Integrand size = 27, antiderivative size = 117 \[ \int (e x)^m \left (a+b x^n\right ) \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=\frac {a A c (e x)^{1+m}}{e (1+m)}+\frac {(A b c+a B c+a A d) x^n (e x)^{1+m}}{e (1+m+n)}+\frac {(b B c+A b d+a B d) x^{2 n} (e x)^{1+m}}{e (1+m+2 n)}+\frac {b B d x^{3 n} (e x)^{1+m}}{e (1+m+3 n)} \] Output:
a*A*c*(e*x)^(1+m)/e/(1+m)+(A*a*d+A*b*c+B*a*c)*x^n*(e*x)^(1+m)/e/(1+m+n)+(A *b*d+B*a*d+B*b*c)*x^(2*n)*(e*x)^(1+m)/e/(1+m+2*n)+b*B*d*x^(3*n)*(e*x)^(1+m )/e/(1+m+3*n)
Time = 0.19 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.72 \[ \int (e x)^m \left (a+b x^n\right ) \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=x (e x)^m \left (\frac {a A c}{1+m}+\frac {(A b c+a B c+a A d) x^n}{1+m+n}+\frac {(b B c+A b d+a B d) x^{2 n}}{1+m+2 n}+\frac {b B d x^{3 n}}{1+m+3 n}\right ) \] Input:
Integrate[(e*x)^m*(a + b*x^n)*(A + B*x^n)*(c + d*x^n),x]
Output:
x*(e*x)^m*((a*A*c)/(1 + m) + ((A*b*c + a*B*c + a*A*d)*x^n)/(1 + m + n) + ( (b*B*c + A*b*d + a*B*d)*x^(2*n))/(1 + m + 2*n) + (b*B*d*x^(3*n))/(1 + m + 3*n))
Time = 0.44 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.92, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1040, 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.
| \(\displaystyle \int (e x)^m \left (a+b x^n\right ) \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx\) |
| \(\Big \downarrow \) 1040 |
| \(\displaystyle \int \left (x^{2 n} (e x)^m (a B d+A b d+b B c)+x^n (e x)^m (a A d+a B c+A b c)+a A c (e x)^m+b B d x^{3 n} (e x)^m\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^{n+1} (e x)^m (a A d+a B c+A b c)}{m+n+1}+\frac {x^{2 n+1} (e x)^m (a B d+A b d+b B c)}{m+2 n+1}+\frac {a A c (e x)^{m+1}}{e (m+1)}+\frac {b B d x^{3 n+1} (e x)^m}{m+3 n+1}\) |
Input:
Int[(e*x)^m*(a + b*x^n)*(A + B*x^n)*(c + d*x^n),x]
Output:
((A*b*c + a*B*c + a*A*d)*x^(1 + n)*(e*x)^m)/(1 + m + n) + ((b*B*c + A*b*d + a*B*d)*x^(1 + 2*n)*(e*x)^m)/(1 + m + 2*n) + (b*B*d*x^(1 + 3*n)*(e*x)^m)/ (1 + m + 3*n) + (a*A*c*(e*x)^(1 + m))/(e*(1 + m))
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x_Symbol] :> Int[ExpandIntegrand[ (g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a, b, c , d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.30 (sec) , antiderivative size = 858, normalized size of antiderivative = 7.33
| method | result | size |
| risch | \(\frac {x \left (10 A a d m n \,x^{n}+10 A b c m n \,x^{n}+10 B a c m n \,x^{n}+3 B b d \,m^{2} n \,x^{3 n}+8 B a d m n \,x^{2 n}+8 B b c m n \,x^{2 n}+4 B b c \,m^{2} n \,x^{2 n}+3 B b c m \,n^{2} x^{2 n}+6 B b d m n \,x^{3 n}+3 B a d m \,n^{2} x^{2 n}+8 A b d m n \,x^{2 n}+2 B b d m \,n^{2} x^{3 n}+4 A b d \,m^{2} n \,x^{2 n}+3 A b d m \,n^{2} x^{2 n}+4 B a d \,m^{2} n \,x^{2 n}+A a d \,m^{3} x^{n}+A b c \,m^{3} x^{n}+B a c \,m^{3} x^{n}+A a c +6 A a c n +3 B \,x^{n} a c m +5 B \,x^{n} a c n +3 A \,x^{n} b c m +5 A \,x^{n} b c n +3 A \,x^{n} a d m +5 A \,x^{n} a d n +3 A a c m +3 B a c \,m^{2} x^{n}+6 