Integrand size = 23, antiderivative size = 155 \[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {a^2 A (e x)^{1+m}}{e (1+m)}+\frac {a (2 A b+a B) (e x)^{2+m}}{e^2 (2+m)}+\frac {\left (2 a b B+A \left (b^2+2 a c\right )\right ) (e x)^{3+m}}{e^3 (3+m)}+\frac {\left (b^2 B+2 A b c+2 a B c\right ) (e x)^{4+m}}{e^4 (4+m)}+\frac {c (2 b B+A c) (e x)^{5+m}}{e^5 (5+m)}+\frac {B c^2 (e x)^{6+m}}{e^6 (6+m)} \]
a^2*A*(e*x)^(1+m)/e/(1+m)+a*(2*A*b+B*a)*(e*x)^(2+m)/e^2/(2+m)+(2*a*b*B+A*( 2*a*c+b^2))*(e*x)^(3+m)/e^3/(3+m)+(2*A*b*c+2*B*a*c+B*b^2)*(e*x)^(4+m)/e^4/ (4+m)+c*(A*c+2*B*b)*(e*x)^(5+m)/e^5/(5+m)+B*c^2*(e*x)^(6+m)/e^6/(6+m)
Time = 0.42 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.86 \[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {(e x)^m \left (x (2 b B+A c (6+m)+B c (5+m) x) (a+x (b+c x))^2+\frac {2 x \left (-\frac {2 a^2 c (4+m) (b B (1+m)-2 A c (6+m))}{1+m}+a b \left (b^2 B (3+m)-2 a B c (5+m)-A b c (6+m)\right )-\frac {a b c (4+m) (b B (1+m)-2 A c (6+m)) x}{2+m}+\frac {\left (b^2 (2+m)-2 a c (3+m)\right ) \left (b^2 B (3+m)-2 a B c (5+m)-A b c (6+m)\right ) x}{2+m}-\left (a c (4+m) (b B (1+m)-2 A c (6+m))+b \left (b^2 B (3+m)-2 a B c (5+m)-A b c (6+m)\right )+c (3+m) \left (b^2 B (3+m)-2 a B c (5+m)-A b c (6+m)\right ) x\right ) (a+x (b+c x))\right )}{c (3+m) (4+m)}\right )}{c (5+m) (6+m)} \]
((e*x)^m*(x*(2*b*B + A*c*(6 + m) + B*c*(5 + m)*x)*(a + x*(b + c*x))^2 + (2 *x*((-2*a^2*c*(4 + m)*(b*B*(1 + m) - 2*A*c*(6 + m)))/(1 + m) + a*b*(b^2*B* (3 + m) - 2*a*B*c*(5 + m) - A*b*c*(6 + m)) - (a*b*c*(4 + m)*(b*B*(1 + m) - 2*A*c*(6 + m))*x)/(2 + m) + ((b^2*(2 + m) - 2*a*c*(3 + m))*(b^2*B*(3 + m) - 2*a*B*c*(5 + m) - A*b*c*(6 + m))*x)/(2 + m) - (a*c*(4 + m)*(b*B*(1 + m) - 2*A*c*(6 + m)) + b*(b^2*B*(3 + m) - 2*a*B*c*(5 + m) - A*b*c*(6 + m)) + c*(3 + m)*(b^2*B*(3 + m) - 2*a*B*c*(5 + m) - A*b*c*(6 + m))*x)*(a + x*(b + c*x))))/(c*(3 + m)*(4 + m))))/(c*(5 + m)*(6 + m))
Time = 0.33 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, 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.
