Integrand size = 26, antiderivative size = 270 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right )^2 \, dx=-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{1+m}}{e^6 (1+m)}+\frac {2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{2+m}}{e^6 (2+m)}-\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{3+m}}{e^6 (3+m)}+\frac {4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{4+m}}{e^6 (4+m)}-\frac {5 c^2 (2 c d-b e) (d+e x)^{5+m}}{e^6 (5+m)}+\frac {2 c^3 (d+e x)^{6+m}}{e^6 (6+m)} \]
-(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^(1+m)/e^6/(1+m)+2*(a*e^2-b*d*e +c*d^2)*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*(e*x+d)^(2+m)/e^6/(2+m)-(-b*e +2*c*d)*(10*c^2*d^2+b^2*e^2-2*c*e*(-3*a*e+5*b*d))*(e*x+d)^(3+m)/e^6/(3+m)+ 4*c*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*(e*x+d)^(4+m)/e^6/(4+m)-5*c^2*(-b *e+2*c*d)*(e*x+d)^(5+m)/e^6/(5+m)+2*c^3*(e*x+d)^(6+m)/e^6/(6+m)
Time = 0.81 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.51 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right )^2 \, dx=\frac {(d+e x)^{1+m} \left (c (a+x (b+c x))^2 (b e (10+m)+2 c (-5 d+e (5+m) x))+\frac {2 \left (-\frac {(2 c d-b e) \left (c d^2+e (-b d+a e)\right ) \left (60 c^2 d^2+b^2 e^2 m (1+m)-4 c e \left (15 b d+a e \left (-15+m+m^2\right )\right )\right )}{e^2 (1+m)}+\frac {\left (120 c^4 d^4+b^4 e^4 m (2+m)-2 b^2 c e^3 m (b d (-4+m)+3 a e (4+m))-8 c^3 d^2 e \left (30 b d+a e \left (-30-4 m+m^2\right )\right )+2 c^2 e^2 \left (4 a b d e \left (-30-4 m+m^2\right )+b^2 d^2 \left (60-4 m+m^2\right )+4 a^2 e^2 \left (15+8 m+m^2\right )\right )\right ) (d+e x)}{e^2 (2+m)}-(a+x (b+c x)) \left (b^3 e^3 m+20 c^3 d^2 (3 d-e (3+m) x)+b c e^2 \left (-2 a e (30+7 m)+b d \left (60+11 m+m^2\right )+b e m (3+m) x\right )-2 c^2 e \left (5 b d (d (12+m)-2 e (3+m) x)+2 a e \left (d \left (-15+m+m^2\right )+e \left (15+8 m+m^2\right ) x\right )\right )\right )\right )}{e^2 (3+m) (4+m)}\right )}{c e^2 (5+m) (6+m)} \]
((d + e*x)^(1 + m)*(c*(a + x*(b + c*x))^2*(b*e*(10 + m) + 2*c*(-5*d + e*(5 + m)*x)) + (2*(-(((2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))*(60*c^2*d^2 + b^2*e^2*m*(1 + m) - 4*c*e*(15*b*d + a*e*(-15 + m + m^2))))/(e^2*(1 + m))) + ((120*c^4*d^4 + b^4*e^4*m*(2 + m) - 2*b^2*c*e^3*m*(b*d*(-4 + m) + 3*a*e* (4 + m)) - 8*c^3*d^2*e*(30*b*d + a*e*(-30 - 4*m + m^2)) + 2*c^2*e^2*(4*a*b *d*e*(-30 - 4*m + m^2) + b^2*d^2*(60 - 4*m + m^2) + 4*a^2*e^2*(15 + 8*m + m^2)))*(d + e*x))/(e^2*(2 + m)) - (a + x*(b + c*x))*(b^3*e^3*m + 20*c^3*d^ 2*(3*d - e*(3 + m)*x) + b*c*e^2*(-2*a*e*(30 + 7*m) + b*d*(60 + 11*m + m^2) + b*e*m*(3 + m)*x) - 2*c^2*e*(5*b*d*(d*(12 + m) - 2*e*(3 + m)*x) + 2*a*e* (d*(-15 + m + m^2) + e*(15 + 8*m + m^2)*x)))))/(e^2*(3 + m)*(4 + m))))/(c* e^2*(5 + m)*(6 + m))
Time = 0.49 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, 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.