B a c \,n^{2} x^{n}+d a \,x^{n} A +c A \,x^{n} b +c B \,x^{n} a +A a c \,m^{3}+3 A a c \,m^{2}+11 A a c \,n^{2}+6 A a c \,n^{3}+3 A a d \,m^{2} x^{n}+6 A a d \,n^{2} x^{n}+3 A b c \,m^{2} x^{n}+6 A b c \,n^{2} x^{n}+12 A a c m n +6 A a c \,m^{2} n +11 A a c m \,n^{2}+5 A a d \,m^{2} n \,x^{n}+6 A a d m \,n^{2} x^{n}+5 A b c \,m^{2} n \,x^{n}+6 A b c m \,n^{2} x^{n}+5 B a c \,m^{2} n \,x^{n}+6 B a c m \,n^{2} x^{n}+A b d \,x^{2 n}+B a d \,x^{2 n}+B b c \,x^{2 n}+B b d \,x^{3 n}+B b c \,m^{3} x^{2 n}+3 B b d \,m^{2} x^{3 n}+2 B b d \,n^{2} x^{3 n}+3 A b d \,m^{2} x^{2 n}+3 A b d \,n^{2} x^{2 n}+3 B a d \,m^{2} x^{2 n}+3 B a d \,n^{2} x^{2 n}+4 B \,x^{2 n} b c n +3 B \,x^{3 n} b d n +3 A \,x^{2 n} b d m +4 A \,x^{2 n} b d n +3 B \,x^{2 n} a d m +4 B \,x^{2 n} a d n +3 B \,x^{2 n} b c m +3 B \,x^{3 n} b d m +3 B b c \,m^{2} x^{2 n}+3 B b c \,n^{2} x^{2 n}+B b d \,m^{3} x^{3 n}+A b d \,m^{3} x^{2 n}+B a d \,m^{3} x^{2 n}\right ) e^{m} x^{m} {\mathrm e}^{\frac {i \operatorname {csgn}\left (i e x \right ) \pi m \left (\operatorname {csgn}\left (i e x \right )-\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i e x \right )+\operatorname {csgn}\left (i e \right )\right )}{2}}}{\left (1+m \right ) \left (1+m +n \right ) \left (1+m +2 n \right ) \left (1+m +3 n \right )}\) | \(858\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1249\) |
| orering | \(\text {Expression too large to display}\) | \(1838\) |
Input:
int((e*x)^m*(a+b*x^n)*(A+B*x^n)*(c+d*x^n),x,method=_RETURNVERBOSE)
Output:
x*(10*A*a*d*m*n*x^n+10*A*b*c*m*n*x^n+10*B*a*c*m*n*x^n+3*B*b*d*m^2*n*(x^n)^ 3+B*b*c*m^3*(x^n)^2+3*B*b*d*m^2*(x^n)^3+2*B*b*d*n^2*(x^n)^3+A*a*d*m^3*x^n+ A*b*c*m^3*x^n+3*A*b*d*m^2*(x^n)^2+3*A*b*d*n^2*(x^n)^2+B*a*c*m^3*x^n+3*B*a* d*m^2*(x^n)^2+3*B*a*d*n^2*(x^n)^2+A*a*c+8*B*a*d*m*n*(x^n)^2+8*B*b*c*m*n*(x ^n)^2+6*A*a*c*n+3*B*x^n*a*c*m+5*B*x^n*a*c*n+3*A*x^n*b*c*m+5*A*x^n*b*c*n+3* A*x^n*a*d*m+5*A*x^n*a*d*n+4*B*(x^n)^2*b*c*n+3*B*(x^n)^3*b*d*n+3*A*(x^n)^2* b*d*m+4*A*(x^n)^2*b*d*n+3*B*(x^n)^2*a*d*m+4*B*(x^n)^2*a*d*n+3*B*(x^n)^2*b* c*m+3*B*(x^n)^3*b*d*m+3*A*a*c*m+3*B*a*c*m^2*x^n+6*B*a*c*n^2*x^n+B*a*d*(x^n )^2+B*b*c*(x^n)^2+d*a*x^n*A+c*A*x^n*b+c*B*x^n*a+B*b*d*(x^n)^3+A*b*d*(x^n)^ 2+A*a*c*m^3+3*A*a*c*m^2+11*A*a*c*n^2+6*A*a*c*n^3+3*B*b*c*m^2*(x^n)^2+3*B*b *c*n^2*(x^n)^2+3*A*a*d*m^2*x^n+6*A*a*d*n^2*x^n+3*A*b*c*m^2*x^n+6*A*b*c*n^2 *x^n+12*A*a*c*m*n+4*B*b*c*m^2*n*(x^n)^2+3*B*b*c*m*n^2*(x^n)^2+6*B*b*d*m*n* (x^n)^3+6*A*a*c*m^2*n+11*A*a*c*m*n^2+B*b*d*m^3*(x^n)^3+A*b*d*m^3*(x^n)^2+B *a*d*m^3*(x^n)^2+3*B*a*d*m*n^2*(x^n)^2+5*A*a*d*m^2*n*x^n+6*A*a*d*m*n^2*x^n +5*A*b*c*m^2*n*x^n+6*A*b*c*m*n^2*x^n+8*A*b*d*m*n*(x^n)^2+5*B*a*c*m^2*n*x^n +6*B*a*c*m*n^2*x^n+2*B*b*d*m*n^2*(x^n)^3+4*A*b*d*m^2*n*(x^n)^2+3*A*b*d*m*n ^2*(x^n)^2+4*B*a*d*m^2*n*(x^n)^2)/(1+m)/(1+m+n)/(1+m+2*n)/(1+m+3*n)*e^m*x^ m*exp(1/2*I*Pi*csgn(I*e*x)*m*(csgn(I*e*x)-csgn(I*x))*(-csgn(I*e*x)+csgn(I* e)))
Leaf count of result is larger than twice the leaf count of optimal. 562 vs. \(2 (117) = 234\).