| \(\displaystyle \int (A+B x) (e x)^m \left (a+b x+c x^2\right )^2 \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (a^2 A (e x)^m+\frac {(e x)^{m+3} \left (2 a B c+2 A b c+b^2 B\right )}{e^3}+\frac {(e x)^{m+2} \left (A \left (2 a c+b^2\right )+2 a b B\right )}{e^2}+\frac {a (e x)^{m+1} (a B+2 A b)}{e}+\frac {c (e x)^{m+4} (A c+2 b B)}{e^4}+\frac {B c^2 (e x)^{m+5}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^2 A (e x)^{m+1}}{e (m+1)}+\frac {(e x)^{m+4} \left (2 a B c+2 A b c+b^2 B\right )}{e^4 (m+4)}+\frac {(e x)^{m+3} \left (A \left (2 a c+b^2\right )+2 a b B\right )}{e^3 (m+3)}+\frac {a (e x)^{m+2} (a B+2 A b)}{e^2 (m+2)}+\frac {c (e x)^{m+5} (A c+2 b B)}{e^5 (m+5)}+\frac {B c^2 (e x)^{m+6}}{e^6 (m+6)}\) |
(a^2*A*(e*x)^(1 + m))/(e*(1 + m)) + (a*(2*A*b + a*B)*(e*x)^(2 + m))/(e^2*( 2 + m)) + ((2*a*b*B + A*(b^2 + 2*a*c))*(e*x)^(3 + m))/(e^3*(3 + m)) + ((b^ 2*B + 2*A*b*c + 2*a*B*c)*(e*x)^(4 + m))/(e^4*(4 + m)) + (c*(2*b*B + A*c)*( e*x)^(5 + m))/(e^5*(5 + m)) + (B*c^2*(e*x)^(6 + m))/(e^6*(6 + m))
3.11.84.3.1 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 = 0.62 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.99
| method | result | size |
| norman | \(\frac {\left (2 A b c +2 B a c +B \,b^{2}\right ) x^{4} {\mathrm e}^{m \ln \left (e x \right )}}{4+m}+\frac {\left (2 A a c +A \,b^{2}+2 a b B \right ) x^{3} {\mathrm e}^{m \ln \left (e x \right )}}{3+m}+\frac {A \,a^{2} x \,{\mathrm e}^{m \ln \left (e x \right )}}{1+m}+\frac {B \,c^{2} x^{6} {\mathrm e}^{m \ln \left (e x \right )}}{6+m}+\frac {a \left (2 A b +B a \right ) x^{2} {\mathrm e}^{m \ln \left (e x \right )}}{2+m}+\frac {c \left (A c +2 B b \right ) x^{5} {\mathrm e}^{m \ln \left (e x \right )}}{5+m}\) | \(154\) |
| gosper | \(\frac {x \left (B \,c^{2} m^{5} x^{5}+A \,c^{2} m^{5} x^{4}+2 B b c \,m^{5} x^{4}+15 B \,c^{2} m^{4} x^{5}+2 A b c \,m^{5} x^{3}+16 A \,c^{2} m^{4} x^{4}+2 B a c \,m^{5} x^{3}+B \,b^{2} m^{5} x^{3}+32 B b c \,m^{4} x^{4}+85 B \,c^{2} m^{3} x^{5}+2 A a c \,m^{5} x^{2}+A \,b^{2} m^{5} x^{2}+34 A b c \,m^{4} x^{3}+95 A \,c^{2} m^{3} x^{4}+2 B a b \,m^{5} x^{2}+34 B a c \,m^{4} x^{3}+17 B \,b^{2} m^{4} x^{3}+190 B b c \,m^{3} x^{4}+225 B \,c^{2} m^{2} x^{5}+2 A a b \,m^{5} x +36 A a c \,m^{4} x^{2}+18 A \,b^{2} m^{4} x^{2}+214 A b c \,m^{3} x^{3}+260 A \,c^{2} m^{2} x^{4}+B \,a^{2} m^{5} x +36 B a b \,m^{4} x^{2}+214 B a c \,m^{3} x^{3}+107 B \,b^{2} m^{3} x^{3}+520 B b c \,m^{2} x^{4}+274 m \,x^{5} B \,c^{2}+A \,a^{2} m^{5}+38 A a b \,m^{4} x +242 A a c \,m^{3} x^{2}+121 A \,b^{2} m^{3} x^{2}+614 