Integrand size = 24, antiderivative size = 484 \[ \int (A+B x) (d+e x)^m \left (b x+c x^2\right )^3 \, dx=-\frac {d^3 (B d-A e) (c d-b e)^3 (d+e x)^{1+m}}{e^8 (1+m)}+\frac {d^2 (c d-b e)^2 (B d (7 c d-4 b e)-3 A e (2 c d-b e)) (d+e x)^{2+m}}{e^8 (2+m)}+\frac {3 d (c d-b e) \left (A e \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )-B d \left (7 c^2 d^2-8 b c d e+2 b^2 e^2\right )\right ) (d+e x)^{3+m}}{e^8 (3+m)}+\frac {\left (B d \left (35 c^3 d^3-60 b c^2 d^2 e+30 b^2 c d e^2-4 b^3 e^3\right )-A e \left (20 c^3 d^3-30 b c^2 d^2 e+12 b^2 c d e^2-b^3 e^3\right )\right ) (d+e x)^{4+m}}{e^8 (4+m)}+\frac {\left (3 A c e \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )-B \left (35 c^3 d^3-45 b c^2 d^2 e+15 b^2 c d e^2-b^3 e^3\right )\right ) (d+e x)^{5+m}}{e^8 (5+m)}-\frac {3 c \left (A c e (2 c d-b e)-B \left (7 c^2 d^2-6 b c d e+b^2 e^2\right )\right ) (d+e x)^{6+m}}{e^8 (6+m)}-\frac {c^2 (7 B c d-3 b B e-A c e) (d+e x)^{7+m}}{e^8 (7+m)}+\frac {B c^3 (d+e x)^{8+m}}{e^8 (8+m)} \]
-d^3*(-A*e+B*d)*(-b*e+c*d)^3*(e*x+d)^(1+m)/e^8/(1+m)+d^2*(-b*e+c*d)^2*(B*d *(-4*b*e+7*c*d)-3*A*e*(-b*e+2*c*d))*(e*x+d)^(2+m)/e^8/(2+m)+3*d*(-b*e+c*d) *(A*e*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)-B*d*(2*b^2*e^2-8*b*c*d*e+7*c^2*d^2))*( e*x+d)^(3+m)/e^8/(3+m)+(B*d*(-4*b^3*e^3+30*b^2*c*d*e^2-60*b*c^2*d^2*e+35*c ^3*d^3)-A*e*(-b^3*e^3+12*b^2*c*d*e^2-30*b*c^2*d^2*e+20*c^3*d^3))*(e*x+d)^( 4+m)/e^8/(4+m)+(3*A*c*e*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)-B*(-b^3*e^3+15*b^2*c *d*e^2-45*b*c^2*d^2*e+35*c^3*d^3))*(e*x+d)^(5+m)/e^8/(5+m)-3*c*(A*c*e*(-b* e+2*c*d)-B*(b^2*e^2-6*b*c*d*e+7*c^2*d^2))*(e*x+d)^(6+m)/e^8/(6+m)-c^2*(-A* c*e-3*B*b*e+7*B*c*d)*(e*x+d)^(7+m)/e^8/(7+m)+B*c^3*(e*x+d)^(8+m)/e^8/(8+m)
Time = 0.89 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.08 \[ \int (A+B x) (d+e x)^m \left (b x+c x^2\right )^3 \, dx=\frac {(d+e x)^{1+m} \left (A e \left (\frac {d^3 (c d-b e)^3}{1+m}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)}{2+m}+\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^2}{3+m}-\frac {(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^3}{4+m}+\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^4}{5+m}-\frac {3 c^2 (2 c d-b e) (d+e x)^5}{6+m}+\frac {c^3 (d+e x)^6}{7+m}\right )+B \left (-\frac {d^4 (c d-b e)^3}{1+m}+\frac {d^3 (7 c d-4 b e) (c d-b e)^2 (d+e x)}{2+m}-\frac {3 d^2 (c d-b e) \left (7 c^2 d^2-8 b c d e+2 b^2 e^2\right ) (d+e x)^2}{3+m}+\frac {d \left (35 c^3 d^3-60 b c^2 d^2 e+30 b^2 c d e^2-4 b^3 e^3\right ) (d+e x)^3}{4+m}-\frac {\left (35 c^3 d^3-45 b c^2 d^2 e+15 b^2 c d e^2-b^3 e^3\right ) (d+e x)^4}{5+m}+\frac {3 c \left (7 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^5}{6+m}-\frac {c^2 (7 c d-3 b e) (d+e x)^6}{7+m}+\frac {c^3 (d+e x)^7}{8+m}\right )\right )}{e^8} \]
((d + e*x)^(1 + m)*(A*e*((d^3*(c*d - b*e)^3)/(1 + m) - (3*d^2*(c*d - b*e)^ 2*(2*c*d - b*e)*(d + e*x))/(2 + m) + (3*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d *e + b^2*e^2)*(d + e*x)^2)/(3 + m) - ((2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d *e + b^2*e^2)*(d + e*x)^3)/(4 + m) + (3*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2 )*(d + e*x)^4)/(5 + m) - (3*c^2*(2*c*d - b*e)*(d + e*x)^5)/(6 + m) + (c^3* (d + e*x)^6)/(7 + m)) + B*(-((d^4*(c*d - b*e)^3)/(1 + m)) + (d^3*(7*c*d - 4*b*e)*(c*d - b*e)^2*(d + e*x))/(2 + m) - (3*d^2*(c*d - b*e)*(7*c^2*d^2 - 8*b*c*d*e + 2*b^2*e^2)*(d + e*x)^2)/(3 + m) + (d*(35*c^3*d^3 - 60*b*c^2*d^ 2*e + 30*b^2*c*d*e^2 - 4*b^3*e^3)*(d + e*x)^3)/(4 + m) - ((35*c^3*d^3 - 45 *b*c^2*d^2*e + 15*b^2*c*d*e^2 - b^3*e^3)*(d + e*x)^4)/(5 + m) + (3*c*(7*c^ 2*d^2 - 6*b*c*d*e + b^2*e^2)*(d + e*x)^5)/(6 + m) - (c^2*(7*c*d - 3*b*e)*( d + e*x)^6)/(7 + m) + (c^3*(d + e*x)^7)/(8 + m))))/e^8
Time = 0.75 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, 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.
| \(\displaystyle \int (A+B x) \left (b x+c x^2\right )^3 (d+e x)^m \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {3 d (c d-b e) (d+e x)^{m+2} \left (A e \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-B d \left (2 b^2 e^2-8 b c d e+7 c^2 d^2\right )\right )}{e^7}+\frac {3 c (d+e x)^{m+5} \left (B \left (b^2 e^2-6 b c d e+7 c^2 d^2\right )-A c e (2 c d-b e)\right )}{e^7}+\frac {(d+e x)^{m+3} \left (B d \left (-4 b^3 e^3+30 b^2 c d e^2-60 b c^2 d^2 e+35 c^3 d^3\right )-A e \left (-b^3 e^3+12 b^2 c d e^2-30 b c^2 d^2 e+20 c^3 d^3\right )\right )}{e^7}+\frac {(d+e x)^{m+4} \left (3 A c e \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-B \left (-b^3 e^3+15 b^2 c d e^2-45 b c^2 d^2 e+35 c^3 d^3\right )\right )}{e^7}+\frac {c^2 (d+e x)^{m+6} (A c e+3 b B e-7 B c d)}{e^7}-\frac {d^3 (B d-A e) (c d-b e)^3 (d+e x)^m}{e^7}+\frac {d^2 (c d-b e)^2 (d+e x)^{m+1} (B d (7 c d-4 b e)-3 