Time = 0.14 (sec) , antiderivative size = 562, normalized size of antiderivative = 4.80 \[ \int (e x)^m \left (a+b x^n\right ) \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=\frac {{\left (B b d m^{3} + 3 \, B b d m^{2} + 3 \, B b d m + B b d + 2 \, {\left (B b d m + B b d\right )} n^{2} + 3 \, {\left (B b d m^{2} + 2 \, B b d m + B b d\right )} n\right )} x x^{3 \, n} e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} + {\left ({\left (B b c + {\left (B a + A b\right )} d\right )} m^{3} + B b c + 3 \, {\left (B b c + {\left (B a + A b\right )} d\right )} m^{2} + 3 \, {\left (B b c + {\left (B a + A b\right )} d + {\left (B b c + {\left (B a + A b\right )} d\right )} m\right )} n^{2} + {\left (B a + A b\right )} d + 3 \, {\left (B b c + {\left (B a + A b\right )} d\right )} m + 4 \, {\left (B b c + {\left (B b c + {\left (B a + A b\right )} d\right )} m^{2} + {\left (B a + A b\right )} d + 2 \, {\left (B b c + {\left (B a + A b\right )} d\right )} m\right )} n\right )} x x^{2 \, n} e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} + {\left ({\left (A a d + {\left (B a + A b\right )} c\right )} m^{3} + A a d + 3 \, {\left (A a d + {\left (B a + A b\right )} c\right )} m^{2} + 6 \, {\left (A a d + {\left (B a + A b\right )} c + {\left (A a d + {\left (B a + A b\right )} c\right )} m\right )} n^{2} + {\left (B a + A b\right )} c + 3 \, {\left (A a d + {\left (B a + A b\right )} c\right )} m + 5 \, {\left (A a d + {\left (A a d + {\left (B a + A b\right )} c\right )} m^{2} + {\left (B a + A b\right )} c + 2 \, {\left (A a d + {\left (B a + A b\right )} c\right )} m\right )} n\right )} x x^{n} e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} + {\left (A a c m^{3} + 6 \, A a c n^{3} + 3 \, A a c m^{2} + 3 \, A a c m + A a c + 11 \, {\left (A a c m + A a c\right )} n^{2} + 6 \, {\left (A a c m^{2} + 2 \, A a c m + A a c\right )} n\right )} x e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )}}{m^{4} + 6 \, {\left (m + 1\right )} n^{3} + 4 \, m^{3} + 11 \, {\left (m^{2} + 2 \, m + 1\right )} n^{2} + 6 \, m^{2} + 6 \, {\left (m^{3} + 3 \, m^{2} + 3 \, m + 1\right )} n + 4 \, m + 1} \] Input:
integrate((e*x)^m*(a+b*x^n)*(A+B*x^n)*(c+d*x^n),x, algorithm="fricas")
Output:
