Integrand size = 25, antiderivative size = 521 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx=-\frac {\left (B (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-A c e \left (10 c^2 d^2+3 b^2 e^2-3 c e (4 b d-a e)\right )\right ) x}{e^7}-\frac {c \left (A c e (4 c d-3 b e)-B \left (10 c^2 d^2+3 b^2 e^2-3 c e (4 b d-a e)\right )\right ) x^2}{2 e^6}-\frac {c^2 (4 B c d-3 b B e-A c e) x^3}{3 e^5}+\frac {B c^3 x^4}{4 e^4}+\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3}{3 e^8 (d+e x)^3}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right )}{2 e^8 (d+e x)^2}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )-A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right )}{e^8 (d+e x)}-\frac {\left (A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )\right ) \log (d+e x)}{e^8} \]
-(B*(-b*e+2*c*d)*(10*c^2*d^2+b^2*e^2-2*c*e*(-3*a*e+5*b*d))-A*c*e*(10*c^2*d ^2+3*b^2*e^2-3*c*e*(-a*e+4*b*d)))*x/e^7-1/2*c*(A*c*e*(-3*b*e+4*c*d)-B*(10* c^2*d^2+3*b^2*e^2-3*c*e*(-a*e+4*b*d)))*x^2/e^6-1/3*c^2*(-A*c*e-3*B*b*e+4*B *c*d)*x^3/e^5+1/4*B*c^3*x^4/e^4+1/3*(-A*e+B*d)*(a*e^2-b*d*e+c*d^2)^3/e^8/( e*x+d)^3+1/2*(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)^2+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)))/e^8/(e *x+d)-(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)))*ln(e*x+d)/e^8
Time = 0.17 (sec) , antiderivative size = 488, normalized size of antiderivative = 0.94 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {12 e \left (A c e \left (10 c^2 d^2+3 b^2 e^2+3 c e (-4 b d+a e)\right )-B (2 c d-b e) \left (10 c^2 d^2+b^2 e^2+2 c e (-5 b d+3 a e)\right )\right ) x+6 c e^2 \left (A c e (-4 c d+3 b e)+B \left (10 c^2 d^2+3 b^2 e^2+3 c e (-4 b d+a e)\right )\right ) x^2+4 c^2 e^3 (-4 B c d+3 b B e+A c e) x^3+3 B c^3 e^4 x^4+\frac {4 (B d-A e) \left (c d^2+e (-b d+a e)\right )^3}{(d+e x)^3}-\frac {6 \left (c d^2+e (-b d+a e)\right )^2 \left (7 B c d^2+B e (-4 b d+a e)+3 A e (-2 c d+b e)\right )}{(d+e x)^2}+\frac {36 \left (c d^2+e (-b d+a e)\right ) \left (-A e \left (5 c^2 d^2+b^2 e^2+c e (-5 b d+a e)\right )+B \left (7 c^2 d^3+b e^2 (2 b d-a e)+c d e (-8 b d+3 a e)\right )\right )}{d+e x}+12 \left (A e (-2 c d+b e) \left (10 c^2 d^2+b^2 e^2+2 c e (-5 b d+3 a e)\right )+B \left (35 c^3 d^4+30 c^2 d^2 e (-2 b d+a e)+b^2 e^3 (-4 b d+3 a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )\right ) \log (d+e x)}{12 e^8} \]
(12*e*(A*c*e*(10*c^2*d^2 + 3*b^2*e^2 + 3*c*e*(-4*b*d + a*e)) - B*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 + 2*c*e*(-5*b*d + 3*a*e)))*x + 6*c*e^2*(A*c*e*( -4*c*d + 3*b*e) + B*(10*c^2*d^2 + 3*b^2*e^2 + 3*c*e*(-4*b*d + a*e)))*x^2 + 4*c^2*e^3*(-4*B*c*d + 3*b*B*e + A*c*e)*x^3 + 3*B*c^3*e^4*x^4 + (4*(B*d - A*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x)^3 - (6*(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)^ 2 + (36*(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) + 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)))*Log[d + e*x])/(1 2*e^8)
Time = 1.25 (sec) , antiderivative size = 519, 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.