A e (2 c d-b e))}{e^7}+\frac {B c^3 (d+e x)^{m+7}}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 d (c d-b e) (d+e x)^{m+3} \left (A e \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-B d \left (2 b^2 e^2-8 b c d e+7 c^2 d^2\right )\right )}{e^8 (m+3)}-\frac {3 c (d+e x)^{m+6} \left (A c e (2 c d-b e)-B \left (b^2 e^2-6 b c d e+7 c^2 d^2\right )\right )}{e^8 (m+6)}+\frac {(d+e x)^{m+4} \left (B d \left (-4 b^3 e^3+30 b^2 c d e^2-60 b c^2 d^2 e+35 c^3 d^3\right )-A e \left (-b^3 e^3+12 b^2 c d e^2-30 b c^2 d^2 e+20 c^3 d^3\right )\right )}{e^8 (m+4)}+\frac {(d+e x)^{m+5} \left (3 A c e \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-B \left (-b^3 e^3+15 b^2 c d e^2-45 b c^2 d^2 e+35 c^3 d^3\right )\right )}{e^8 (m+5)}-\frac {c^2 (d+e x)^{m+7} (-A c e-3 b B e+7 B c d)}{e^8 (m+7)}-\frac {d^3 (B d-A e) (c d-b e)^3 (d+e x)^{m+1}}{e^8 (m+1)}+\frac {d^2 (c d-b e)^2 (d+e x)^{m+2} (B d (7 c d-4 b e)-3 A e (2 c d-b e))}{e^8 (m+2)}+\frac {B c^3 (d+e x)^{m+8}}{e^8 (m+8)}\) |
-((d^3*(B*d - A*e)*(c*d - b*e)^3*(d + e*x)^(1 + m))/(e^8*(1 + m))) + (d^2* (c*d - b*e)^2*(B*d*(7*c*d - 4*b*e) - 3*A*e*(2*c*d - b*e))*(d + e*x)^(2 + m ))/(e^8*(2 + m)) + (3*d*(c*d - b*e)*(A*e*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2) - B*d*(7*c^2*d^2 - 8*b*c*d*e + 2*b^2*e^2))*(d + e*x)^(3 + m))/(e^8*(3 + m )) + ((B*d*(35*c^3*d^3 - 60*b*c^2*d^2*e + 30*b^2*c*d*e^2 - 4*b^3*e^3) - A* e*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 - b^3*e^3))*(d + e*x)^(4 + m))/(e^8*(4 + m)) + ((3*A*c*e*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2) - B*(35*c ^3*d^3 - 45*b*c^2*d^2*e + 15*b^2*c*d*e^2 - b^3*e^3))*(d + e*x)^(5 + m))/(e ^8*(5 + m)) - (3*c*(A*c*e*(2*c*d - b*e) - B*(7*c^2*d^2 - 6*b*c*d*e + b^2*e ^2))*(d + e*x)^(6 + m))/(e^8*(6 + m)) - (c^2*(7*B*c*d - 3*b*B*e - A*c*e)*( d + e*x)^(7 + m))/(e^8*(7 + m)) + (B*c^3*(d + e*x)^(8 + m))/(e^8*(8 + m))
3.12.27.3.1 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. \(2223\) vs. \(2(484)=968\).
Time = 0.45 (sec) , antiderivative size = 2224, normalized size of antiderivative = 4.60
| method | result | size |
| norman | \(\text {Expression too large to display}\) | \(2224\) |
| gosper | \(\text {Expression too large to display}\) | \(4138\) |
| risch | \(\text {Expression too large to display}\) | \(4678\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(6636\) |
