Integrand size = 24, antiderivative size = 282 \[ \int (A+B x) (d+e x)^m \left (b x+c x^2\right )^2 \, dx=-\frac {d^2 (B d-A e) (c d-b e)^2 (d+e x)^{1+m}}{e^6 (1+m)}+\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) (d+e x)^{2+m}}{e^6 (2+m)}+\frac {\left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{3+m}}{e^6 (3+m)}-\frac {\left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{4+m}}{e^6 (4+m)}-\frac {c (5 B c d-2 b B e-A c e) (d+e x)^{5+m}}{e^6 (5+m)}+\frac {B c^2 (d+e x)^{6+m}}{e^6 (6+m)} \]
-d^2*(-A*e+B*d)*(-b*e+c*d)^2*(e*x+d)^(1+m)/e^6/(1+m)+d*(-b*e+c*d)*(B*d*(-3 *b*e+5*c*d)-2*A*e*(-b*e+2*c*d))*(e*x+d)^(2+m)/e^6/(2+m)+(A*e*(b^2*e^2-6*b* c*d*e+6*c^2*d^2)-B*d*(3*b^2*e^2-12*b*c*d*e+10*c^2*d^2))*(e*x+d)^(3+m)/e^6/ (3+m)-(2*A*c*e*(-b*e+2*c*d)-B*(b^2*e^2-8*b*c*d*e+10*c^2*d^2))*(e*x+d)^(4+m )/e^6/(4+m)-c*(-A*c*e-2*B*b*e+5*B*c*d)*(e*x+d)^(5+m)/e^6/(5+m)+B*c^2*(e*x+ d)^(6+m)/e^6/(6+m)
Time = 0.41 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.10 \[ \int (A+B x) (d+e x)^m \left (b x+c x^2\right )^2 \, dx=\frac {(d+e x)^{1+m} \left (A e \left (\frac {d^2 (c d-b e)^2}{1+m}-\frac {2 d (c d-b e) (2 c d-b e) (d+e x)}{2+m}+\frac {\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^2}{3+m}-\frac {2 c (2 c d-b e) (d+e x)^3}{4+m}+\frac {c^2 (d+e x)^4}{5+m}\right )+B \left (-\frac {d^3 (c d-b e)^2}{1+m}+\frac {d^2 (5 c d-3 b e) (c d-b e) (d+e x)}{2+m}-\frac {d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right ) (d+e x)^2}{3+m}+\frac {\left (10 c^2 d^2-8 b c d e+b^2 e^2\right ) (d+e x)^3}{4+m}-\frac {c (5 c d-2 b e) (d+e x)^4}{5+m}+\frac {c^2 (d+e x)^5}{6+m}\right )\right )}{e^6} \]
((d + e*x)^(1 + m)*(A*e*((d^2*(c*d - b*e)^2)/(1 + m) - (2*d*(c*d - b*e)*(2 *c*d - b*e)*(d + e*x))/(2 + m) + ((6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(d + e *x)^2)/(3 + m) - (2*c*(2*c*d - b*e)*(d + e*x)^3)/(4 + m) + (c^2*(d + e*x)^ 4)/(5 + m)) + B*(-((d^3*(c*d - b*e)^2)/(1 + m)) + (d^2*(5*c*d - 3*b*e)*(c* d - b*e)*(d + e*x))/(2 + m) - (d*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2)*(d + e*x)^2)/(3 + m) + ((10*c^2*d^2 - 8*b*c*d*e + b^2*e^2)*(d + e*x)^3)/(4 + m) - (c*(5*c*d - 2*b*e)*(d + e*x)^4)/(5 + m) + (c^2*(d + e*x)^5)/(6 + m))) )/e^6
Time = 0.50 (sec) , antiderivative size = 282, 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.