((B*b*d*m^3 + 3*B*b*d*m^2 + 3*B*b*d*m + B*b*d + 2*(B*b*d*m + B*b*d)*n^2 + 3*(B*b*d*m^2 + 2*B*b*d*m + B*b*d)*n)*x*x^(3*n)*e^(m*log(e) + m*log(x)) + ( (B*b*c + (B*a + A*b)*d)*m^3 + B*b*c + 3*(B*b*c + (B*a + A*b)*d)*m^2 + 3*(B *b*c + (B*a + A*b)*d + (B*b*c + (B*a + A*b)*d)*m)*n^2 + (B*a + A*b)*d + 3* (B*b*c + (B*a + A*b)*d)*m + 4*(B*b*c + (B*b*c + (B*a + A*b)*d)*m^2 + (B*a + A*b)*d + 2*(B*b*c + (B*a + A*b)*d)*m)*n)*x*x^(2*n)*e^(m*log(e) + m*log(x )) + ((A*a*d + (B*a + A*b)*c)*m^3 + A*a*d + 3*(A*a*d + (B*a + A*b)*c)*m^2 + 6*(A*a*d + (B*a + A*b)*c + (A*a*d + (B*a + A*b)*c)*m)*n^2 + (B*a + A*b)* c + 3*(A*a*d + (B*a + A*b)*c)*m + 5*(A*a*d + (A*a*d + (B*a + A*b)*c)*m^2 + (B*a + A*b)*c + 2*(A*a*d + (B*a + A*b)*c)*m)*n)*x*x^n*e^(m*log(e) + m*log (x)) + (A*a*c*m^3 + 6*A*a*c*n^3 + 3*A*a*c*m^2 + 3*A*a*c*m + A*a*c + 11*(A* a*c*m + A*a*c)*n^2 + 6*(A*a*c*m^2 + 2*A*a*c*m + A*a*c)*n)*x*e^(m*log(e) + m*log(x)))/(m^4 + 6*(m + 1)*n^3 + 4*m^3 + 11*(m^2 + 2*m + 1)*n^2 + 6*m^2 + 6*(m^3 + 3*m^2 + 3*m + 1)*n + 4*m + 1)
Leaf count of result is larger than twice the leaf count of optimal. 7796 vs. \(2 (109) = 218\).
Time = 5.21 (sec) , antiderivative size = 7796, normalized size of antiderivative = 66.63 \[ \int (e x)^m \left (a+b x^n\right ) \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=\text {Too large to display} \] Input:
integrate((e*x)**m*(a+b*x**n)*(A+B*x**n)*(c+d*x**n),x)
Output:
Piecewise(((A + B)*(a + b)*(c + d)*log(x)/e, Eq(m, -1) & Eq(n, 0)), ((A*a* c*log(x) + A*a*d*x**n/n + A*b*c*x**n/n + A*b*d*x**(2*n)/(2*n) + B*a*c*x**n /n + B*a*d*x**(2*n)/(2*n) + B*b*c*x**(2*n)/(2*n) + B*b*d*x**(3*n)/(3*n))/e , Eq(m, -1)), (A*a*c*Piecewise((0**(-3*n - 1)*x, Eq(e, 0)), (Piecewise((-1 /(3*n*(e*x)**(3*n)), Ne(n, 0)), (log(e*x), True))/e, True)) + A*a*d*Piecew ise((-x*x**n*(e*x)**(-3*n - 1)/(2*n), Ne(n, 0)), (x*x**n*(e*x)**(-3*n - 1) *log(x), True)) + A*b*c*Piecewise((-x*x**n*(e*x)**(-3*n - 1)/(2*n), Ne(n, 0)), (x*x**n*(e*x)**(-3*n - 1)*log(x), True)) + A*b*d*Piecewise((-x*x**(2* n)*(e*x)**(-3*n - 1)/n, Ne(n, 0)), (x*x**(2*n)*(e*x)**(-3*n - 1)*log(x), T rue)) + B*a*c*Piecewise((-x*x**n*(e*x)**(-3*n - 1)/(2*n), Ne(n, 0)), (x*x* *n*(e*x)**(-3*n - 1)*log(x), True)) + B*a*d*Piecewise((-x*x**(2*n)*(e*x)** (-3*n - 1)/n, Ne(n, 0)), (x*x**(2*n)*(e*x)**(-3*n - 1)*log(x), True)) + B* b*c*Piecewise((-x*x**(2*n)*(e*x)**(-3*n - 1)/n, Ne(n, 0)), (x*x**(2*n)*(e* x)**(-3*n - 1)*log(x), True)) + B*b*d*x*x**(3*n)*(e*x)**(-3*n - 1)*log(x), Eq(m, -3*n - 1)), (A*a*c*Piecewise((0**(-2*n - 1)*x, Eq(e, 0)), (Piecewis e((-1/(2*n*(e*x)**(2*n)), Ne(n, 0)), (log(e*x), True))/e, True)) + A*a*d*P iecewise((-x*x**n*(e*x)**(-2*n - 1)/n, Ne(n, 0)), (x*x**n*(e*x)**(-2*n - 1 )*log(x), True)) + A*b*c*Piecewise((-x*x**n*(e*x)**(-2*n - 1)/n, Ne(n, 0)) , (x*x**n*(e*x)**(-2*n - 1)*log(x), True)) + A*b*d*x*x**(2*n)*(e*x)**(-2*n - 1)*log(x) + B*a*c*Piecewise((-x*x**n*(e*x)**(-2*n - 1)/n, Ne(n, 0)),...
