Integrand size = 23, antiderivative size = 240 \[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^3 \, dx=\frac {a^3 A (e x)^{1+m}}{e (1+m)}+\frac {a^2 (3 A b+a B) (e x)^{2+m}}{e^2 (2+m)}+\frac {3 a \left (a b B+A \left (b^2+a c\right )\right ) (e x)^{3+m}}{e^3 (3+m)}+\frac {\left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) (e x)^{4+m}}{e^4 (4+m)}+\frac {\left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) (e x)^{5+m}}{e^5 (5+m)}+\frac {3 c \left (b^2 B+A b c+a B c\right ) (e x)^{6+m}}{e^6 (6+m)}+\frac {c^2 (3 b B+A c) (e x)^{7+m}}{e^7 (7+m)}+\frac {B c^3 (e x)^{8+m}}{e^8 (8+m)} \]
a^3*A*(e*x)^(1+m)/e/(1+m)+a^2*(3*A*b+B*a)*(e*x)^(2+m)/e^2/(2+m)+3*a*(a*b*B +A*(a*c+b^2))*(e*x)^(3+m)/e^3/(3+m)+(3*a*B*(a*c+b^2)+A*(6*a*b*c+b^3))*(e*x )^(4+m)/e^4/(4+m)+(3*A*a*c^2+3*A*b^2*c+6*B*a*b*c+B*b^3)*(e*x)^(5+m)/e^5/(5 +m)+3*c*(A*b*c+B*a*c+B*b^2)*(e*x)^(6+m)/e^6/(6+m)+c^2*(A*c+3*B*b)*(e*x)^(7 +m)/e^7/(7+m)+B*c^3*(e*x)^(8+m)/e^8/(8+m)
Leaf count is larger than twice the leaf count of optimal. \(672\) vs. \(2(240)=480\).
Time = 1.86 (sec) , antiderivative size = 672, normalized size of antiderivative = 2.80 \[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^3 \, dx=\frac {(e x)^m \left (x (3 b B+A c (8+m)+B c (7+m) x) (a+x (b+c x))^3+\frac {3 \left (-x \left (a c (6+m) (b B (1+m)-2 A c (8+m))+2 b \left (b^2 B (4+m)-2 a B c (7+m)-A b c (8+m)\right )+c (5+m) \left (b^2 B (4+m)-2 a B c (7+m)-A b c (8+m)\right ) x\right ) (a+x (b+c x))^2+\frac {2 x \left (-\frac {2 a^2 c (4+m) \left (2 a c (6+m) (b B (1+m)-2 A c (8+m))-b (1+m) \left (b^2 B (4+m)-2 a B c (7+m)-A b c (8+m)\right )\right )}{1+m}+a b \left (a b c (6+m) (b B (1+m)-2 A c (8+m))-\left (b^2 (3+m)-2 a c (5+m)\right ) \left (b^2 B (4+m)-2 a B c (7+m)-A b c (8+m)\right )\right )-\frac {a b c (4+m) \left (2 a c (6+m) (b B (1+m)-2 A c (8+m))-b (1+m) \left (b^2 B (4+m)-2 a B c (7+m)-A b c (8+m)\right )\right ) x}{2+m}+\frac {\left (b^2 (2+m)-2 a c (3+m)\right ) \left (a b c (6+m) (b B (1+m)-2 A c (8+m))-\left (b^2 (3+m)-2 a c (5+m)\right ) \left (b^2 B (4+m)-2 a B c (7+m)-A b c (8+m)\right )\right ) x}{2+m}+\left (-a c (4+m) \left (2 a c (6+m) (b B (1+m)-2 A c (8+m))-b (1+m) \left (b^2 B (4+m)-2 a B c (7+m)-A b c (8+m)\right )\right )+b \left (-a b c (6+m) (b B (1+m)-2 A c (8+m))+\left (b^2 (3+m)-2 a c (5+m)\right ) \left (b^2 B (4+m)-2 a B c (7+m)-A b c (8+m)\right )\right )+c (3+m) \left (-a b c (6+m) (b B (1+m)-2 A c (8+m))+\left (b^2 (3+m)-2 a c (5+m)\right ) \left (b^2 B (4+m)-2 a B c (7+m)-A b c (8+m)\right )\right ) x\right ) (a+x (b+c x))\right )}{c (3+m) (4+m)}\right )}{c (5+m) (6+m)}\right )}{c (7+m) (8+m)} \]
((e*x)^m*(x*(3*b*B + A*c*(8 + m) + B*c*(7 + m)*x)*(a + x*(b + c*x))^3 + (3 *(-(x*(a*c*(6 + m)*(b*B*(1 + m) - 2*A*c*(8 + m)) + 2*b*(b^2*B*(4 + m) - 2* a*B*c*(7 + m) - A*b*c*(8 + m)) + c*(5 + m)*(b^2*B*(4 + m) - 2*a*B*c*(7 + m ) - A*b*c*(8 + m))*x)*(a + x*(b + c*x))^2) + (2*x*((-2*a^2*c*(4 + m)*(2*a* c*(6 + m)*(b*B*(1 + m) - 2*A*c*(8 + m)) - b*(1 + m)*(b^2*B*(4 + m) - 2*a*B *c*(7 + m) - A*b*c*(8 + m))))/(1 + m) + a*b*(a*b*c*(6 + m)*(b*B*(1 + m) - 2*A*c*(8 + m)) - (b^2*(3 + m) - 2*a*c*(5 + m))*(b^2*B*(4 + m) - 2*a*B*c*(7 + m) - A*b*c*(8 + m))) - (a*b*c*(4 + m)*(2*a*c*(6 + m)*(b*B*(1 + m) - 2*A *c*(8 + m)) - b*(1 + m)*(b^2*B*(4 + m) - 2*a*B*c*(7 + m) - A*b*c*(8 + m))) *x)/(2 + m) + ((b^2*(2 + m) - 2*a*c*(3 + m))*(a*b*c*(6 + m)*(b*B*(1 + m) - 2*A*c*(8 + m)) - (b^2*(3 + m) - 2*a*c*(5 + m))*(b^2*B*(4 + m) - 2*a*B*c*( 7 + m) - A*b*c*(8 + m)))*x)/(2 + m) + (-(a*c*(4 + m)*(2*a*c*(6 + m)*(b*B*( 1 + m) - 2*A*c*(8 + m)) - b*(1 + m)*(b^2*B*(4 + m) - 2*a*B*c*(7 + m) - A*b *c*(8 + m)))) + b*(-(a*b*c*(6 + m)*(b*B*(1 + m) - 2*A*c*(8 + m))) + (b^2*( 3 + m) - 2*a*c*(5 + m))*(b^2*B*(4 + m) - 2*a*B*c*(7 + m) - A*b*c*(8 + m))) + c*(3 + m)*(-(a*b*c*(6 + m)*(b*B*(1 + m) - 2*A*c*(8 + m))) + (b^2*(3 + m ) - 2*a*c*(5 + m))*(b^2*B*(4 + m) - 2*a*B*c*(7 + m) - A*b*c*(8 + m)))*x)*( a + x*(b + c*x))))/(c*(3 + m)*(4 + m))))/(c*(5 + m)*(6 + m))))/(c*(7 + m)* (8 + m))
Time = 0.44 (sec) , antiderivative size = 240, 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.