| \(\displaystyle \int (A+B x) \left (b x+c x^2\right )^2 (d+e x)^m \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {(d+e x)^{m+2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{e^5}+\frac {(d+e x)^{m+3} \left (B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )-2 A c e (2 c d-b e)\right )}{e^5}-\frac {d^2 (B d-A e) (c d-b e)^2 (d+e x)^m}{e^5}+\frac {d (c d-b e) (d+e x)^{m+1} (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^5}+\frac {c (d+e x)^{m+4} (A c e+2 b B e-5 B c d)}{e^5}+\frac {B c^2 (d+e x)^{m+5}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^{m+3} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{e^6 (m+3)}-\frac {(d+e x)^{m+4} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{e^6 (m+4)}-\frac {d^2 (B d-A e) (c d-b e)^2 (d+e x)^{m+1}}{e^6 (m+1)}+\frac {d (c d-b e) (d+e x)^{m+2} (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6 (m+2)}-\frac {c (d+e x)^{m+5} (-A c e-2 b B e+5 B c d)}{e^6 (m+5)}+\frac {B c^2 (d+e x)^{m+6}}{e^6 (m+6)}\) |
-((d^2*(B*d - A*e)*(c*d - b*e)^2*(d + e*x)^(1 + m))/(e^6*(1 + m))) + (d*(c *d - b*e)*(B*d*(5*c*d - 3*b*e) - 2*A*e*(2*c*d - b*e))*(d + e*x)^(2 + m))/( e^6*(2 + m)) + ((A*e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2))*(d + e*x)^(3 + m))/(e^6*(3 + m)) - ((2*A*c*e*(2* c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2))*(d + e*x)^(4 + m))/(e^6 *(4 + m)) - (c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(5 + m))/(e^6*(5 + m) ) + (B*c^2*(d + e*x)^(6 + m))/(e^6*(6 + m))
3.12.14.3.1 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. \(1029\) vs. \(2(282)=564\).
Time = 0.34 (sec) , antiderivative size = 1030, normalized size of antiderivative = 3.65
| method | result | size |
| norman | \(\text {Expression too large to display}\) | \(1030\) |
| gosper | \(\text {Expression too large to display}\) | \(1616\) |
| risch | \(\text {Expression too large to display}\) | \(1922\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(2903\) |
B*c^2/(6+m)*x^6*exp(m*ln(e*x+d))+(2*A*b*c*e^2*m^2+A*c^2*d*e*m^2+B*b^2*e^2* m^2+2*B*b*c*d*e*m^2+22*A*b*c*e^2*m+6*A*c^2*d*e*m+11*B*b^2*e^2*m+12*B*b*c*d *e*m-5*B*c^2*d^2*m+60*A*b*c*e^2+30*B*b^2*e^2)/e^2/(m^3+15*m^2+74*m+120)*x^ 4*exp(m*ln(e*x+d))+(A*b^2*e^3*m^3+2*A*b*c*d*e^2*m^3+B*b^2*d*e^2*m^3+15*A*b ^2*e^3*m^2+22*A*b*c*d*e^2*m^2-4*A*c^2*d^2*e*m^2+11*B*b^2*d*e^2*m^2-8*B*b*c *d^2*e*m^2+74*A*b^2*e^3*m+60*A*b*c*d*e^2*m-24*A*c^2*d^2*e*m+30*B*b^2*d*e^2 *m-48*B*b*c*d^2*e*m+20*B*c^2*d^3*m+120*A*b^2*e^3)/e^3/(m^4+18*m^3+119*m^2+ 342*m+360)*x^3*exp(m*ln(e*x+d))+(A*c*e*m+2*B*b*e*m+B*c*d*m+6*A*c*e+12*B*b* e)*c/e/(m^2+11*m+30)*x^5*exp(m*ln(e*x+d))+(A*b^2*e^3*m^3+15*A*b^2*e^3*m^2- 6*A*b*c*d*e^2*m^2-3*B*b^2*d*e^2*m^2+74*A*b^2*e^3*m-66*A*b*c*d*e^2*m+12*A*c ^2*d^2*e*m-33*B*b^2*d*e^2*m+24*B*b*c*d^2*e*m+120*A*b^2*e^3-180*A*b*c*d*e^2 +72*A*c^2*d^2*e-90*B*b^2*d*e^2+144*B*b*c*d^2*e-60*B*c^2*d^3)*d/e^4*m/(m^5+ 20*m^4+155*m^3+580*m^2+1044*m+720)*x^2*exp(m*ln(e*x+d))+2*d^3*(A*b^2*e^3*m ^3+15*A*b^2*e^3*m^2-6*A*b*c*d*e^2*m^2-3*B*b^2*d*e^2*m^2+74*A*b^2*e^3*m-66* A*b*c*d*e^2*m+12*A*c^2*d^2*e*m-33*B*b^2*d*e^2*m+24*B*b*c*d^2*e*m+120*A*b^2 *e^3-180*A*b*c*d*e^2+72*A*c^2*d^2*e-90*B*b^2*d*e^2+144*B*b*c*d^2*e-60*B*c^ 2*d^3)/e^6/(m^6+21*m^5+175*m^4+735*m^3+1624*m^2+1764*m+720)*exp(m*ln(e*x+d ))-2/e^5*m*d^2*(A*b^2*e^3*m^3+15*A*b^2*e^3*m^2-6*A*b*c*d*e^2*m^2-3*B*b^2*d *e^2*m^2+74*A*b^2*e^3*m-66*A*b*c*d*e^2*m+12*A*c^2*d^2*e*m-33*B*b^2*d*e^2*m +24*B*b*c*d^2*e*m+120*A*b^2*e^3-180*A*b*c*d*e^2+72*A*c^2*d^2*e-90*B*b^2...
Leaf count of result is larger than twice the leaf count of optimal. 1417 vs. \(2 (283) = 566\).
Time = 0.54 (sec) , antiderivative size = 1417, normalized size of antiderivative = 5.02 \[ \int (A+B x) (d+e x)^m \left (b x+c x^2\right )^2 \, dx=\text {Too large to display} \]
(2*A*b^2*d^3*e^3*m^3 - 120*B*c^2*d^6 + 240*A*b^2*d^3*e^3 + 144*(2*B*b*c + A*c^2)*d^5*e - 180*(B*b^2 + 2*A*b*c)*d^4*e^2 + (B*c^2*e^6*m^5 + 15*B*c^2*e ^6*m^4 + 85*B*c^2*e^6*m^3 + 225*B*c^2*e^6*m^2 + 274*B*c^2*e^6*m + 120*B*c^ 2*e^6)*x^6 + (144*(2*B*b*c + A*c^2)*e^6 + (B*c^2*d*e^5 + (2*B*b*c + A*c^2) *e^6)*m^5 + 2*(5*B*c^2*d*e^5 + 8*(2*B*b*c + A*c^2)*e^6)*m^4 + 5*(7*B*c^2*d *e^5 + 19*(2*B*b*c + A*c^2)*e^6)*m^3 + 10*(5*B*c^2*d*e^5 + 26*(2*B*b*c + A *c^2)*e^6)*m^2 + 12*(2*B*c^2*d*e^5 + 27*(2*B*b*c + A*c^2)*e^6)*m)*x^5 + (1 80*(B*b^2 + 2*A*b*c)*e^6 + ((2*B*b*c + A*c^2)*d*e^5 + (B*b^2 + 2*A*b*c)*e^ 6)*m^5 - (5*B*c^2*d^2*e^4 - 12*(2*B*b*c + A*c^2)*d*e^5 - 17*(B*b^2 + 2*A*b *c)*e^6)*m^4 - (30*B*c^2*d^2*e^4 - 47*(2*B*b*c + A*c^2)*d*e^5 - 107*(B*b^2 + 2*A*b*c)*e^6)*m^3 - (55*B*c^2*d^2*e^4 - 72*(2*B*b*c + A*c^2)*d*e^5 - 30 7*(B*b^2 + 2*A*b*c)*e^6)*m^2 - 6*(5*B*c^2*d^2*e^4 - 6*(2*B*b*c + A*c^2)*d* e^5 - 66*(B*b^2 + 2*A*b*c)*e^6)*m)*x^4 + (240*A*b^2*e^6 + (A*b^2*e^6 + (B* b^2 + 2*A*b*c)*d*e^5)*m^5 + 2*(9*A*b^2*e^6 - 2*(2*B*b*c + A*c^2)*d^2*e^4 + 7*(B*b^2 + 2*A*b*c)*d*e^5)*m^4 + (20*B*c^2*d^3*e^3 + 121*A*b^2*e^6 - 36*( 2*B*b*c + A*c^2)*d^2*e^4 + 65*(B*b^2 + 2*A*b*c)*d*e^5)*m^3 + 4*(15*B*c^2*d ^3*e^3 + 93*A*b^2*e^6 - 20*(2*B*b*c + A*c^2)*d^2*e^4 + 28*(B*b^2 + 2*A*b*c )*d*e^5)*m^2 + 4*(10*B*c^2*d^3*e^3 + 127*A*b^2*e^6 - 12*(2*B*b*c + A*c^2)* d^2*e^4 + 15*(B*b^2 + 2*A*b*c)*d*e^5)*m)*x^3 + 6*(5*A*b^2*d^3*e^3 - (B*b^2 + 2*A*b*c)*d^4*e^2)*m^2 + (A*b^2*d*e^5*m^5 + (16*A*b^2*d*e^5 - 3*(B*b^...
Leaf count of result is larger than twice the leaf count of optimal. 20284 vs. \(2 (267) = 534\).
Time = 3.77 (sec) , antiderivative size = 20284, normalized size of antiderivative = 71.93 \[ \int (A+B x) (d+e x)^m \left (b x+c x^2\right )^2 \, dx=\text {Too large to display} \]
Piecewise((d**m*(A*b**2*x**3/3 + A*b*c*x**4/2 + A*c**2*x**5/5 + B*b**2*x** 4/4 + 2*B*b*c*x**5/5 + B*c**2*x**6/6), Eq(e, 0)), (-2*A*b**2*d**2*e**3/(60 *d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 3 00*d*e**10*x**4 + 60*e**11*x**5) - 10*A*b**2*d*e**4*x/(60*d**5*e**6 + 300* d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 20*A*b**2*e**5*x**2/(60*d**5*e**6 + 300*d**4*e**7*x + 60 0*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 6*A*b*c*d**3*e**2/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 