| \(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )-A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7 (d+e x)}+\frac {A c e \left (-3 c e (4 b d-a e)+3 b^2 e^2+10 c^2 d^2\right )-B (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7}+\frac {c x \left (B \left (-3 c e (4 b d-a e)+3 b^2 e^2+10 c^2 d^2\right )-A c e (4 c d-3 b e)\right )}{e^6}+\frac {3 \left (a e^2-b d e+c d^2\right ) \left (A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )-B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )\right )}{e^7 (d+e x)^2}+\frac {(A e-B d) \left (a e^2-b d e+c d^2\right )^3}{e^7 (d+e x)^4}+\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{e^7 (d+e x)^3}+\frac {c^2 x^2 (A c e+3 b B e-4 B c d)}{e^5}+\frac {B c^3 x^3}{e^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\log (d+e x) \left (A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )\right )}{e^8}-\frac {x \left (B (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-A c e \left (-3 c e (4 b d-a e)+3 b^2 e^2+10 c^2 d^2\right )\right )}{e^7}-\frac {c x^2 \left (A c e (4 c d-3 b e)-B \left (-3 c e (4 b d-a e)+3 b^2 e^2+10 c^2 d^2\right )\right )}{2 e^6}+\frac {3 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{e^8 (d+e x)}-\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{2 e^8 (d+e x)^2}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8 (d+e x)^3}-\frac {c^2 x^3 (-A c e-3 b B e+4 B c d)}{3 e^5}+\frac {B c^3 x^4}{4 e^4}\) |
-(((B*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e)) - A*c*e *(10*c^2*d^2 + 3*b^2*e^2 - 3*c*e*(4*b*d - a*e)))*x)/e^7) - (c*(A*c*e*(4*c* d - 3*b*e) - B*(10*c^2*d^2 + 3*b^2*e^2 - 3*c*e*(4*b*d - a*e)))*x^2)/(2*e^6 ) - (c^2*(4*B*c*d - 3*b*B*e - A*c*e)*x^3)/(3*e^5) + (B*c^3*x^4)/(4*e^4) + ((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^3)/(3*e^8*(d + e*x)^3) - ((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)))/(2*e ^8*(d + e*x)^2) + (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))))/(e^8*(d + e*x)) - ((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)))*Log[d + e*x ])/e^8
3.24.42.3.1 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]
Leaf count of result is larger than twice the leaf count of optimal. \(1044\) vs. \(2(511)=1022\).
Time = 0.36 (sec) , antiderivative size = 1045, normalized size of antiderivative = 2.01
| method | result | size |
| norman | \(\text {Expression too large to display}\) | \(1045\) |
| default | \(\text {Expression too large to display}\) | \(1089\) |
| risch | \(\text {Expression too large to display}\) | \(1168\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(2116\) |