B*c^3/(8+m)*x^8*exp(m*ln(e*x+d))+(3*A*b^2*c*e^3*m^3+3*A*b*c^2*d*e^2*m^3+B* b^3*e^3*m^3+3*B*b^2*c*d*e^2*m^3+63*A*b^2*c*e^3*m^2+45*A*b*c^2*d*e^2*m^2-6* A*c^3*d^2*e*m^2+21*B*b^3*e^3*m^2+45*B*b^2*c*d*e^2*m^2-18*B*b*c^2*d^2*e*m^2 +438*A*b^2*c*e^3*m+168*A*b*c^2*d*e^2*m-48*A*c^3*d^2*e*m+146*B*b^3*e^3*m+16 8*B*b^2*c*d*e^2*m-144*B*b*c^2*d^2*e*m+42*B*c^3*d^3*m+1008*A*b^2*c*e^3+336* B*b^3*e^3)/e^3/(m^4+26*m^3+251*m^2+1066*m+1680)*x^5*exp(m*ln(e*x+d))+(A*b^ 3*e^4*m^4+3*A*b^2*c*d*e^3*m^4+B*b^3*d*e^3*m^4+26*A*b^3*e^4*m^3+63*A*b^2*c* d*e^3*m^3-15*A*b*c^2*d^2*e^2*m^3+21*B*b^3*d*e^3*m^3-15*B*b^2*c*d^2*e^2*m^3 +251*A*b^3*e^4*m^2+438*A*b^2*c*d*e^3*m^2-225*A*b*c^2*d^2*e^2*m^2+30*A*c^3* d^3*e*m^2+146*B*b^3*d*e^3*m^2-225*B*b^2*c*d^2*e^2*m^2+90*B*b*c^2*d^3*e*m^2 +1066*A*b^3*e^4*m+1008*A*b^2*c*d*e^3*m-840*A*b*c^2*d^2*e^2*m+240*A*c^3*d^3 *e*m+336*B*b^3*d*e^3*m-840*B*b^2*c*d^2*e^2*m+720*B*b*c^2*d^3*e*m-210*B*c^3 *d^4*m+1680*A*b^3*e^4)/e^4/(m^5+30*m^4+355*m^3+2070*m^2+5944*m+6720)*x^4*e xp(m*ln(e*x+d))+(A*c*e*m+3*B*b*e*m+B*c*d*m+8*A*c*e+24*B*b*e)*c^2/e/(m^2+15 *m+56)*x^7*exp(m*ln(e*x+d))+(3*A*b*c*e^2*m^2+A*c^2*d*e*m^2+3*B*b^2*e^2*m^2 +3*B*b*c*d*e*m^2+45*A*b*c*e^2*m+8*A*c^2*d*e*m+45*B*b^2*e^2*m+24*B*b*c*d*e* m-7*B*c^2*d^2*m+168*A*b*c*e^2+168*B*b^2*e^2)*c/e^2/(m^3+21*m^2+146*m+336)* x^6*exp(m*ln(e*x+d))+m*d*(A*b^3*e^4*m^4+26*A*b^3*e^4*m^3-12*A*b^2*c*d*e^3* m^3-4*B*b^3*d*e^3*m^3+251*A*b^3*e^4*m^2-252*A*b^2*c*d*e^3*m^2+60*A*b*c^2*d ^2*e^2*m^2-84*B*b^3*d*e^3*m^2+60*B*b^2*c*d^2*e^2*m^2+1066*A*b^3*e^4*m-1...
Leaf count of result is larger than twice the leaf count of optimal. 3254 vs. \(2 (484) = 968\).
Time = 0.34 (sec) , antiderivative size = 3254, normalized size of antiderivative = 6.72 \[ \int (A+B x) (d+e x)^m \left (b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
-(6*A*b^3*d^4*e^4*m^4 + 5040*B*c^3*d^8 + 10080*A*b^3*d^4*e^4 - 5760*(3*B*b *c^2 + A*c^3)*d^7*e + 20160*(B*b^2*c + A*b*c^2)*d^6*e^2 - 8064*(B*b^3 + 3* A*b^2*c)*d^5*e^3 - (B*c^3*e^8*m^7 + 28*B*c^3*e^8*m^6 + 322*B*c^3*e^8*m^5 + 1960*B*c^3*e^8*m^4 + 6769*B*c^3*e^8*m^3 + 13132*B*c^3*e^8*m^2 + 13068*B*c ^3*e^8*m + 5040*B*c^3*e^8)*x^8 - (5760*(3*B*b*c^2 + A*c^3)*e^8 + (B*c^3*d* e^7 + (3*B*b*c^2 + A*c^3)*e^8)*m^7 + (21*B*c^3*d*e^7 + 29*(3*B*b*c^2 + A*c ^3)*e^8)*m^6 + 7*(25*B*c^3*d*e^7 + 49*(3*B*b*c^2 + A*c^3)*e^8)*m^5 + 35*(2 1*B*c^3*d*e^7 + 61*(3*B*b*c^2 + A*c^3)*e^8)*m^4 + 56*(29*B*c^3*d*e^7 + 134 *(3*B*b*c^2 + A*c^3)*e^8)*m^3 + 