A b c \,m^{2} x^{3}+324 A \,c^{2} x^{4} m +19 B \,a^{2} m^{4} x +242 B a b \,m^{3} x^{2}+614 B a c \,m^{2} x^{3}+307 B \,b^{2} m^{2} x^{3}+648 x^{4} B b c m +120 B \,c^{2} x^{5}+20 A \,a^{2} m^{4}+274 A a b \,m^{3} x +744 A a c \,m^{2} x^{2}+372 A \,b^{2} m^{2} x^{2}+792 x^{3} A b c m +144 A \,c^{2} x^{4}+137 B \,a^{2} m^{3} x +744 B a b \,m^{2} x^{2}+792 a B c \,x^{3} m +396 B \,b^{2} x^{3} m +288 x^{4} B b c +155 A \,a^{2} m^{3}+922 A a b \,m^{2} x +1016 a A c \,x^{2} m +508 A \,b^{2} x^{2} m +360 x^{3} A b c +461 B \,a^{2} m^{2} x +1016 B a b \,x^{2} m +360 a B c \,x^{3}+180 B \,b^{2} x^{3}+580 A \,a^{2} m^{2}+1404 a A b x m +480 a A c \,x^{2}+240 A \,b^{2} x^{2}+702 a^{2} B x m +480 B a b \,x^{2}+1044 A \,a^{2} m +720 a A b x +360 a^{2} B x +720 A \,a^{2}\right ) \left (e x \right )^{m}}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(759\) |
| risch | \(\frac {x \left (B \,c^{2} m^{5} x^{5}+A \,c^{2} m^{5} x^{4}+2 B b c \,m^{5} x^{4}+15 B \,c^{2} m^{4} x^{5}+2 A b c \,m^{5} x^{3}+16 A \,c^{2} m^{4} x^{4}+2 B a c \,m^{5} x^{3}+B \,b^{2} m^{5} x^{3}+32 B b c \,m^{4} x^{4}+85 B \,c^{2} m^{3} x^{5}+2 A a c \,m^{5} x^{2}+A \,b^{2} m^{5} x^{2}+34 A b c \,m^{4} x^{3}+95 A \,c^{2} m^{3} x^{4}+2 B a b \,m^{5} x^{2}+34 B a c \,m^{4} x^{3}+17 B \,b^{2} m^{4} x^{3}+190 B b c \,m^{3} x^{4}+225 B \,c^{2} m^{2} x^{5}+2 A a b \,m^{5} x +36 A a c \,m^{4} x^{2}+18 A \,b^{2} m^{4} x^{2}+214 A b c \,m^{3} x^{3}+260 A \,c^{2} m^{2} x^{4}+B \,a^{2} m^{5} x +36 B a b \,m^{4} x^{2}+214 B a c \,m^{3} x^{3}+107 B \,b^{2} m^{3} x^{3}+520 B b c \,m^{2} x^{4}+274 m \,x^{5} B \,c^{2}+A \,a^{2} m^{5}+38 A a b \,m^{4} x +242 A a c \,m^{3} x^{2}+121 A \,b^{2} m^{3} x^{2}+614 A b c \,m^{2} x^{3}+324 A \,c^{2} x^{4} m +19 B \,a^{2} m^{4} x +242 B a b \,m^{3} x^{2}+614 B a c \,m^{2} x^{3}+307 B \,b^{2} m^{2} x^{3}+648 x^{4} B b c m +120 B \,c^{2} x^{5}+20 A \,a^{2} m^{4}+274 A a b \,m^{3} x +744 A a c \,m^{2} x^{2}+372 A \,b^{2} m^{2} x^{2}+792 x^{3} A b c m +144 A \,c^{2} x^{4}+137 B \,a^{2} m^{3} x +744 B a b \,m^{2} x^{2}+792 a B c \,x^{3} m +396 B \,b^{2} x^{3} m +288 x^{4} B b c +155 A \,a^{2} m^{3}+922 A a b \,m^{2} x +1016 a A c \,x^{2} m +508 A \,b^{2} x^{2} m +360 x^{3} A b c +461 B \,a^{2} m^{2} x +1016 B a b \,x^{2} m +360 a B c \,x^{3}+180 B \,b^{2} x^{3}+580 A \,a^{2} m^{2}+1404 a A b x m +480 a A c \,x^{2}+240 A \,b^{2} x^{2}+702 a^{2} B x m +480 B a b \,x^{2}+1044 A \,a^{2} m +720 a A b x +360 a^{2} B x +720 A \,a^{2}\right ) \left (e x \right )^{m}}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(759\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1143\) |