(-1/6*(2*A*a^3*e^7+3*A*a^2*b*d*e^6+6*A*a^2*c*d^2*e^5+6*A*a*b^2*d^2*e^5-66* A*a*b*c*d^3*e^4+132*A*a*c^2*d^4*e^3-11*A*b^3*d^3*e^4+132*A*b^2*c*d^4*e^3-3 30*A*b*c^2*d^5*e^2+220*A*c^3*d^6*e+B*a^3*d*e^6+6*B*a^2*b*d^2*e^5-33*B*a^2* c*d^3*e^4-33*B*a*b^2*d^3*e^4+264*B*a*b*c*d^4*e^3-330*B*a*c^2*d^5*e^2+44*B* b^3*d^4*e^3-330*B*b^2*c*d^5*e^2+660*B*b*c^2*d^6*e-385*B*c^3*d^7)/e^8+1/4*( 12*A*a*c^2*e^3+12*A*b^2*c*e^3-30*A*b*c^2*d*e^2+20*A*c^3*d^2*e+24*B*a*b*c*e ^3-30*B*a*c^2*d*e^2+4*B*b^3*e^3-30*B*b^2*c*d*e^2+60*B*b*c^2*d^2*e-35*B*c^3 *d^3)/e^4*x^4-3*(A*a^2*c*e^5+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^3-30*A*b*c^2*d^3*e^2+20*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+24*B*a*b*c*d^2*e^3-30*B*a*c^2*d^3*e^2 +4*B*b^3*d^2*e^3-30*B*b^2*c*d^3*e^2+60*B*b*c^2*d^4*e-35*B*c^3*d^5)/e^6*x^2 -1/2*(3*A*a^2*b*e^6+6*A*a^2*c*d*e^5+6*A*a*b^2*d*e^5-54*A*a*b*c*d^2*e^4+108 *A*a*c^2*d^3*e^3-9*A*b^3*d^2*e^4+108*A*b^2*c*d^3*e^3-270*A*b*c^2*d^4*e^2+1 80*A*c^3*d^5*e+B*a^3*e^6+6*B*a^2*b*d*e^5-27*B*a^2*c*d^2*e^4-27*B*a*b^2*d^2 *e^4+216*B*a*b*c*d^3*e^3-270*B*a*c^2*d^4*e^2+36*B*b^3*d^3*e^3-270*B*b^2*c* d^4*e^2+540*B*b*c^2*d^5*e-315*B*c^3*d^6)/e^7*x+1/4*B*c^3*x^7/e+1/4*c*(6*A* b*c*e^2-4*A*c^2*d*e+6*B*a*c*e^2+6*B*b^2*e^2-12*B*b*c*d*e+7*B*c^2*d^2)/e^3* x^5+1/12*c^2*(4*A*c*e+12*B*b*e-7*B*c*d)/e^2*x^6)/(e*x+d)^3+1/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+30*A*b*c^2*d^2*e^2-20*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+30*B*a*c^2*d^2*e...
Leaf count of result is larger than twice the leaf count of optimal. 1369 vs. \(2 (510) = 1020\).
Time = 0.34 (sec) , antiderivative size = 1369, normalized size of antiderivative = 2.63 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx=\text {Too large to display} \]
1/12*(3*B*c^3*e^7*x^7 + 214*B*c^3*d^7 - 4*A*a^3*e^7 - 148*(3*B*b*c^2 + A*c ^3)*d^6*e + 282*(B*b^2*c + (B*a + A*b)*c^2)*d^5*e^2 - 52*(B*b^3 + 3*A*a*c^ 2 + 3*(2*B*a*b + A*b^2)*c)*d^4*e^3 + 22*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2* A*a*b)*c)*d^3*e^4 - 12*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^2*e^5 - 2*(B*a^3 + 3*A*a^2*b)*d*e^6 - (7*B*c^3*d*e^6 - 4*(3*B*b*c^2 + A*c^3)*e^7)*x^6 + 3*(7* B*c^3*d^2*e^5 - 4*(3*B*b*c^2 + A*c^3)*d*e^6 + 6*(B*b^2*c + (B*a + A*b)*c^2 )*e^7)*x^5 - 3*(35*B*c^3*d^3*e^4 - 20*(3*B*b*c^2 + A*c^3)*d^2*e^5 + 30*(B* b^2*c + (B*a + A*b)*c^2)*d*e^6 - 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2 )*c)*e^7)*x^4 - 2*(278*B*c^3*d^4*e^3 - 146*(3*B*b*c^2 + A*c^3)*d^3*e^4 + 1 89*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^5 - 18*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a* b + A*b^2)*c)*d*e^6)*x^3 - 6*(68*B*c^3*d^5*e^2 - 26*(3*B*b*c^2 + A*c^3)*d^ 4*e^3 + 9*(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 - 6*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c )*d*e^6 + 6*(B*a^2*b + A*a*b^2 + A*a^2*c)*e^7)*x^2 + 6*(37*B*c^3*d^6*e - 3 4*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 81*(B*b^2*c + (B*a + A*b)*c^2)*d^4*e^3 - 1 8*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^3*e^4 + 9*(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 + 12*(35*B*c^3*d^7 - 20*(3*B*b*c^2 + A*c^3) *d^6*e + 30*(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...
Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 875, normalized size of antiderivative = 1.68 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {107 \, B c^{3} d^{7} - 2 \, A a^{3} e^{7} - 74 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 141 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{5} e^{2} - 26 \, {\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} + 11 \, {\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} - 6 \, {\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} + 18 \, {\left (7 \, B c^{3} d^{5} e^{2} - 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} + 10 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} e^{4} - 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^{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} + 3 \, {\left (77 \, B c^{3} d^{6} e - 54 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 105 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} e^{3} - 20 \, {\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} + 9 \, {\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}{6 \, {\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} e^{8}\right )}} + \frac {3 \, B c^{3} e^{3} x^{4} - 4 \, {\left (4 \, B c^{3} d e^{2} - {\left (3 \, B b c^{2} + A c^{3}\right )} e^{3}\right )} x^{3} + 6 \, {\left (10 \, B c^{3} d^{2} e - 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{2} + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} e^{3}\right )} x^{2} - 12 \, {\left (20 \, B c^{3} d^{3} - 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e + 12 \, {\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 )} x}{12 \, e^{7}} + \frac {{\left (35 \, B c^{3} d^{4} - 20 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e + 30 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} 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 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 )} \log \left (e x + d\right )}{e^{8}} \]
1/6*(107*B*c^3*d^7 - 2*A*a^3*e^7 - 74*(3*B*b*c^2 + A*c^3)*d^6*e + 141*(B*b ^2*c + (B*a + A*b)*c^2)*d^5*e^2 - 26*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b ^2)*c)*d^4*e^3 + 11*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^3*e^4 - 6*(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 + 18*( 7*B*c^3*d^5*e^2 - 5*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 10*(B*b^2*c + (B*a + A*b )*c^2)*d^3*e^4 - 2*(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 + 3*(77*B*c^3*d^6*e - 54*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 105 *(B*b^2*c + (B*a + A*b)*c^2)*d^4*e^3 - 20*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^3*e^4 + 9*(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 ^11*x^3 + 3*d*e^10*x^2 + 3*d^2*e^9*x + d^3*e^8) + 1/12*(3*B*c^3*e^3*x^4 - 4*(4*B*c^3*d*e^2 - (3*B*b*c^2 + A*c^3)*e^3)*x^3 + 6*(10*B*c^3*d^2*e - 4*(3 *B*b*c^2 + A*c^3)*d*e^2 + 3*(B*b^2*c + (B*a + A*b)*c^2)*e^3)*x^2 - 12*(20* B*c^3*d^3 - 10*(3*B*b*c^2 + A*c^3)*d^2*e + 12*(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)*x)/e^7 + (35*B*c^ 3*d^4 - 20*(3*B*b*c^2 + A*c^3)*d^3*e + 30*(B*b^2*c + (B*a + A*b)*c^2)*d^2* e^2 - 4*(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)*log(e*x + d)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 1100 vs. \(2 (510) = 1020\).
