b*c^2)/(4*e^3) - (3*B*c^3*d)/(4*e^4)) + x^2*((B*b^3 + 3...