30*A*b*c*d**2*e* *3*x/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9* x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 60*A*b*c*d*e**4*x**2/(60*d**5*e **6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e* *10*x**4 + 60*e**11*x**5) - 60*A*b*c*e**5*x**3/(60*d**5*e**6 + 300*d**4*e* *7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e** 11*x**5) - 12*A*c**2*d**4*e/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e** 8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 60*A*c** 2*d**3*e**2*x/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d **2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 120*A*c**2*d**2*e**3*x **2/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x **3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 120*A*c**2*d*e**4*x**3/(60*d*...
Leaf count of result is larger than twice the leaf count of optimal. 755 vs. \(2 (283) = 566\).
Time = 0.21 (sec) , antiderivative size = 755, normalized size of antiderivative = 2.68 \[ \int (A+B x) (d+e x)^m \left (b x+c x^2\right )^2 \, dx=\frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} A b^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} B b^{2}}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {2 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} A b c}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {2 \, {\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )} {\left (e x + d\right )}^{m} B b c}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{5}} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )} {\left (e x + d\right )}^{m} A c^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{5}} + \frac {{\left ({\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{6} x^{6} + {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d e^{5} x^{5} - 5 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{2} e^{4} x^{4} + 20 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{3} e^{3} x^{3} - 60 \, {\left (m^{2} + m\right )} d^{4} e^{2} x^{2} + 120 \, d^{5} e m x - 120 \, d^{6}\right )} {\left (e x + d\right )}^{m} B c^{2}}{{\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} e^{6}} \]
((m^2 + 3*m + 2)*e^3*x^3 + (m^2 + m)*d*e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e*x + d)^m*A*b^2/((m^3 + 6*m^2 + 11*m + 6)*e^3) + ((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*B*b^2/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + 2*((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*c/((m^4 + 10* m^3 + 35*m^2 + 50*m + 24)*e^4) + 2*((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*c/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5) + ((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*c^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5) + ((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*c^2/((m^6 + 21*m^5 + 175 *m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*e^6)
Leaf count of result is larger than twice the leaf count of optimal. 2823 vs. \(2 (283) = 566\).