28*(63*B*c^3*d*e^7 + 527*(3*B*b*c^2 + A*c^ 3)*e^8)*m^2 + 144*(5*B*c^3*d*e^7 + 103*(3*B*b*c^2 + A*c^3)*e^8)*m)*x^7 - ( 20160*(B*b^2*c + A*b*c^2)*e^8 + ((3*B*b*c^2 + A*c^3)*d*e^7 + 3*(B*b^2*c + A*b*c^2)*e^8)*m^7 - (7*B*c^3*d^2*e^6 - 23*(3*B*b*c^2 + A*c^3)*d*e^7 - 90*( B*b^2*c + A*b*c^2)*e^8)*m^6 - (105*B*c^3*d^2*e^6 - 205*(3*B*b*c^2 + A*c^3) *d*e^7 - 1098*(B*b^2*c + A*b*c^2)*e^8)*m^5 - 5*(119*B*c^3*d^2*e^6 - 181*(3 *B*b*c^2 + A*c^3)*d*e^7 - 1404*(B*b^2*c + A*b*c^2)*e^8)*m^4 - (1575*B*c^3* d^2*e^6 - 2074*(3*B*b*c^2 + A*c^3)*d*e^7 - 25227*(B*b^2*c + A*b*c^2)*e^8)* m^3 - 2*(959*B*c^3*d^2*e^6 - 1156*(3*B*b*c^2 + A*c^3)*d*e^7 - 25245*(B*b^2 *c + A*b*c^2)*e^8)*m^2 - 24*(35*B*c^3*d^2*e^6 - 40*(3*B*b*c^2 + A*c^3)*d*e ^7 - 2143*(B*b^2*c + A*b*c^2)*e^8)*m)*x^6 - (8064*(B*b^3 + 3*A*b^2*c)*e^8 + (3*(B*b^2*c + A*b*c^2)*d*e^7 + (B*b^3 + 3*A*b^2*c)*e^8)*m^7 - (6*(3*B...
Leaf count of result is larger than twice the leaf count of optimal. 59148 vs. \(2 (473) = 946\).
Time = 11.27 (sec) , antiderivative size = 59148, normalized size of antiderivative = 122.21 \[ \int (A+B x) (d+e x)^m \left (b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
Piecewise((d**m*(A*b**3*x**4/4 + 3*A*b**2*c*x**5/5 + A*b*c**2*x**6/2 + A*c **3*x**7/7 + B*b**3*x**5/5 + B*b**2*c*x**6/2 + 3*B*b*c**2*x**7/7 + B*c**3* x**8/8), Eq(e, 0)), (-3*A*b**3*d**3*e**4/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 21*A*b**3*d** 2*e**5*x/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700* d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e* *14*x**6 + 420*e**15*x**7) - 63*A*b**3*d*e**6*x**2/(420*d**7*e**8 + 2940*d **6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e** 12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 105 *A*b**3*e**7*x**3/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 12*A*b**2*c*d**4*e**3/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700 *d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x* *7) - 84*A*b**2*c*d**3*e**4*x/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d** 5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e **13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 252*A*b**2*c*d**2*e**5*x **2/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4* e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**1...
Leaf count of result is larger than twice the leaf count of optimal. 1527 vs. \(2 (484) = 968\).