(2*A*b*c+2*B*a*c+B*b^2)/(4+m)*x^4*exp(m*ln(e*x))+(2*A*a*c+A*b^2+2*B*a*b)/( 3+m)*x^3*exp(m*ln(e*x))+A*a^2/(1+m)*x*exp(m*ln(e*x))+B*c^2/(6+m)*x^6*exp(m *ln(e*x))+a*(2*A*b+B*a)/(2+m)*x^2*exp(m*ln(e*x))+c*(A*c+2*B*b)/(5+m)*x^5*e xp(m*ln(e*x))
Leaf count of result is larger than twice the leaf count of optimal. 573 vs. \(2 (155) = 310\).
Time = 0.28 (sec) , antiderivative size = 573, normalized size of antiderivative = 3.70 \[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {{\left ({\left (B c^{2} m^{5} + 15 \, B c^{2} m^{4} + 85 \, B c^{2} m^{3} + 225 \, B c^{2} m^{2} + 274 \, B c^{2} m + 120 \, B c^{2}\right )} x^{6} + {\left ({\left (2 \, B b c + A c^{2}\right )} m^{5} + 16 \, {\left (2 \, B b c + A c^{2}\right )} m^{4} + 95 \, {\left (2 \, B b c + A c^{2}\right )} m^{3} + 288 \, B b c + 144 \, A c^{2} + 260 \, {\left (2 \, B b c + A c^{2}\right )} m^{2} + 324 \, {\left (2 \, B b c + A c^{2}\right )} m\right )} x^{5} + {\left ({\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} m^{5} + 17 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} m^{4} + 107 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} m^{3} + 180 \, B b^{2} + 307 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} m^{2} + 360 \, {\left (B a + A b\right )} c + 396 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} m\right )} x^{4} + {\left ({\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} m^{5} + 18 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} m^{4} + 121 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} m^{3} + 480 \, B a b + 240 \, A b^{2} + 480 \, A a c + 372 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} m^{2} + 508 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} m\right )} x^{3} + {\left ({\left (B a^{2} + 2 \, A a b\right )} m^{5} + 19 \, {\left (B a^{2} + 2 \, A a b\right )} m^{4} + 137 \, {\left (B a^{2} + 2 \, A a b\right )} m^{3} + 360 \, B a^{2} + 720 \, A a b + 461 \, {\left (B a^{2} + 2 \, A a b\right )} m^{2} + 702 \, {\left (B a^{2} + 2 \, A a b\right )} m\right )} x^{2} + {\left (A a^{2} m^{5} + 20 \, A a^{2} m^{4} + 155 \, A a^{2} m^{3} + 580 \, A a^{2} m^{2} + 1044 \, A a^{2} m + 720 \, A a^{2}\right )} x\right )} \left (e x\right )^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \]