Time = 0.27 (sec) , antiderivative size = 1100, normalized size of antiderivative = 2.11 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {{\left (35 \, B c^{3} d^{4} - 60 \, B b c^{2} d^{3} e - 20 \, A c^{3} d^{3} e + 30 \, B b^{2} c d^{2} e^{2} + 30 \, B a c^{2} d^{2} e^{2} + 30 \, A b c^{2} d^{2} e^{2} - 4 \, B b^{3} d e^{3} - 24 \, B a b c d e^{3} - 12 \, A b^{2} c d e^{3} - 12 \, A a c^{2} d e^{3} + 3 \, B a b^{2} e^{4} + A b^{3} e^{4} + 3 \, B a^{2} c e^{4} + 6 \, A a b c e^{4}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{8}} + \frac {107 \, B c^{3} d^{7} - 222 \, B b c^{2} d^{6} e - 74 \, A c^{3} d^{6} e + 141 \, B b^{2} c d^{5} e^{2} + 141 \, B a c^{2} d^{5} e^{2} + 141 \, A b c^{2} d^{5} e^{2} - 26 \, B b^{3} d^{4} e^{3} - 156 \, B a b c d^{4} e^{3} - 78 \, A b^{2} c d^{4} e^{3} - 78 \, A a c^{2} d^{4} e^{3} + 33 \, B a b^{2} d^{3} e^{4} + 11 \, A b^{3} d^{3} e^{4} + 33 \, B a^{2} c d^{3} e^{4} + 66 \, A a b c d^{3} e^{4} - 6 \, B a^{2} b d^{2} e^{5} - 6 \, A a b^{2} d^{2} e^{5} - 6 \, A a^{2} c d^{2} e^{5} - B a^{3} d e^{6} - 3 \, A a^{2} b d e^{6} - 2 \, A a^{3} e^{7} + 18 \, {\left (7 \, B c^{3} d^{5} e^{2} - 15 \, B b c^{2} d^{4} e^{3} - 5 \, A c^{3} d^{4} e^{3} + 10 \, B b^{2} c d^{3} e^{4} + 10 \, B a c^{2} d^{3} e^{4} + 10 \, A b c^{2} d^{3} e^{4} - 2 \, B b^{3} d^{2} e^{5} - 12 \, B a b c d^{2} e^{5} - 6 \, A b^{2} c d^{2} e^{5} - 6 \, 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} - B a^{2} b e^{7} - A a b^{2} e^{7} - A a^{2} c e^{7}\right )} x^{2} + 3 \, {\left (77 \, B c^{3} d^{6} e - 162 \, B b c^{2} d^{5} e^{2} - 54 \, A c^{3} d^{5} e^{2} + 105 \, B b^{2} c d^{4} e^{3} + 105 \, B a c^{2} d^{4} e^{3} + 105 \, A b c^{2} d^{4} e^{3} - 20 \, B b^{3} d^{3} e^{4} - 120 \, B a b c d^{3} e^{4} - 60 \, A b^{2} c d^{3} e^{4} - 60 \, A a c^{2} d^{3} e^{4} + 27 \, B a b^{2} d^{2} e^{5} + 9 \, A b^{3} d^{2} e^{5} + 27 \, B a^{2} c d^{2} e^{5} + 54 \, 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}{6 \, {\left (e x + d\right )}^{3} e^{8}} + \frac {3 \, B c^{3} e^{12} x^{4} - 16 \, B c^{3} d e^{11} x^{3} + 12 \, B b c^{2} e^{12} x^{3} + 4 \, A c^{3} e^{12} x^{3} + 60 \, B c^{3} d^{2} e^{10} x^{2} - 72 \, B b c^{2} d e^{11} x^{2} - 24 \, A c^{3} d e^{11} x^{2} + 18 \, B b^{2} c e^{12} x^{2} + 18 \, B a c^{2} e^{12} x^{2} + 18 \, A b c^{2} e^{12} x^{2} - 240 \, B c^{3} d^{3} e^{9} x + 360 \, B b c^{2} d^{2} e^{10} x + 120 \, A c^{3} d^{2} e^{10} x - 144 \, B b^{2} c d e^{11} x - 144 \, B a c^{2} d e^{11} x - 144 \, A b c^{2} d e^{11} x + 12 \, B b^{3} e^{12} x + 72 \, B a b c e^{12} x + 36 \, A b^{2} c e^{12} x + 36 \, A a c^{2} e^{12} x}{12 \, e^{16}} \]
(35*B*c^3*d^4 - 60*B*b*c^2*d^3*e - 20*A*c^3*d^3*e + 30*B*b^2*c*d^2*e^2 + 3 0*B*a*c^2*d^2*e^2 + 30*A*b*c^2*d^2*e^2 - 4*B*b^3*d*e^3 - 24*B*a*b*c*d*e^3 - 12*A*b^2*c*d*e^3 - 12*A*a*c^2*d*e^3 + 3*B*a*b^2*e^4 + A*b^3*e^4 + 3*B*a^ 2*c*e^4 + 6*A*a*b*c*e^4)*log(abs(e*x + d))/e^8 + 1/6*(107*B*c^3*d^7 - 222* B*b*c^2*d^6*e - 74*A*c^3*d^6*e + 141*B*b^2*c*d^5*e^2 + 141*B*a*c^2*d^5*e^2 + 141*A*b*c^2*d^5*e^2 - 26*B*b^3*d^4*e^3 - 156*B*a*b*c*d^4*e^3 - 78*A*b^2 *c*d^4*e^3 - 78*A*a*c^2*d^4*e^3 + 33*B*a*b^2*d^3*e^4 + 11*A*b^3*d^3*e^4 + 33*B*a^2*c*d^3*e^4 + 66*A*a*b*c*d^3*e^4 - 6*B*a^2*b*d^2*e^5 - 6*A*a*b^2*d^ 2*e^5 - 6*A*a^2*c*d^2*e^5 - B*a^3*d*e^6 - 3*A*a^2*b*d*e^6 - 2*A*a^3*e^7 + 18*(7*B*c^3*d^5*e^2 - 15*B*b*c^2*d^4*e^3 - 5*A*c^3*d^4*e^3 + 10*B*b^2*c*d^ 3*e^4 + 10*B*a*c^2*d^3*e^4 + 10*A*b*c^2*d^3*e^4 - 2*B*b^3*d^2*e^5 - 12*B*a *b*c*d^2*e^5 - 6*A*b^2*c*d^2*e^5 - 6*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 - B*a^2*b*e^7 - A*a*b^2*e^7 - A*a^2*c*e^7)*x^2 + 3*(77*B*c^3*d^6*e - 162*B*b*c^2*d^5*e^2 - 54*A*c^3*d ^5*e^2 + 105*B*b^2*c*d^4*e^3 + 105*B*a*c^2*d^4*e^3 + 105*A*b*c^2*d^4*e^3 - 20*B*b^3*d^3*e^4 - 120*B*a*b*c*d^3*e^4 - 60*A*b^2*c*d^3*e^4 - 60*A*a*c^2* d^3*e^4 + 27*B*a*b^2*d^2*e^5 + 9*A*b^3*d^2*e^5 + 27*B*a^2*c*d^2*e^5 + 54*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)^3*e^8) + 1/12*(3*B*c^3*e^12*x^4 - 16 *B*c^3*d*e^11*x^3 + 12*B*b*c^2*e^12*x^3 + 4*A*c^3*e^12*x^3 + 60*B*c^3*d...