| \(\displaystyle \int (A+B x) (e x)^m \left (a+b x+c x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (a^3 A (e x)^m+\frac {a^2 (e x)^{m+1} (a B+3 A b)}{e}+\frac {3 c (e x)^{m+5} \left (a B c+A b c+b^2 B\right )}{e^5}+\frac {3 a (e x)^{m+2} \left (A \left (a c+b^2\right )+a b B\right )}{e^2}+\frac {(e x)^{m+4} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )}{e^4}+\frac {(e x)^{m+3} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )}{e^3}+\frac {c^2 (e x)^{m+6} (A c+3 b B)}{e^6}+\frac {B c^3 (e x)^{m+7}}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3 A (e x)^{m+1}}{e (m+1)}+\frac {a^2 (e x)^{m+2} (a B+3 A b)}{e^2 (m+2)}+\frac {3 c (e x)^{m+6} \left (a B c+A b c+b^2 B\right )}{e^6 (m+6)}+\frac {3 a (e x)^{m+3} \left (A \left (a c+b^2\right )+a b B\right )}{e^3 (m+3)}+\frac {(e x)^{m+5} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )}{e^5 (m+5)}+\frac {(e x)^{m+4} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )}{e^4 (m+4)}+\frac {c^2 (e x)^{m+7} (A c+3 b B)}{e^7 (m+7)}+\frac {B c^3 (e x)^{m+8}}{e^8 (m+8)}\) |
(a^3*A*(e*x)^(1 + m))/(e*(1 + m)) + (a^2*(3*A*b + a*B)*(e*x)^(2 + m))/(e^2 *(2 + m)) + (3*a*(a*b*B + A*(b^2 + a*c))*(e*x)^(3 + m))/(e^3*(3 + m)) + (( 3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*(e*x)^(4 + m))/(e^4*(4 + m)) + ((b^ 3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*(e*x)^(5 + m))/(e^5*(5 + m)) + (3 *c*(b^2*B + A*b*c + a*B*c)*(e*x)^(6 + m))/(e^6*(6 + m)) + (c^2*(3*b*B + A* c)*(e*x)^(7 + m))/(e^7*(7 + m)) + (B*c^3*(e*x)^(8 + m))/(e^8*(8 + m))
3.11.83.3.1 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 = 0.36 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.01
| method | result | size |
| norman | \(\frac {\left (3 A a \,c^{2}+3 A \,b^{2} c +6 B a b c +B \,b^{3}\right ) x^{5} {\mathrm e}^{m \ln \left (e x \right )}}{5+m}+\frac {\left (6 A a b c +A \,b^{3}+3 B \,a^{2} c +3 B a \,b^{2}\right ) x^{4} {\mathrm e}^{m \ln \left (e x \right )}}{4+m}+\frac {A \,a^{3} x \,{\mathrm e}^{m \ln \left (e x \right )}}{1+m}+\frac {B \,c^{3} x^{8} {\mathrm e}^{m \ln \left (e x \right )}}{8+m}+\frac {a^{2} \left (3 A b +B a \right ) x^{2} {\mathrm e}^{m \ln \left (e x \right )}}{2+m}+\frac {c^{2} \left (A c +3 B b \right ) x^{7} {\mathrm e}^{m \ln \left (e x \right )}}{7+m}+\frac {3 a \left (A a c +A \,b^{2}+a b B \right ) x^{3} {\mathrm e}^{m \ln \left (e x \right )}}{3+m}+\frac {3 c \left (A b c +B a c +B \,b^{2}\right ) x^{6} {\mathrm e}^{m \ln \left (e x \right )}}{6+m}\) | \(242\) |
| gosper | \(\text {Expression too large to display}\) | \(1903\) |
| risch | \(\text {Expression too large to display}\) | \(1903\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(2737\) |
(3*A*a*c^2+3*A*b^2*c+6*B*a*b*c+B*b^3)/(5+m)*x^5*exp(m*ln(e*x))+(6*A*a*b*c+ A*b^3+3*B*a^2*c+3*B*a*b^2)/(4+m)*x^4*exp(m*ln(e*x))+A*a^3/(1+m)*x*exp(m*ln (e*x))+B*c^3/(8+m)*x^8*exp(m*ln(e*x))+a^2*(3*A*b+B*a)/(2+m)*x^2*exp(m*ln(e *x))+c^2*(A*c+3*B*b)/(7+m)*x^7*exp(m*ln(e*x))+3*a*(A*a*c+A*b^2+B*a*b)/(3+m )*x^3*exp(m*ln(e*x))+3*c*(A*b*c+B*a*c+B*b^2)/(6+m)*x^6*exp(m*ln(e*x))
Leaf count of result is larger than twice the leaf count of optimal. 1350 vs. \(2 (240) = 480\).