Time = 0.29 (sec) , antiderivative size = 2823, normalized size of antiderivative = 10.01 \[ \int (A+B x) (d+e x)^m \left (b x+c x^2\right )^2 \, dx=\text {Too large to display} \]
((e*x + d)^m*B*c^2*e^6*m^5*x^6 + (e*x + d)^m*B*c^2*d*e^5*m^5*x^5 + 2*(e*x + d)^m*B*b*c*e^6*m^5*x^5 + (e*x + d)^m*A*c^2*e^6*m^5*x^5 + 15*(e*x + d)^m* B*c^2*e^6*m^4*x^6 + 2*(e*x + d)^m*B*b*c*d*e^5*m^5*x^4 + (e*x + d)^m*A*c^2* d*e^5*m^5*x^4 + (e*x + d)^m*B*b^2*e^6*m^5*x^4 + 2*(e*x + d)^m*A*b*c*e^6*m^ 5*x^4 + 10*(e*x + d)^m*B*c^2*d*e^5*m^4*x^5 + 32*(e*x + d)^m*B*b*c*e^6*m^4* x^5 + 16*(e*x + d)^m*A*c^2*e^6*m^4*x^5 + 85*(e*x + d)^m*B*c^2*e^6*m^3*x^6 + (e*x + d)^m*B*b^2*d*e^5*m^5*x^3 + 2*(e*x + d)^m*A*b*c*d*e^5*m^5*x^3 + (e *x + d)^m*A*b^2*e^6*m^5*x^3 - 5*(e*x + d)^m*B*c^2*d^2*e^4*m^4*x^4 + 24*(e* x + d)^m*B*b*c*d*e^5*m^4*x^4 + 12*(e*x + d)^m*A*c^2*d*e^5*m^4*x^4 + 17*(e* x + d)^m*B*b^2*e^6*m^4*x^4 + 34*(e*x + d)^m*A*b*c*e^6*m^4*x^4 + 35*(e*x + d)^m*B*c^2*d*e^5*m^3*x^5 + 190*(e*x + d)^m*B*b*c*e^6*m^3*x^5 + 95*(e*x + d )^m*A*c^2*e^6*m^3*x^5 + 225*(e*x + d)^m*B*c^2*e^6*m^2*x^6 + (e*x + d)^m*A* b^2*d*e^5*m^5*x^2 - 8*(e*x + d)^m*B*b*c*d^2*e^4*m^4*x^3 - 4*(e*x + d)^m*A* c^2*d^2*e^4*m^4*x^3 + 14*(e*x + d)^m*B*b^2*d*e^5*m^4*x^3 + 28*(e*x + d)^m* A*b*c*d*e^5*m^4*x^3 + 18*(e*x + d)^m*A*b^2*e^6*m^4*x^3 - 30*(e*x + d)^m*B* c^2*d^2*e^4*m^3*x^4 + 94*(e*x + d)^m*B*b*c*d*e^5*m^3*x^4 + 47*(e*x + d)^m* A*c^2*d*e^5*m^3*x^4 + 107*(e*x + d)^m*B*b^2*e^6*m^3*x^4 + 214*(e*x + d)^m* A*b*c*e^6*m^3*x^4 + 50*(e*x + d)^m*B*c^2*d*e^5*m^2*x^5 + 520*(e*x + d)^m*B *b*c*e^6*m^2*x^5 + 260*(e*x + d)^m*A*c^2*e^6*m^2*x^5 + 274*(e*x + d)^m*B*c ^2*e^6*m*x^6 - 3*(e*x + d)^m*B*b^2*d^2*e^4*m^4*x^2 - 6*(e*x + d)^m*A*b*...