| \(\displaystyle \int (b+2 c x) \left (a+b x+c x^2\right )^2 (d+e x)^m \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {2 (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right ) \left (a c e^2+b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^5}+\frac {(2 c d-b e) (d+e x)^{m+2} \left (2 c e (5 b d-3 a e)-b^2 e^2-10 c^2 d^2\right )}{e^5}+\frac {4 c (d+e x)^{m+3} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^5}+\frac {(b e-2 c d) (d+e x)^m \left (a e^2-b d e+c d^2\right )^2}{e^5}-\frac {5 c^2 (2 c d-b e) (d+e x)^{m+4}}{e^5}+\frac {2 c^3 (d+e x)^{m+5}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 (d+e x)^{m+2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6 (m+2)}-\frac {(2 c d-b e) (d+e x)^{m+3} \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^6 (m+3)}+\frac {4 c (d+e x)^{m+4} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6 (m+4)}-\frac {(2 c d-b e) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^2}{e^6 (m+1)}-\frac {5 c^2 (2 c d-b e) (d+e x)^{m+5}}{e^6 (m+5)}+\frac {2 c^3 (d+e x)^{m+6}}{e^6 (m+6)}\) |
-(((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(1 + m))/(e^6*(1 + m) )) + (2*(c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))* (d + e*x)^(2 + m))/(e^6*(2 + m)) - ((2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*(d + e*x)^(3 + m))/(e^6*(3 + m)) + (4*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(4 + m))/(e^6*(4 + m)) - (5*c^2*( 2*c*d - b*e)*(d + e*x)^(5 + m))/(e^6*(5 + m)) + (2*c^3*(d + e*x)^(6 + m))/ (e^6*(6 + m))
3.17.55.3.1 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. \(1667\) vs. \(2(270)=540\).
Time = 0.33 (sec) , antiderivative size = 1668, normalized size of antiderivative = 6.18
| method | result | size |
| norman | \(\text {Expression too large to display}\) | \(1668\) |
| gosper | \(\text {Expression too large to display}\) | \(1852\) |
| risch | \(\text {Expression too large to display}\) | \(2445\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(3763\) |