((B*c^2*m^5 + 15*B*c^2*m^4 + 85*B*c^2*m^3 + 225*B*c^2*m^2 + 274*B*c^2*m + 120*B*c^2)*x^6 + ((2*B*b*c + A*c^2)*m^5 + 16*(2*B*b*c + A*c^2)*m^4 + 95*(2 *B*b*c + A*c^2)*m^3 + 288*B*b*c + 144*A*c^2 + 260*(2*B*b*c + A*c^2)*m^2 + 324*(2*B*b*c + A*c^2)*m)*x^5 + ((B*b^2 + 2*(B*a + A*b)*c)*m^5 + 17*(B*b^2 + 2*(B*a + A*b)*c)*m^4 + 107*(B*b^2 + 2*(B*a + A*b)*c)*m^3 + 180*B*b^2 + 3 07*(B*b^2 + 2*(B*a + A*b)*c)*m^2 + 360*(B*a + A*b)*c + 396*(B*b^2 + 2*(B*a + A*b)*c)*m)*x^4 + ((2*B*a*b + A*b^2 + 2*A*a*c)*m^5 + 18*(2*B*a*b + A*b^2 + 2*A*a*c)*m^4 + 121*(2*B*a*b + A*b^2 + 2*A*a*c)*m^3 + 480*B*a*b + 240*A* b^2 + 480*A*a*c + 372*(2*B*a*b + A*b^2 + 2*A*a*c)*m^2 + 508*(2*B*a*b + A*b ^2 + 2*A*a*c)*m)*x^3 + ((B*a^2 + 2*A*a*b)*m^5 + 19*(B*a^2 + 2*A*a*b)*m^4 + 137*(B*a^2 + 2*A*a*b)*m^3 + 360*B*a^2 + 720*A*a*b + 461*(B*a^2 + 2*A*a*b) *m^2 + 702*(B*a^2 + 2*A*a*b)*m)*x^2 + (A*a^2*m^5 + 20*A*a^2*m^4 + 155*A*a^ 2*m^3 + 580*A*a^2*m^2 + 1044*A*a^2*m + 720*A*a^2)*x)*(e*x)^m/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)
Leaf count of result is larger than twice the leaf count of optimal. 4027 vs. \(2 (144) = 288\).
Time = 0.56 (sec) , antiderivative size = 4027, normalized size of antiderivative = 25.98 \[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\text {Too large to display} \]
Piecewise(((-A*a**2/(5*x**5) - A*a*b/(2*x**4) - 2*A*a*c/(3*x**3) - A*b**2/ (3*x**3) - A*b*c/x**2 - A*c**2/x - B*a**2/(4*x**4) - 2*B*a*b/(3*x**3) - B* a*c/x**2 - B*b**2/(2*x**2) - 2*B*b*c/x + B*c**2*log(x))/e**6, Eq(m, -6)), ((-A*a**2/(4*x**4) - 2*A*a*b/(3*x**3) - A*a*c/x**2 - A*b**2/(2*x**2) - 2*A *b*c/x + A*c**2*log(x) - B*a**2/(3*x**3) - B*a*b/x**2 - 2*B*a*c/x - B*b**2 /x + 2*B*b*c*log(x) + B*c**2*x)/e**5, Eq(m, -5)), ((-A*a**2/(3*x**3) - A*a *b/x**2 - 2*A*a*c/x - A*b**2/x + 2*A*b*c*log(x) + A*c**2*x - B*a**2/(2*x** 2) - 2*B*a*b/x + 2*B*a*c*log(x) + B*b**2*log(x) + 2*B*b*c*x + B*c**2*x**2/ 2)/e**4, Eq(m, -4)), ((-A*a**2/(2*x**2) - 2*A*a*b/x + 2*A*a*c*log(x) + A*b **2*log(x) + 2*A*b*c*x + A*c**2*x**2/2 - B*a**2/x + 2*B*a*b*log(x) + 2*B*a *c*x + B*b**2*x + B*b*c*x**2 + B*c**2*x**3/3)/e**3, Eq(m, -3)), ((-A*a**2/ x + 2*A*a*b*log(x) + 2*A*a*c*x + A*b**2*x + A*b*c*x**2 + A*c**2*x**3/3 + B *a**2*log(x) + 2*B*a*b*x + B*a*c*x**2 + B*b**2*x**2/2 + 2*B*b*c*x**3/3 + B *c**2*x**4/4)/e**2, Eq(m, -2)), ((A*a**2*log(x) + 2*A*a*b*x + A*a*c*x**2 + A*b**2*x**2/2 + 2*A*b*c*x**3/3 + A*c**2*x**4/4 + B*a**2*x + B*a*b*x**2 + 2*B*a*c*x**3/3 + B*b**2*x**3/3 + B*b*c*x**4/2 + B*c**2*x**5/5)/e, Eq(m, -1 )), (A*a**2*m**5*x*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m **2 + 1764*m + 720) + 20*A*a**2*m**4*x*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 155*A*a**2*m**3*x*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 580*A*a*...