Time = 0.28 (sec) , antiderivative size = 1350, normalized size of antiderivative = 5.62 \[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
((B*c^3*m^7 + 28*B*c^3*m^6 + 322*B*c^3*m^5 + 1960*B*c^3*m^4 + 6769*B*c^3*m ^3 + 13132*B*c^3*m^2 + 13068*B*c^3*m + 5040*B*c^3)*x^8 + ((3*B*b*c^2 + A*c ^3)*m^7 + 29*(3*B*b*c^2 + A*c^3)*m^6 + 343*(3*B*b*c^2 + A*c^3)*m^5 + 2135* (3*B*b*c^2 + A*c^3)*m^4 + 17280*B*b*c^2 + 5760*A*c^3 + 7504*(3*B*b*c^2 + A *c^3)*m^3 + 14756*(3*B*b*c^2 + A*c^3)*m^2 + 14832*(3*B*b*c^2 + A*c^3)*m)*x ^7 + 3*((B*b^2*c + (B*a + A*b)*c^2)*m^7 + 30*(B*b^2*c + (B*a + A*b)*c^2)*m ^6 + 366*(B*b^2*c + (B*a + A*b)*c^2)*m^5 + 2340*(B*b^2*c + (B*a + A*b)*c^2 )*m^4 + 6720*B*b^2*c + 8409*(B*b^2*c + (B*a + A*b)*c^2)*m^3 + 6720*(B*a + A*b)*c^2 + 16830*(B*b^2*c + (B*a + A*b)*c^2)*m^2 + 17144*(B*b^2*c + (B*a + A*b)*c^2)*m)*x^6 + ((B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*m^7 + 31* (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*m^6 + 391*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*m^5 + 2581*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2 )*c)*m^4 + 8064*B*b^3 + 24192*A*a*c^2 + 9544*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a *b + A*b^2)*c)*m^3 + 19564*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*m^2 + 24192*(2*B*a*b + A*b^2)*c + 20304*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b ^2)*c)*m)*x^5 + ((3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*m^7 + 32*(3*B *a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*m^6 + 418*(3*B*a*b^2 + A*b^3 + 3*( B*a^2 + 2*A*a*b)*c)*m^5 + 2864*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c) *m^4 + 30240*B*a*b^2 + 10080*A*b^3 + 10993*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*m^3 + 23312*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*m^...
Leaf count of result is larger than twice the leaf count of optimal. 11116 vs. \(2 (230) = 460\).
Time = 0.95 (sec) , antiderivative size = 11116, normalized size of antiderivative = 46.32 \[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
Piecewise(((-A*a**3/(7*x**7) - A*a**2*b/(2*x**6) - 3*A*a**2*c/(5*x**5) - 3 *A*a*b**2/(5*x**5) - 3*A*a*b*c/(2*x**4) - A*a*c**2/x**3 - A*b**3/(4*x**4) - A*b**2*c/x**3 - 3*A*b*c**2/(2*x**2) - A*c**3/x - B*a**3/(6*x**6) - 3*B*a **2*b/(5*x**5) - 3*B*a**2*c/(4*x**4) - 3*B*a*b**2/(4*x**4) - 2*B*a*b*c/x** 3 - 3*B*a*c**2/(2*x**2) - B*b**3/(3*x**3) - 3*B*b**2*c/(2*x**2) - 3*B*b*c* *2/x + B*c**3*log(x))/e**8, Eq(m, -8)), ((-A*a**3/(6*x**6) - 3*A*a**2*b/(5 *x**5) - 3*A*a**2*c/(4*x**4) - 3*A*a*b**2/(4*x**4) - 2*A*a*b*c/x**3 - 3*A* a*c**2/(2*x**2) - A*b**3/(3*x**3) - 3*A*b**2*c/(2*x**2) - 3*A*b*c**2/x + A *c**3*log(x) - B*a**3/(5*x**5) - 3*B*a**2*b/(4*x**4) - B*a**2*c/x**3 - B*a *b**2/x**3 - 3*B*a*b*c/x**2 - 3*B*a*c**2/x - B*b**3/(2*x**2) - 3*B*b**2*c/ x + 3*B*b*c**2*log(x) + B*c**3*x)/e**7, Eq(m, -7)), ((-A*a**3/(5*x**5) - 3 *A*a**2*b/(4*x**4) - A*a**2*c/x**3 - A*a*b**2/x**3 - 3*A*a*b*c/x**2 - 3*A* a*c**2/x - A*b**3/(2*x**2) - 3*A*b**2*c/x + 3*A*b*c**2*log(x) + A*c**3*x - B*a**3/(4*x**4) - B*a**2*b/x**3 - 3*B*a**2*c/(2*x**2) - 3*B*a*b**2/(2*x** 2) - 6*B*a*b*c/x + 3*B*a*c**2*log(x) - B*b**3/x + 3*B*b**2*c*log(x) + 3*B* b*c**2*x + B*c**3*x**2/2)/e**6, Eq(m, -6)), ((-A*a**3/(4*x**4) - A*a**2*b/ x**3 - 3*A*a**2*c/(2*x**2) - 3*A*a*b**2/(2*x**2) - 6*A*a*b*c/x + 3*A*a*c** 2*log(x) - A*b**3/x + 3*A*b**2*c*log(x) + 3*A*b*c**2*x + A*c**3*x**2/2 - B *a**3/(3*x**3) - 3*B*a**2*b/(2*x**2) - 3*B*a**2*c/x - 3*B*a*b**2/x + 6*B*a *b*c*log(x) + 3*B*a*c**2*x + B*b**3*log(x) + 3*B*b**2*c*x + 3*B*b*c**2*...
Time = 0.23 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.70 \[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^3 \, dx=\frac {B c^{3} e^{m} x^{8} x^{m}}{m + 8} + \frac {3 \, B b c^{2} e^{m} x^{7} x^{m}}{m + 7} + \frac {A c^{3} e^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, B b^{2} c e^{m} x^{6} x^{m}}{m + 6} + \frac {3 \, B a c^{2} e^{m} x^{6} x^{m}}{m + 6} + \frac {3 \, A b c^{2} e^{m} x^{6} x^{m}}{m + 6} + \frac {B b^{3} e^{m} x^{5} x^{m}}{m + 5} + \frac {6 \, B a b c e^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, A b^{2} c e^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, A a c^{2} e^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, B a b^{2} e^{m} x^{4} x^{m}}{m + 4} + \frac {A b^{3} e^{m} x^{4} x^{m}}{m + 4} + \frac {3 \, B a^{2} c e^{m} x^{4} x^{m}}{m + 4} + \frac {6 \, A a b c e^{m} x^{4} x^{m}}{m + 4} + \frac {3 \, B a^{2} b e^{m} x^{3} x^{m}}{m + 3} + \frac {3 \, A a b^{2} e^{m} x^{3} x^{m}}{m + 3} + \frac {3 \, A a^{2} c e^{m} x^{3} x^{m}}{m + 3} + \frac {B a^{3} e^{m} x^{2} x^{m}}{m + 2} + \frac {3 \, A a^{2} b e^{m} x^{2} x^{m}}{m + 2} + \frac {\left (e x\right )^{m + 1} A a^{3}}{e {\left (m + 1\right )}} \]
B*c^3*e^m*x^8*x^m/(m + 8) + 3*B*b*c^2*e^m*x^7*x^m/(m + 7) + A*c^3*e^m*x^7* x^m/(m + 7) + 3*B*b^2*c*e^m*x^6*x^m/(m + 6) + 3*B*a*c^2*e^m*x^6*x^m/(m + 6 ) + 3*A*b*c^2*e^m*x^6*x^m/(m + 6) + B*b^3*e^m*x^5*x^m/(m + 5) + 6*B*a*b*c* e^m*x^5*x^m/(m + 5) + 3*A*b^2*c*e^m*x^5*x^m/(m + 5) + 3*A*a*c^2*e^m*x^5*x^ m/(m + 5) + 3*B*a*b^2*e^m*x^4*x^m/(m + 4) + A*b^3*e^m*x^4*x^m/(m + 4) + 3* B*a^2*c*e^m*x^4*x^m/(m + 4) + 6*A*a*b*c*e^m*x^4*x^m/(m + 4) + 3*B*a^2*b*e^ m*x^3*x^m/(m + 3) + 3*A*a*b^2*e^m*x^3*x^m/(m + 3) + 3*A*a^2*c*e^m*x^3*x^m/ (m + 3) + B*a^3*e^m*x^2*x^m/(m + 2) + 3*A*a^2*b*e^m*x^2*x^m/(m + 2) + (e*x )^(m + 1)*A*a^3/(e*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 2736 vs. \(2 (240) = 480\).