d*(a^2*b*e^5*m^5+20*a^2*b*e^5*m^4-2*a^2*c*d*e^4*m^4-2*a*b^2*d*e^4*m^4+155* a^2*b*e^5*m^3-36*a^2*c*d*e^4*m^3-36*a*b^2*d*e^4*m^3+12*a*b*c*d^2*e^3*m^3+2 *b^3*d^2*e^3*m^3+580*a^2*b*e^5*m^2-238*a^2*c*d*e^4*m^2-238*a*b^2*d*e^4*m^2 +180*a*b*c*d^2*e^3*m^2-24*a*c^2*d^3*e^2*m^2+30*b^3*d^2*e^3*m^2-24*b^2*c*d^ 3*e^2*m^2+1044*a^2*b*e^5*m-684*a^2*c*d*e^4*m-684*a*b^2*d*e^4*m+888*a*b*c*d ^2*e^3*m-264*a*c^2*d^3*e^2*m+148*b^3*d^2*e^3*m-264*b^2*c*d^3*e^2*m+120*b*c ^2*d^4*e*m+720*a^2*b*e^5-720*a^2*c*d*e^4-720*a*b^2*d*e^4+1440*a*b*c*d^2*e^ 3-720*a*c^2*d^3*e^2+240*b^3*d^2*e^3-720*b^2*c*d^3*e^2+720*b*c^2*d^4*e-240* c^3*d^5)/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))+(6*a*b*c*e^3*m^3+4*a*c^2*d*e^2*m^3+b^3*e^3*m^3+4*b^2*c*d*e^2*m^3+90*a *b*c*e^3*m^2+44*a*c^2*d*e^2*m^2+15*b^3*e^3*m^2+44*b^2*c*d*e^2*m^2-20*b*c^2 *d^2*e*m^2+444*a*b*c*e^3*m+120*a*c^2*d*e^2*m+74*b^3*e^3*m+120*b^2*c*d*e^2* m-120*b*c^2*d^2*e*m+40*c^3*d^3*m+720*a*b*c*e^3+120*b^3*e^3)/e^3/(m^4+18*m^ 3+119*m^2+342*m+360)*x^3*exp(m*ln(e*x+d))+(2*a^2*c*e^4*m^4+2*a*b^2*e^4*m^4 +6*a*b*c*d*e^3*m^4+b^3*d*e^3*m^4+36*a^2*c*e^4*m^3+36*a*b^2*e^4*m^3+90*a*b* c*d*e^3*m^3-12*a*c^2*d^2*e^2*m^3+15*b^3*d*e^3*m^3-12*b^2*c*d^2*e^2*m^3+238 *a^2*c*e^4*m^2+238*a*b^2*e^4*m^2+444*a*b*c*d*e^3*m^2-132*a*c^2*d^2*e^2*m^2 +74*b^3*d*e^3*m^2-132*b^2*c*d^2*e^2*m^2+60*b*c^2*d^3*e*m^2+684*a^2*c*e^4*m +684*a*b^2*e^4*m+720*a*b*c*d*e^3*m-360*a*c^2*d^2*e^2*m+120*b^3*d*e^3*m-360 *b^2*c*d^2*e^2*m+360*b*c^2*d^3*e*m-120*c^3*d^4*m+720*a^2*c*e^4+720*a*b^...
Leaf count of result is larger than twice the leaf count of optimal. 1750 vs. \(2 (270) = 540\).
Time = 0.41 (sec) , antiderivative size = 1750, normalized size of antiderivative = 6.48 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right )^2 \, dx=\text {Too large to display} \]
(a^2*b*d*e^5*m^5 - 240*c^3*d^6 + 720*b*c^2*d^5*e + 720*a^2*b*d*e^5 - 720*( b^2*c + a*c^2)*d^4*e^2 + 240*(b^3 + 6*a*b*c)*d^3*e^3 - 720*(a*b^2 + a^2*c) *d^2*e^4 + 2*(c^3*e^6*m^5 + 15*c^3*e^6*m^4 + 85*c^3*e^6*m^3 + 225*c^3*e^6* m^2 + 274*c^3*e^6*m + 120*c^3*e^6)*x^6 + (720*b*c^2*e^6 + (2*c^3*d*e^5 + 5 *b*c^2*e^6)*m^5 + 20*(c^3*d*e^5 + 4*b*c^2*e^6)*m^4 + 5*(14*c^3*d*e^5 + 95* b*c^2*e^6)*m^3 + 100*(c^3*d*e^5 + 13*b*c^2*e^6)*m^2 + 12*(4*c^3*d*e^5 + 13 5*b*c^2*e^6)*m)*x^5 + 2*(10*a^2*b*d*e^5 - (a*b^2 + a^2*c)*d^2*e^4)*m^4 + ( 720*(b^2*c + a*c^2)*e^6 + (5*b*c^2*d*e^5 + 4*(b^2*c + a*c^2)*e^6)*m^5 - 2* (5*c^3*d^2*e^4 - 30*b*c^2*d*e^5 - 34*(b^2*c + a*c^2)*e^6)*m^4 - (60*c^3*d^ 2*e^4 - 235*b*c^2*d*e^5 - 428*(b^2*c + a*c^2)*e^6)*m^3 - 2*(55*c^3*d^2*e^4 - 180*b*c^2*d*e^5 - 614*(b^2*c + a*c^2)*e^6)*m^2 - 12*(5*c^3*d^2*e^4 - 15 *b*c^2*d*e^5 - 132*(b^2*c + a*c^2)*e^6)*m)*x^4 + (155*a^2*b*d*e^5 + 2*(b^3 + 6*a*b*c)*d^3*e^3 - 36*(a*b^2 + a^2*c)*d^2*e^4)*m^3 + (240*(b^3 + 6*a*b* c)*e^6 + (4*(b^2*c + a*c^2)*d*e^5 + (b^3 + 6*a*b*c)*e^6)*m^5 - 2*(10*b*c^2 *d^2*e^4 - 28*(b^2*c + a*c^2)*d*e^5 - 9*(b^3 + 6*a*b*c)*e^6)*m^4 + (40*c^3 *d^3*e^3 - 180*b*c^2*d^2*e^4 + 260*(b^2*c + a*c^2)*d*e^5 + 121*(b^3 + 6*a* b*c)*e^6)*m^3 + 4*(30*c^3*d^3*e^3 - 100*b*c^2*d^2*e^4 + 112*(b^2*c + a*c^2 )*d*e^5 + 93*(b^3 + 6*a*b*c)*e^6)*m^2 + 4*(20*c^3*d^3*e^3 - 60*b*c^2*d^2*e ^4 + 60*(b^2*c + a*c^2)*d*e^5 + 127*(b^3 + 6*a*b*c)*e^6)*m)*x^3 + 2*(290*a ^2*b*d*e^5 - 12*(b^2*c + a*c^2)*d^4*e^2 + 15*(b^3 + 6*a*b*c)*d^3*e^3 - ...
Leaf count of result is larger than twice the leaf count of optimal. 23525 vs. \(2 (252) = 504\).
Time = 4.21 (sec) , antiderivative size = 23525, normalized size of antiderivative = 87.13 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right )^2 \, dx=\text {Too large to display} \]
Piecewise((d**m*(a**2*b*x + a**2*c*x**2 + a*b**2*x**2 + 2*a*b*c*x**3 + a*c **2*x**4 + b**3*x**3/3 + b**2*c*x**4 + b*c**2*x**5 + c**3*x**6/3), Eq(e, 0 )), (-6*a**2*b*e**5/(30*d**5*e**6 + 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) - 3*a**2*c*d*e**4/ (30*d**5*e**6 + 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) - 15*a**2*c*e**5*x/(30*d**5*e**6 + 150 *d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) - 3*a*b**2*d*e**4/(30*d**5*e**6 + 150*d**4*e**7*x + 300*d **3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) - 1 5*a*b**2*e**5*x/(30*d**5*e**6 + 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 300 *d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) - 6*a*b*c*d**2*e**3/(3 0*d**5*e**6 + 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) - 30*a*b*c*d*e**4*x/(30*d**5*e**6 + 150* d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) - 60*a*b*c*e**5*x**2/(30*d**5*e**6 + 150*d**4*e**7*x + 300 *d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) - 6*a*c**2*d**3*e**2/(30*d**5*e**6 + 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) - 30*a*c**2*d**2*e **3*x/(30*d**5*e**6 + 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9 *x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) - 60*a*c**2*d*e**4*x**2/(30*d...
Leaf count of result is larger than twice the leaf count of optimal. 802 vs. \(2 (270) = 540\).