Time = 0.24 (sec) , antiderivative size = 1527, normalized size of antiderivative = 3.15 \[ \int (A+B x) (d+e x)^m \left (b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*d^2*e^2*x^2 + 6*d^3*e*m*x - 6*d^4)*(e*x + d)^m*A*b^3/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + ((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^5*x^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*e^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*e^3*x ^3 + 12*(m^2 + m)*d^3*e^2*x^2 - 24*d^4*e*m*x + 24*d^5)*(e*x + d)^m*B*b^3/( (m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5) + 3*((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^5*x^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*e^4*x^4 - 4*( m^3 + 3*m^2 + 2*m)*d^2*e^3*x^3 + 12*(m^2 + m)*d^3*e^2*x^2 - 24*d^4*e*m*x + 24*d^5)*(e*x + d)^m*A*b^2*c/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 1 20)*e^5) + 3*((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^6*x^6 + (m ^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d*e^5*x^5 - 5*(m^4 + 6*m^3 + 11*m^2 + 6*m)*d^2*e^4*x^4 + 20*(m^3 + 3*m^2 + 2*m)*d^3*e^3*x^3 - 60*(m^2 + m)*d^4 *e^2*x^2 + 120*d^5*e*m*x - 120*d^6)*(e*x + d)^m*B*b^2*c/((m^6 + 21*m^5 + 1 75*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*e^6) + 3*((m^5 + 15*m^4 + 85*m ^3 + 225*m^2 + 274*m + 120)*e^6*x^6 + (m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24 *m)*d*e^5*x^5 - 5*(m^4 + 6*m^3 + 11*m^2 + 6*m)*d^2*e^4*x^4 + 20*(m^3 + 3*m ^2 + 2*m)*d^3*e^3*x^3 - 60*(m^2 + m)*d^4*e^2*x^2 + 120*d^5*e*m*x - 120*d^6 )*(e*x + d)^m*A*b*c^2/((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764 *m + 720)*e^6) + 3*((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*e^7*x^7 + (m^6 + 15*m^5 + 85*m^4 + 225*m^3 + 274*m^2 + 120*m)*d*...
Leaf count of result is larger than twice the leaf count of optimal. 6661 vs. \(2 (484) = 968\).
Time = 0.32 (sec) , antiderivative size = 6661, normalized size of antiderivative = 13.76 \[ \int (A+B x) (d+e x)^m \left (b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
((e*x + d)^m*B*c^3*e^8*m^7*x^8 + (e*x + d)^m*B*c^3*d*e^7*m^7*x^7 + 3*(e*x + d)^m*B*b*c^2*e^8*m^7*x^7 + (e*x + d)^m*A*c^3*e^8*m^7*x^7 + 28*(e*x + d)^ m*B*c^3*e^8*m^6*x^8 + 3*(e*x + d)^m*B*b*c^2*d*e^7*m^7*x^6 + (e*x + d)^m*A* c^3*d*e^7*m^7*x^6 + 3*(e*x + d)^m*B*b^2*c*e^8*m^7*x^6 + 3*(e*x + d)^m*A*b* c^2*e^8*m^7*x^6 + 21*(e*x + d)^m*B*c^3*d*e^7*m^6*x^7 + 87*(e*x + d)^m*B*b* c^2*e^8*m^6*x^7 + 29*(e*x + d)^m*A*c^3*e^8*m^6*x^7 + 322*(e*x + d)^m*B*c^3 *e^8*m^5*x^8 + 3*(e*x + d)^m*B*b^2*c*d*e^7*m^7*x^5 + 3*(e*x + d)^m*A*b*c^2 *d*e^7*m^7*x^5 + (e*x + d)^m*B*b^3*e^8*m^7*x^5 + 3*(e*x + d)^m*A*b^2*c*e^8 *m^7*x^5 - 7*(e*x + d)^m*B*c^3*d^2*e^6*m^6*x^6 + 69*(e*x + d)^m*B*b*c^2*d* e^7*m^6*x^6 + 23*(e*x + d)^m*A*c^3*d*e^7*m^6*x^6 + 90*(e*x + d)^m*B*b^2*c* e^8*m^6*x^6 + 90*(e*x + d)^m*A*b*c^2*e^8*m^6*x^6 + 175*(e*x + d)^m*B*c^3*d *e^7*m^5*x^7 + 1029*(e*x + d)^m*B*b*c^2*e^8*m^5*x^7 + 343*(e*x + d)^m*A*c^ 3*e^8*m^5*x^7 + 1960*(e*x + d)^m*B*c^3*e^8*m^4*x^8 + (e*x + d)^m*B*b^3*d*e ^7*m^7*x^4 + 3*(e*x + d)^m*A*b^2*c*d*e^7*m^7*x^4 + (e*x + d)^m*A*b^3*e^8*m ^7*x^4 - 18*(e*x + d)^m*B*b*c^2*d^2*e^6*m^6*x^5 - 6*(e*x + d)^m*A*c^3*d^2* e^6*m^6*x^5 + 75*(e*x + d)^m*B*b^2*c*d*e^7*m^6*x^5 + 75*(e*x + d)^m*A*b*c^ 2*d*e^7*m^6*x^5 + 31*(e*x + d)^m*B*b^3*e^8*m^6*x^5 + 93*(e*x + d)^m*A*b^2* c*e^8*m^6*x^5 - 105*(e*x + d)^m*B*c^3*d^2*e^6*m^5*x^6 + 615*(e*x + d)^m*B* b*c^2*d*e^7*m^5*x^6 + 205*(e*x + d)^m*A*c^3*d*e^7*m^5*x^6 + 1098*(e*x + d) ^m*B*b^2*c*e^8*m^5*x^6 + 1098*(e*x + d)^m*A*b*c^2*e^8*m^5*x^6 + 735*(e*...