Time = 0.22 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.48 \[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {B c^{2} e^{m} x^{6} x^{m}}{m + 6} + \frac {2 \, B b c e^{m} x^{5} x^{m}}{m + 5} + \frac {A c^{2} e^{m} x^{5} x^{m}}{m + 5} + \frac {B b^{2} e^{m} x^{4} x^{m}}{m + 4} + \frac {2 \, B a c e^{m} x^{4} x^{m}}{m + 4} + \frac {2 \, A b c e^{m} x^{4} x^{m}}{m + 4} + \frac {2 \, B a b e^{m} x^{3} x^{m}}{m + 3} + \frac {A b^{2} e^{m} x^{3} x^{m}}{m + 3} + \frac {2 \, A a c e^{m} x^{3} x^{m}}{m + 3} + \frac {B a^{2} e^{m} x^{2} x^{m}}{m + 2} + \frac {2 \, A a b e^{m} x^{2} x^{m}}{m + 2} + \frac {\left (e x\right )^{m + 1} A a^{2}}{e {\left (m + 1\right )}} \]
B*c^2*e^m*x^6*x^m/(m + 6) + 2*B*b*c*e^m*x^5*x^m/(m + 5) + A*c^2*e^m*x^5*x^ m/(m + 5) + B*b^2*e^m*x^4*x^m/(m + 4) + 2*B*a*c*e^m*x^4*x^m/(m + 4) + 2*A* b*c*e^m*x^4*x^m/(m + 4) + 2*B*a*b*e^m*x^3*x^m/(m + 3) + A*b^2*e^m*x^3*x^m/ (m + 3) + 2*A*a*c*e^m*x^3*x^m/(m + 3) + B*a^2*e^m*x^2*x^m/(m + 2) + 2*A*a* b*e^m*x^2*x^m/(m + 2) + (e*x)^(m + 1)*A*a^2/(e*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 1142 vs. \(2 (155) = 310\).
Time = 0.28 (sec) , antiderivative size = 1142, normalized size of antiderivative = 7.37 \[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\text {Too large to display} \]
((e*x)^m*B*c^2*m^5*x^6 + 2*(e*x)^m*B*b*c*m^5*x^5 + (e*x)^m*A*c^2*m^5*x^5 + 15*(e*x)^m*B*c^2*m^4*x^6 + (e*x)^m*B*b^2*m^5*x^4 + 2*(e*x)^m*B*a*c*m^5*x^ 4 + 2*(e*x)^m*A*b*c*m^5*x^4 + 32*(e*x)^m*B*b*c*m^4*x^5 + 16*(e*x)^m*A*c^2* m^4*x^5 + 85*(e*x)^m*B*c^2*m^3*x^6 + 2*(e*x)^m*B*a*b*m^5*x^3 + (e*x)^m*A*b ^2*m^5*x^3 + 2*(e*x)^m*A*a*c*m^5*x^3 + 17*(e*x)^m*B*b^2*m^4*x^4 + 34*(e*x) ^m*B*a*c*m^4*x^4 + 34*(e*x)^m*A*b*c*m^4*x^4 + 190*(e*x)^m*B*b*c*m^3*x^5 + 95*(e*x)^m*A*c^2*m^3*x^5 + 225*(e*x)^m*B*c^2*m^2*x^6 + (e*x)^m*B*a^2*m^5*x ^2 + 2*(e*x)^m*A*a*b*m^5*x^2 + 36*(e*x)^m*B*a*b*m^4*x^3 + 18*(e*x)^m*A*b^2 *m^4*x^3 + 36*(e*x)^m*A*a*c*m^4*x^3 + 107*(e*x)^m*B*b^2*m^3*x^4 + 214*(e*x )^m*B*a*c*m^3*x^4 + 214*(e*x)^m*A*b*c*m^3*x^4 + 520*(e*x)^m*B*b*c*m^2*x^5 + 260*(e*x)^m*A*c^2*m^2*x^5 + 274*(e*x)^m*B*c^2*m*x^6 + (e*x)^m*A*a^2*m^5* x + 19*(e*x)^m*B*a^2*m^4*x^2 + 38*(e*x)^m*A*a*b*m^4*x^2 + 242*(e*x)^m*B*a* b*m^3*x^3 + 121*(e*x)^m*A*b^2*m^3*x^3 + 242*(e*x)^m*A*a*c*m^3*x^3 + 