Time = 0.23 (sec) , antiderivative size = 802, normalized size of antiderivative = 2.97 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} a b^{2}}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {2 \, {\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} a^{2} c}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} a^{2} b}{e {\left (m + 1\right )}} + \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} b^{3}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {6 \, {\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 c}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {4 \, {\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^{2} c}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {4 \, {\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 c^{2}}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {5 \, {\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 c^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{5}} + \frac {2 \, {\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} c^{3}}{{\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} e^{6}} \]
2*(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*a*b^2/((m^2 + 3*m + 2)*e^2 ) + 2*(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*a^2*c/((m^2 + 3*m + 2) *e^2) + (e*x + d)^(m + 1)*a^2*b/(e*(m + 1)) + ((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*b^3/((m^3 + 6*m^2 + 11*m + 6)*e^3) + 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*c/((m^3 + 6*m^2 + 11*m + 6)*e^3) + 4*((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^2*c/((m^4 + 10*m^3 + 35*m ^2 + 50*m + 24)*e^4) + 4*((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*c^2/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + 5*((m^4 + 10*m^3 + 3 5*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*c^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120) *e^5) + 2*((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*c^3/((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. 3633 vs. \(2 (270) = 540\).
Time = 0.31 (sec) , antiderivative size = 3633, normalized size of antiderivative = 13.46 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right )^2 \, dx=\text {Too large to display} \]
(2*(e*x + d)^m*c^3*e^6*m^5*x^6 + 2*(e*x + d)^m*c^3*d*e^5*m^5*x^5 + 5*(e*x + d)^m*b*c^2*e^6*m^5*x^5 + 30*(e*x + d)^m*c^3*e^6*m^4*x^6 + 5*(e*x + d)^m* b*c^2*d*e^5*m^5*x^4 + 4*(e*x + d)^m*b^2*c*e^6*m^5*x^4 + 4*(e*x + d)^m*a*c^ 2*e^6*m^5*x^4 + 20*(e*x + d)^m*c^3*d*e^5*m^4*x^5 + 80*(e*x + d)^m*b*c^2*e^ 6*m^4*x^5 + 170*(e*x + d)^m*c^3*e^6*m^3*x^6 + 4*(e*x + d)^m*b^2*c*d*e^5*m^ 5*x^3 + 4*(e*x + d)^m*a*c^2*d*e^5*m^5*x^3 + (e*x + d)^m*b^3*e^6*m^5*x^3 + 6*(e*x + d)^m*a*b*c*e^6*m^5*x^3 - 10*(e*x + d)^m*c^3*d^2*e^4*m^4*x^4 + 60* (e*x + d)^m*b*c^2*d*e^5*m^4*x^4 + 68*(e*x + d)^m*b^2*c*e^6*m^4*x^4 + 68*(e *x + d)^m*a*c^2*e^6*m^4*x^4 + 70*(e*x + d)^m*c^3*d*e^5*m^3*x^5 + 475*(e*x + d)^m*b*c^2*e^6*m^3*x^5 + 450*(e*x + d)^m*c^3*e^6*m^2*x^6 + (e*x + d)^m*b ^3*d*e^5*m^5*x^2 + 6*(e*x + d)^m*a*b*c*d*e^5*m^5*x^2 + 2*(e*x + d)^m*a*b^2 *e^6*m^5*x^2 + 2*(e*x + d)^m*a^2*c*e^6*m^5*x^2 - 20*(e*x + d)^m*b*c^2*d^2* e^4*m^4*x^3 + 56*(e*x + d)^m*b^2*c*d*e^5*m^4*x^3 + 56*(e*x + d)^m*a*c^2*d* e^5*m^4*x^3 + 18*(e*x + d)^m*b^3*e^6*m^4*x^3 + 108*(e*x + d)^m*a*b*c*e^6*m ^4*x^3 - 60*(e*x + d)^m*c^3*d^2*e^4*m^3*x^4 + 235*(e*x + d)^m*b*c^2*d*e^5* m^3*x^4 + 428*(e*x + d)^m*b^2*c*e^6*m^3*x^4 + 428*(e*x + d)^m*a*c^2*e^6*m^ 3*x^4 + 100*(e*x + d)^m*c^3*d*e^5*m^2*x^5 + 1300*(e*x + d)^m*b*c^2*e^6*m^2 *x^5 + 548*(e*x + d)^m*c^3*e^6*m*x^6 + 2*(e*x + d)^m*a*b^2*d*e^5*m^5*x + 2 *(e*x + d)^m*a^2*c*d*e^5*m^5*x + (e*x + d)^m*a^2*b*e^6*m^5*x - 12*(e*x + d )^m*b^2*c*d^2*e^4*m^4*x^2 - 12*(e*x + d)^m*a*c^2*d^2*e^4*m^4*x^2 + 16*(...