Time = 0.30 (sec) , antiderivative size = 2736, normalized size of antiderivative = 11.40 \[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
((e*x)^m*B*c^3*m^7*x^8 + 3*(e*x)^m*B*b*c^2*m^7*x^7 + (e*x)^m*A*c^3*m^7*x^7 + 28*(e*x)^m*B*c^3*m^6*x^8 + 3*(e*x)^m*B*b^2*c*m^7*x^6 + 3*(e*x)^m*B*a*c^ 2*m^7*x^6 + 3*(e*x)^m*A*b*c^2*m^7*x^6 + 87*(e*x)^m*B*b*c^2*m^6*x^7 + 29*(e *x)^m*A*c^3*m^6*x^7 + 322*(e*x)^m*B*c^3*m^5*x^8 + (e*x)^m*B*b^3*m^7*x^5 + 6*(e*x)^m*B*a*b*c*m^7*x^5 + 3*(e*x)^m*A*b^2*c*m^7*x^5 + 3*(e*x)^m*A*a*c^2* m^7*x^5 + 90*(e*x)^m*B*b^2*c*m^6*x^6 + 90*(e*x)^m*B*a*c^2*m^6*x^6 + 90*(e* x)^m*A*b*c^2*m^6*x^6 + 1029*(e*x)^m*B*b*c^2*m^5*x^7 + 343*(e*x)^m*A*c^3*m^ 5*x^7 + 1960*(e*x)^m*B*c^3*m^4*x^8 + 3*(e*x)^m*B*a*b^2*m^7*x^4 + (e*x)^m*A *b^3*m^7*x^4 + 3*(e*x)^m*B*a^2*c*m^7*x^4 + 6*(e*x)^m*A*a*b*c*m^7*x^4 + 31* (e*x)^m*B*b^3*m^6*x^5 + 186*(e*x)^m*B*a*b*c*m^6*x^5 + 93*(e*x)^m*A*b^2*c*m ^6*x^5 + 93*(e*x)^m*A*a*c^2*m^6*x^5 + 1098*(e*x)^m*B*b^2*c*m^5*x^6 + 1098* (e*x)^m*B*a*c^2*m^5*x^6 + 1098*(e*x)^m*A*b*c^2*m^5*x^6 + 6405*(e*x)^m*B*b* c^2*m^4*x^7 + 2135*(e*x)^m*A*c^3*m^4*x^7 + 6769*(e*x)^m*B*c^3*m^3*x^8 + 3* (e*x)^m*B*a^2*b*m^7*x^3 + 3*(e*x)^m*A*a*b^2*m^7*x^3 + 3*(e*x)^m*A*a^2*c*m^ 7*x^3 + 96*(e*x)^m*B*a*b^2*m^6*x^4 + 32*(e*x)^m*A*b^3*m^6*x^4 + 96*(e*x)^m *B*a^2*c*m^6*x^4 + 192*(e*x)^m*A*a*b*c*m^6*x^4 + 391*(e*x)^m*B*b^3*m^5*x^5 + 2346*(e*x)^m*B*a*b*c*m^5*x^5 + 1173*(e*x)^m*A*b^2*c*m^5*x^5 + 1173*(e*x )^m*A*a*c^2*m^5*x^5 + 7020*(e*x)^m*B*b^2*c*m^4*x^6 + 7020*(e*x)^m*B*a*c^2* m^4*x^6 + 7020*(e*x)^m*A*b*c^2*m^4*x^6 + 22512*(e*x)^m*B*b*c^2*m^3*x^7 + 7 504*(e*x)^m*A*c^3*m^3*x^7 + 13132*(e*x)^m*B*c^3*m^2*x^8 + (e*x)^m*B*a^3...