307*(e *x)^m*B*b^2*m^2*x^4 + 614*(e*x)^m*B*a*c*m^2*x^4 + 614*(e*x)^m*A*b*c*m^2*x^ 4 + 648*(e*x)^m*B*b*c*m*x^5 + 324*(e*x)^m*A*c^2*m*x^5 + 120*(e*x)^m*B*c^2* x^6 + 20*(e*x)^m*A*a^2*m^4*x + 137*(e*x)^m*B*a^2*m^3*x^2 + 274*(e*x)^m*A*a *b*m^3*x^2 + 744*(e*x)^m*B*a*b*m^2*x^3 + 372*(e*x)^m*A*b^2*m^2*x^3 + 744*( e*x)^m*A*a*c*m^2*x^3 + 396*(e*x)^m*B*b^2*m*x^4 + 792*(e*x)^m*B*a*c*m*x^4 + 792*(e*x)^m*A*b*c*m*x^4 + 288*(e*x)^m*B*b*c*x^5 + 144*(e*x)^m*A*c^2*x^5 + 155*(e*x)^m*A*a^2*m^3*x + 461*(e*x)^m*B*a^2*m^2*x^2 + 922*(e*x)^m*A*a*...
Time = 10.50 (sec) , antiderivative size = 405, normalized size of antiderivative = 2.61 \[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^2 \, dx={\left (e\,x\right )}^m\,\left (\frac {x^3\,\left (A\,b^2+2\,B\,a\,b+2\,A\,a\,c\right )\,\left (m^5+18\,m^4+121\,m^3+372\,m^2+508\,m+240\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {x^4\,\left (B\,b^2+2\,A\,c\,b+2\,B\,a\,c\right )\,\left (m^5+17\,m^4+107\,m^3+307\,m^2+396\,m+180\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {A\,a^2\,x\,\left (m^5+20\,m^4+155\,m^3+580\,m^2+1044\,m+720\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {a\,x^2\,\left (2\,A\,b+B\,a\right )\,\left (m^5+19\,m^4+137\,m^3+461\,m^2+702\,m+360\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {c\,x^5\,\left (A\,c+2\,B\,b\right )\,\left (m^5+16\,m^4+95\,m^3+260\,m^2+324\,m+144\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {B\,c^2\,x^6\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}\right ) \]
(e*x)^m*((x^3*(A*b^2 + 2*A*a*c + 2*B*a*b)*(508*m + 372*m^2 + 121*m^3 + 18* m^4 + m^5 + 240))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) + (x^4*(B*b^2 + 2*A*b*c + 2*B*a*c)*(396*m + 307*m^2 + 107*m^3 + 17*m^ 4 + m^5 + 180))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 72 0) + (A*a^2*x*(1044*m + 580*m^2 + 155*m^3 + 20*m^4 + m^5 + 720))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) + (a*x^2*(2*A*b + B*a) *(702*m + 461*m^2 + 137*m^3 + 19*m^4 + m^5 + 360))/(1764*m + 1624*m^2 + 73 5*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) + (c*x^5*(A*c + 2*B*b)*(324*m + 260* m^2 + 95*m^3 + 16*m^4 + m^5 + 144))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) + (B*c^2*x^6*(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))