Time = 11.66 (sec) , antiderivative size = 1825, normalized size of antiderivative = 6.76 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right )^2 \, dx=\text {Too large to display} \]
(2*c^3*x^6*(d + e*x)^m*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120))/(1 764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) - ((d + e*x)^m* (240*c^3*d^6 - 240*b^3*d^3*e^3 + 720*a*b^2*d^2*e^4 + 720*a*c^2*d^4*e^2 + 7 20*a^2*c*d^2*e^4 + 720*b^2*c*d^4*e^2 - 148*b^3*d^3*e^3*m - 30*b^3*d^3*e^3* m^2 - 2*b^3*d^3*e^3*m^3 - 720*a^2*b*d*e^5 - 720*b*c^2*d^5*e - 1440*a*b*c*d ^3*e^3 - 1044*a^2*b*d*e^5*m - 120*b*c^2*d^5*e*m + 684*a*b^2*d^2*e^4*m - 58 0*a^2*b*d*e^5*m^2 - 155*a^2*b*d*e^5*m^3 - 20*a^2*b*d*e^5*m^4 - a^2*b*d*e^5 *m^5 + 264*a*c^2*d^4*e^2*m + 684*a^2*c*d^2*e^4*m + 264*b^2*c*d^4*e^2*m + 2 38*a*b^2*d^2*e^4*m^2 + 36*a*b^2*d^2*e^4*m^3 + 2*a*b^2*d^2*e^4*m^4 + 24*a*c ^2*d^4*e^2*m^2 + 238*a^2*c*d^2*e^4*m^2 + 36*a^2*c*d^2*e^4*m^3 + 2*a^2*c*d^ 2*e^4*m^4 + 24*b^2*c*d^4*e^2*m^2 - 888*a*b*c*d^3*e^3*m - 180*a*b*c*d^3*e^3 *m^2 - 12*a*b*c*d^3*e^3*m^3))/(e^6*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720)) + (x*(d + e*x)^m*(720*a^2*b*e^6 + 580*a^2*b*e^6*m^2 + 155*a^2*b*e^6*m^3 + 20*a^2*b*e^6*m^4 + a^2*b*e^6*m^5 - 240*b^3*d^2*e^4* m - 148*b^3*d^2*e^4*m^2 - 30*b^3*d^2*e^4*m^3 - 2*b^3*d^2*e^4*m^4 + 1044*a^ 2*b*e^6*m + 240*c^3*d^5*e*m + 720*a*b^2*d*e^5*m + 720*a^2*c*d*e^5*m + 684* a*b^2*d*e^5*m^2 + 238*a*b^2*d*e^5*m^3 + 36*a*b^2*d*e^5*m^4 + 2*a*b^2*d*e^5 *m^5 + 720*a*c^2*d^3*e^3*m + 684*a^2*c*d*e^5*m^2 + 238*a^2*c*d*e^5*m^3 + 3 6*a^2*c*d*e^5*m^4 + 2*a^2*c*d*e^5*m^5 - 720*b*c^2*d^4*e^2*m + 720*b^2*c*d^ 3*e^3*m + 264*a*c^2*d^3*e^3*m^2 + 24*a*c^2*d^3*e^3*m^3 - 120*b*c^2*d^4*...