Time = 11.30 (sec) , antiderivative size = 1151, normalized size of antiderivative = 2.21 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx=x\,\left (\frac {B\,b^3+3\,A\,b^2\,c+6\,B\,a\,b\,c+3\,A\,a\,c^2}{e^4}-\frac {6\,d^2\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^4}-\frac {4\,B\,c^3\,d}{e^5}\right )}{e^2}+\frac {4\,d\,\left (\frac {4\,d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^4}-\frac {4\,B\,c^3\,d}{e^5}\right )}{e}-\frac {3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2}{e^4}+\frac {6\,B\,c^3\,d^2}{e^6}\right )}{e}-\frac {4\,B\,c^3\,d^3}{e^7}\right )-x^2\,\left (\frac {2\,d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^4}-\frac {4\,B\,c^3\,d}{e^5}\right )}{e}-\frac {3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2}{2\,e^4}+\frac {3\,B\,c^3\,d^2}{e^6}\right )-\frac {\frac {B\,a^3\,d\,e^6+2\,A\,a^3\,e^7+6\,B\,a^2\,b\,d^2\,e^5+3\,A\,a^2\,b\,d\,e^6-33\,B\,a^2\,c\,d^3\,e^4+6\,A\,a^2\,c\,d^2\,e^5-33\,B\,a\,b^2\,d^3\,e^4+6\,A\,a\,b^2\,d^2\,e^5+156\,B\,a\,b\,c\,d^4\,e^3-66\,A\,a\,b\,c\,d^3\,e^4-141\,B\,a\,c^2\,d^5\,e^2+78\,A\,a\,c^2\,d^4\,e^3+26\,B\,b^3\,d^4\,e^3-11\,A\,b^3\,d^3\,e^4-141\,B\,b^2\,c\,d^5\,e^2+78\,A\,b^2\,c\,d^4\,e^3+222\,B\,b\,c^2\,d^6\,e-141\,A\,b\,c^2\,d^5\,e^2-107\,B\,c^3\,d^7+74\,A\,c^3\,d^6\,e}{6\,e}+x\,\left (\frac {B\,a^3\,e^6}{2}+3\,B\,a^2\,b\,d\,e^5+\frac {3\,A\,a^2\,b\,e^6}{2}-\frac {27\,B\,a^2\,c\,d^2\,e^4}{2}+3\,A\,a^2\,c\,d\,e^5-\frac {27\,B\,a\,b^2\,d^2\,e^4}{2}+3\,A\,a\,b^2\,d\,e^5+60\,B\,a\,b\,c\,d^3\,e^3-27\,A\,a\,b\,c\,d^2\,e^4-\frac {105\,B\,a\,c^2\,d^4\,e^2}{2}+30\,A\,a\,c^2\,d^3\,e^3+10\,B\,b^3\,d^3\,e^3-\frac {9\,A\,b^3\,d^2\,e^4}{2}-\frac {105\,B\,b^2\,c\,d^4\,e^2}{2}+30\,A\,b^2\,c\,d^3\,e^3+81\,B\,b\,c^2\,d^5\,e-\frac {105\,A\,b\,c^2\,d^4\,e^2}{2}-\frac {77\,B\,c^3\,d^6}{2}+27\,A\,c^3\,d^5\,e\right )+x^2\,\left (3\,B\,a^2\,b\,e^6-9\,B\,a^2\,c\,d\,e^5+3\,A\,a^2\,c\,e^6-9\,B\,a\,b^2\,d\,e^5+3\,A\,a\,b^2\,e^6+36\,B\,a\,b\,c\,d^2\,e^4-18\,A\,a\,b\,c\,d\,e^5-30\,B\,a\,c^2\,d^3\,e^3+18\,A\,a\,c^2\,d^2\,e^4+6\,B\,b^3\,d^2\,e^4-3\,A\,b^3\,d\,e^5-30\,B\,b^2\,c\,d^3\,e^3+18\,A\,b^2\,c\,d^2\,e^4+45\,B\,b\,c^2\,d^4\,e^2-30\,A\,b\,c^2\,d^3\,e^3-21\,B\,c^3\,d^5\,e+15\,A\,c^3\,d^4\,e^2\right )}{d^3\,e^7+3\,d^2\,e^8\,x+3\,d\,e^9\,x^2+e^{10}\,x^3}+x^3\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{3\,e^4}-\frac {4\,B\,c^3\,d}{3\,e^5}\right )+\frac {\ln \left (d+e\,x\right )\,\left (3\,B\,a^2\,c\,e^4+3\,B\,a\,b^2\,e^4-24\,B\,a\,b\,c\,d\,e^3+6\,A\,a\,b\,c\,e^4+30\,B\,a\,c^2\,d^2\,e^2-12\,A\,a\,c^2\,d\,e^3-4\,B\,b^3\,d\,e^3+A\,b^3\,e^4+30\,B\,b^2\,c\,d^2\,e^2-12\,A\,b^2\,c\,d\,e^3-60\,B\,b\,c^2\,d^3\,e+30\,A\,b\,c^2\,d^2\,e^2+35\,B\,c^3\,d^4-20\,A\,c^3\,d^3\,e\right )}{e^8}+\frac {B\,c^3\,x^4}{4\,e^4} \]
x*((B*b^3 + 3*A*a*c^2 + 3*A*b^2*c + 6*B*a*b*c)/e^4 - (6*d^2*((A*c^3 + 3*B* b*c^2)/e^4 - (4*B*c^3*d)/e^5))/e^2 + (4*d*((4*d*((A*c^3 + 3*B*b*c^2)/e^4 - (4*B*c^3*d)/e^5))/e - (3*A*b*c^2 + 3*B*a*c^2 + 3*B*b^2*c)/e^4 + (6*B*c^3* d^2)/e^6))/e - (4*B*c^3*d^3)/e^7) - x^2*((2*d*((A*c^3 + 3*B*b*c^2)/e^4 - ( 4*B*c^3*d)/e^5))/e - (3*A*b*c^2 + 3*B*a*c^2 + 3*B*b^2*c)/(2*e^4) + (3*B*c^ 3*d^2)/e^6) - ((2*A*a^3*e^7 - 107*B*c^3*d^7 + B*a^3*d*e^6 + 74*A*c^3*d^6*e - 11*A*b^3*d^3*e^4 + 26*B*b^3*d^4*e^3 + 6*A*a*b^2*d^2*e^5 + 78*A*a*c^2*d^ 4*e^3 + 6*A*a^2*c*d^2*e^5 - 33*B*a*b^2*d^3*e^4 + 6*B*a^2*b*d^2*e^5 - 141*A *b*c^2*d^5*e^2 + 78*A*b^2*c*d^4*e^3 - 141*B*a*c^2*d^5*e^2 - 33*B*a^2*c*d^3 *e^4 - 141*B*b^2*c*d^5*e^2 + 3*A*a^2*b*d*e^6 + 222*B*b*c^2*d^6*e - 66*A*a* b*c*d^3*e^4 + 156*B*a*b*c*d^4*e^3)/(6*e) + x*((B*a^3*e^6)/2 - (77*B*c^3*d^ 6)/2 + (3*A*a^2*b*e^6)/2 + 27*A*c^3*d^5*e - (9*A*b^3*d^2*e^4)/2 + 10*B*b^3 *d^3*e^3 + 30*A*a*c^2*d^3*e^3 - (27*B*a*b^2*d^2*e^4)/2 - (105*A*b*c^2*d^4* e^2)/2 + 30*A*b^2*c*d^3*e^3 - (105*B*a*c^2*d^4*e^2)/2 - (27*B*a^2*c*d^2*e^ 4)/2 - (105*B*b^2*c*d^4*e^2)/2 + 3*A*a*b^2*d*e^5 + 3*A*a^2*c*d*e^5 + 3*B*a ^2*b*d*e^5 + 81*B*b*c^2*d^5*e - 27*A*a*b*c*d^2*e^4 + 60*B*a*b*c*d^3*e^3) + x^2*(3*A*a*b^2*e^6 + 3*A*a^2*c*e^6 + 3*B*a^2*b*e^6 - 3*A*b^3*d*e^5 - 21*B *c^3*d^5*e + 15*A*c^3*d^4*e^2 + 6*B*b^3*d^2*e^4 + 18*A*a*c^2*d^2*e^4 - 30* A*b*c^2*d^3*e^3 + 18*A*b^2*c*d^2*e^4 - 30*B*a*c^2*d^3*e^3 + 45*B*b*c^2*d^4 *e^2 - 30*B*b^2*c*d^3*e^3 - 9*B*a*b^2*d*e^5 - 9*B*a^2*c*d*e^5 + 36*B*a*...