Time = 12.46 (sec) , antiderivative size = 2500, normalized size of antiderivative = 5.17 \[ \int (A+B x) (d+e x)^m \left (b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
(B*c^3*x^8*(d + e*x)^m*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^ 5 + 28*m^6 + m^7 + 5040))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) - ((d + e*x)^m*(5040*B*c^3*d^8 - 5760*A*c^3*d^7*e + 10080*A*b^3*d^4*e^4 - 8064*B*b^3*d^5*e^3 + 20160*A*b *c^2*d^6*e^2 - 24192*A*b^2*c*d^5*e^3 + 20160*B*b^2*c*d^6*e^2 + 6396*A*b^3* d^4*e^4*m - 3504*B*b^3*d^5*e^3*m + 1506*A*b^3*d^4*e^4*m^2 + 156*A*b^3*d^4* e^4*m^3 + 6*A*b^3*d^4*e^4*m^4 - 504*B*b^3*d^5*e^3*m^2 - 24*B*b^3*d^5*e^3*m ^3 - 17280*B*b*c^2*d^7*e - 720*A*c^3*d^7*e*m + 360*A*b*c^2*d^6*e^2*m^2 - 1 512*A*b^2*c*d^5*e^3*m^2 - 72*A*b^2*c*d^5*e^3*m^3 + 360*B*b^2*c*d^6*e^2*m^2 - 2160*B*b*c^2*d^7*e*m + 5400*A*b*c^2*d^6*e^2*m - 10512*A*b^2*c*d^5*e^3*m + 5400*B*b^2*c*d^6*e^2*m))/(e^8*(109584*m + 118124*m^2 + 67284*m^3 + 2244 9*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320)) + (x^5*(d + e*x)^m*(50 *m + 35*m^2 + 10*m^3 + m^4 + 24)*(336*B*b^3*e^3 + 1008*A*b^2*c*e^3 + 146*B *b^3*e^3*m + 42*B*c^3*d^3*m + 21*B*b^3*e^3*m^2 + B*b^3*e^3*m^3 + 63*A*b^2* c*e^3*m^2 + 3*A*b^2*c*e^3*m^3 - 6*A*c^3*d^2*e*m^2 + 438*A*b^2*c*e^3*m - 48 *A*c^3*d^2*e*m + 168*A*b*c^2*d*e^2*m - 144*B*b*c^2*d^2*e*m + 168*B*b^2*c*d *e^2*m + 45*A*b*c^2*d*e^2*m^2 + 3*A*b*c^2*d*e^2*m^3 - 18*B*b*c^2*d^2*e*m^2 + 45*B*b^2*c*d*e^2*m^2 + 3*B*b^2*c*d*e^2*m^3))/(e^3*(109584*m + 118124*m^ 2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320)) + (x^4*(d + e*x)^m*(11*m + 6*m^2 + m^3 + 6)*(1680*A*b^3*e^4 + 1066*A*b^3*...