Time = 11.46 (sec) , antiderivative size = 1176, normalized size of antiderivative = 4.17 \[ \int (A+B x) (d+e x)^m \left (b x+c x^2\right )^2 \, dx=\frac {{\left (d+e\,x\right )}^m\,\left (-6\,B\,b^2\,d^4\,e^2\,m^2-66\,B\,b^2\,d^4\,e^2\,m-180\,B\,b^2\,d^4\,e^2+2\,A\,b^2\,d^3\,e^3\,m^3+30\,A\,b^2\,d^3\,e^3\,m^2+148\,A\,b^2\,d^3\,e^3\,m+240\,A\,b^2\,d^3\,e^3+48\,B\,b\,c\,d^5\,e\,m+288\,B\,b\,c\,d^5\,e-12\,A\,b\,c\,d^4\,e^2\,m^2-132\,A\,b\,c\,d^4\,e^2\,m-360\,A\,b\,c\,d^4\,e^2-120\,B\,c^2\,d^6+24\,A\,c^2\,d^5\,e\,m+144\,A\,c^2\,d^5\,e\right )}{e^6\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {x^3\,{\left (d+e\,x\right )}^m\,\left (m^2+3\,m+2\right )\,\left (B\,b^2\,d\,e^2\,m^3+11\,B\,b^2\,d\,e^2\,m^2+30\,B\,b^2\,d\,e^2\,m+A\,b^2\,e^3\,m^3+15\,A\,b^2\,e^3\,m^2+74\,A\,b^2\,e^3\,m+120\,A\,b^2\,e^3-8\,B\,b\,c\,d^2\,e\,m^2-48\,B\,b\,c\,d^2\,e\,m+2\,A\,b\,c\,d\,e^2\,m^3+22\,A\,b\,c\,d\,e^2\,m^2+60\,A\,b\,c\,d\,e^2\,m+20\,B\,c^2\,d^3\,m-4\,A\,c^2\,d^2\,e\,m^2-24\,A\,c^2\,d^2\,e\,m\right )}{e^3\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {x^4\,{\left (d+e\,x\right )}^m\,\left (m^3+6\,m^2+11\,m+6\right )\,\left (B\,b^2\,e^2\,m^2+11\,B\,b^2\,e^2\,m+30\,B\,b^2\,e^2+2\,B\,b\,c\,d\,e\,m^2+12\,B\,b\,c\,d\,e\,m+2\,A\,b\,c\,e^2\,m^2+22\,A\,b\,c\,e^2\,m+60\,A\,b\,c\,e^2-5\,B\,c^2\,d^2\,m+A\,c^2\,d\,e\,m^2+6\,A\,c^2\,d\,e\,m\right )}{e^2\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {B\,c^2\,x^6\,{\left (d+e\,x\right )}^m\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {c\,x^5\,{\left (d+e\,x\right )}^m\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )\,\left (6\,A\,c\,e+12\,B\,b\,e+A\,c\,e\,m+2\,B\,b\,e\,m+B\,c\,d\,m\right )}{e\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}-\frac {2\,d^2\,m\,x\,{\left (d+e\,x\right )}^m\,\left (-3\,B\,b^2\,d\,e^2\,m^2-33\,B\,b^2\,d\,e^2\,m-90\,B\,b^2\,d\,e^2+A\,b^2\,e^3\,m^3+15\,A\,b^2\,e^3\,m^2+74\,A\,b^2\,e^3\,m+120\,A\,b^2\,e^3+24\,B\,b\,c\,d^2\,e\,m+144\,B\,b\,c\,d^2\,e-6\,A\,b\,c\,d\,e^2\,m^2-66\,A\,b\,c\,d\,e^2\,m-180\,A\,b\,c\,d\,e^2-60\,B\,c^2\,d^3+12\,A\,c^2\,d^2\,e\,m+72\,A\,c^2\,d^2\,e\right )}{e^5\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {d\,m\,x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (-3\,B\,b^2\,d\,e^2\,m^2-33\,B\,b^2\,d\,e^2\,m-90\,B\,b^2\,d\,e^2+A\,b^2\,e^3\,m^3+15\,A\,b^2\,e^3\,m^2+74\,A\,b^2\,e^3\,m+120\,A\,b^2\,e^3+24\,B\,b\,c\,d^2\,e\,m+144\,B\,b\,c\,d^2\,e-6\,A\,b\,c\,d\,e^2\,m^2-66\,A\,b\,c\,d\,e^2\,m-180\,A\,b\,c\,d\,e^2-60\,B\,c^2\,d^3+12\,A\,c^2\,d^2\,e\,m+72\,A\,c^2\,d^2\,e\right )}{e^4\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )} \]
((d + e*x)^m*(144*A*c^2*d^5*e - 120*B*c^2*d^6 + 240*A*b^2*d^3*e^3 - 180*B* b^2*d^4*e^2 + 148*A*b^2*d^3*e^3*m - 66*B*b^2*d^4*e^2*m + 288*B*b*c*d^5*e + 30*A*b^2*d^3*e^3*m^2 + 2*A*b^2*d^3*e^3*m^3 - 6*B*b^2*d^4*e^2*m^2 - 360*A* b*c*d^4*e^2 + 24*A*c^2*d^5*e*m - 132*A*b*c*d^4*e^2*m - 12*A*b*c*d^4*e^2*m^ 2 + 48*B*b*c*d^5*e*m))/(e^6*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^ 5 + m^6 + 720)) + (x^3*(d + e*x)^m*(3*m + m^2 + 2)*(120*A*b^2*e^3 + 74*A*b ^2*e^3*m + 20*B*c^2*d^3*m + 15*A*b^2*e^3*m^2 + A*b^2*e^3*m^3 - 4*A*c^2*d^2 *e*m^2 + 11*B*b^2*d*e^2*m^2 + B*b^2*d*e^2*m^3 - 24*A*c^2*d^2*e*m + 30*B*b^ 2*d*e^2*m + 22*A*b*c*d*e^2*m^2 + 2*A*b*c*d*e^2*m^3 - 8*B*b*c*d^2*e*m^2 + 6 0*A*b*c*d*e^2*m - 48*B*b*c*d^2*e*m))/(e^3*(1764*m + 1624*m^2 + 735*m^3 + 1 75*m^4 + 21*m^5 + m^6 + 720)) + (x^4*(d + e*x)^m*(11*m + 6*m^2 + m^3 + 6)* (30*B*b^2*e^2 + 60*A*b*c*e^2 + 11*B*b^2*e^2*m - 5*B*c^2*d^2*m + B*b^2*e^2* m^2 + 22*A*b*c*e^2*m + 6*A*c^2*d*e*m + 2*A*b*c*e^2*m^2 + A*c^2*d*e*m^2 + 1 2*B*b*c*d*e*m + 2*B*b*c*d*e*m^2))/(e^2*(1764*m + 1624*m^2 + 735*m^3 + 175* m^4 + 21*m^5 + m^6 + 720)) + (B*c^2*x^6*(d + e*x)^m*(274*m + 225*m^2 + 85* m^3 + 15*m^4 + m^5 + 120))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) + (c*x^5*(d + e*x)^m*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)*(6* A*c*e + 12*B*b*e + A*c*e*m + 2*B*b*e*m + B*c*d*m))/(e*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720)) - (2*d^2*m*x*(d + e*x)^m*(120*A* b^2*e^3 - 60*B*c^2*d^3 + 72*A*c^2*d^2*e - 90*B*b^2*d*e^2 + 74*A*b^2*e^3...