Time = 10.93 (sec) , antiderivative size = 769, normalized size of antiderivative = 3.20 \[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^3 \, dx=\frac {x^4\,{\left (e\,x\right )}^m\,\left (3\,B\,c\,a^2+3\,B\,a\,b^2+6\,A\,c\,a\,b+A\,b^3\right )\,\left (m^7+32\,m^6+418\,m^5+2864\,m^4+10993\,m^3+23312\,m^2+24876\,m+10080\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {x^5\,{\left (e\,x\right )}^m\,\left (B\,b^3+3\,A\,b^2\,c+6\,B\,a\,b\,c+3\,A\,a\,c^2\right )\,\left (m^7+31\,m^6+391\,m^5+2581\,m^4+9544\,m^3+19564\,m^2+20304\,m+8064\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {A\,a^3\,x\,{\left (e\,x\right )}^m\,\left (m^7+35\,m^6+511\,m^5+4025\,m^4+18424\,m^3+48860\,m^2+69264\,m+40320\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {3\,a\,x^3\,{\left (e\,x\right )}^m\,\left (A\,b^2+B\,a\,b+A\,a\,c\right )\,\left (m^7+33\,m^6+447\,m^5+3195\,m^4+12864\,m^3+28692\,m^2+32048\,m+13440\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {3\,c\,x^6\,{\left (e\,x\right )}^m\,\left (B\,b^2+A\,c\,b+B\,a\,c\right )\,\left (m^7+30\,m^6+366\,m^5+2340\,m^4+8409\,m^3+16830\,m^2+17144\,m+6720\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {B\,c^3\,x^8\,{\left (e\,x\right )}^m\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {a^2\,x^2\,{\left (e\,x\right )}^m\,\left (3\,A\,b+B\,a\right )\,\left (m^7+34\,m^6+478\,m^5+3580\,m^4+15289\,m^3+36706\,m^2+44712\,m+20160\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {c^2\,x^7\,{\left (e\,x\right )}^m\,\left (A\,c+3\,B\,b\right )\,\left (m^7+29\,m^6+343\,m^5+2135\,m^4+7504\,m^3+14756\,m^2+14832\,m+5760\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320} \]
(x^4*(e*x)^m*(A*b^3 + 3*B*a*b^2 + 3*B*a^2*c + 6*A*a*b*c)*(24876*m + 23312* m^2 + 10993*m^3 + 2864*m^4 + 418*m^5 + 32*m^6 + m^7 + 10080))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 4 0320) + (x^5*(e*x)^m*(B*b^3 + 3*A*a*c^2 + 3*A*b^2*c + 6*B*a*b*c)*(20304*m + 19564*m^2 + 9544*m^3 + 2581*m^4 + 391*m^5 + 31*m^6 + m^7 + 8064))/(10958 4*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m ^8 + 40320) + (A*a^3*x*(e*x)^m*(69264*m + 48860*m^2 + 18424*m^3 + 4025*m^4 + 511*m^5 + 35*m^6 + m^7 + 40320))/(109584*m + 118124*m^2 + 67284*m^3 + 2 2449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (3*a*x^3*(e*x)^m*( A*b^2 + A*a*c + B*a*b)*(32048*m + 28692*m^2 + 12864*m^3 + 3195*m^4 + 447*m ^5 + 33*m^6 + m^7 + 13440))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (3*c*x^6*(e*x)^m*(B*b^2 + A*b*c + B*a*c)*(17144*m + 16830*m^2 + 8409*m^3 + 2340*m^4 + 366*m^5 + 30*m ^6 + m^7 + 6720))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^ 5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (B*c^3*x^8*(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 + 11 8124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 403 20) + (a^2*x^2*(e*x)^m*(3*A*b + B*a)*(44712*m + 36706*m^2 + 15289*m^3 + 35 80*m^4 + 478*m^5 + 34*m^6 + m^7 + 20160))/(109584*m + 118124*m^2 + 67284*m ^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (c^2*x^7*...