Time = 0.06 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.71 \[ \int (e x)^m \left (a+b x^n\right ) \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=\frac {B b d e^{m} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 1} + \frac {B b c e^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {B a d e^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {A b d e^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {B a c e^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 1} + \frac {A b c e^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 1} + \frac {A a d e^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 1} + \frac {\left (e x\right )^{m + 1} A a c}{e {\left (m + 1\right )}} \] Input:
integrate((e*x)^m*(a+b*x^n)*(A+B*x^n)*(c+d*x^n),x, algorithm="maxima")
Output:
B*b*d*e^m*x*e^(m*log(x) + 3*n*log(x))/(m + 3*n + 1) + B*b*c*e^m*x*e^(m*log (x) + 2*n*log(x))/(m + 2*n + 1) + B*a*d*e^m*x*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1) + A*b*d*e^m*x*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1) + B*a*c* e^m*x*e^(m*log(x) + n*log(x))/(m + n + 1) + A*b*c*e^m*x*e^(m*log(x) + n*lo g(x))/(m + n + 1) + A*a*d*e^m*x*e^(m*log(x) + n*log(x))/(m + n + 1) + (e*x )^(m + 1)*A*a*c/(e*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 3764 vs. \(2 (117) = 234\).
Time = 0.16 (sec) , antiderivative size = 3764, normalized size of antiderivative = 32.17 \[ \int (e x)^m \left (a+b x^n\right ) \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=\text {Too large to display} \] Input:
integrate((e*x)^m*(a+b*x^n)*(A+B*x^n)*(c+d*x^n),x, algorithm="giac")
Output:
(B*b*d*m^3*x*x^(3*n)*e^(m*log(e) + m*log(x)) + 3*B*b*d*m^2*n*x*x^(3*n)*e^( m*log(e) + m*log(x)) + 2*B*b*d*m*n^2*x*x^(3*n)*e^(m*log(e) + m*log(x)) + B *b*c*m^3*x*x^(2*n)*e^(m*log(e) + m*log(x)) + B*a*d*m^3*x*x^(2*n)*e^(m*log( e) + m*log(x)) + A*b*d*m^3*x*x^(2*n)*e^(m*log(e) + m*log(x)) + B*b*d*m^3*x *x^(2*n)*e^(m*log(e) + m*log(x)) + 4*B*b*c*m^2*n*x*x^(2*n)*e^(m*log(e) + m *log(x)) + 4*B*a*d*m^2*n*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 4*A*b*d*m^2*n *x*x^(2*n)*e^(m*log(e) + m*log(x)) + 3*B*b*d*m^2*n*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 3*B*b*c*m*n^2*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 3*B*a*d*m*n ^2*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 3*A*b*d*m*n^2*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 2*B*b*d*m*n^2*x*x^(2*n)*e^(m*log(e) + m*log(x)) + B*a*c*m^3 *x*x^n*e^(m*log(e) + m*log(x)) + A*b*c*m^3*x*x^n*e^(m*log(e) + m*log(x)) + B*b*c*m^3*x*x^n*e^(m*log(e) + m*log(x)) + A*a*d*m^3*x*x^n*e^(m*log(e) + m *log(x)) + B*a*d*m^3*x*x^n*e^(m*log(e) + m*log(x)) + A*b*d*m^3*x*x^n*e^(m* log(e) + m*log(x)) + B*b*d*m^3*x*x^n*e^(m*log(e) + m*log(x)) + 5*B*a*c*m^2 *n*x*x^n*e^(m*log(e) + m*log(x)) + 5*A*b*c*m^2*n*x*x^n*e^(m*log(e) + m*log (x)) + 4*B*b*c*m^2*n*x*x^n*e^(m*log(e) + m*log(x)) + 5*A*a*d*m^2*n*x*x^n*e ^(m*log(e) + m*log(x)) + 4*B*a*d*m^2*n*x*x^n*e^(m*log(e) + m*log(x)) + 4*A *b*d*m^2*n*x*x^n*e^(m*log(e) + m*log(x)) + 3*B*b*d*m^2*n*x*x^n*e^(m*log(e) + m*log(x)) + 6*B*a*c*m*n^2*x*x^n*e^(m*log(e) + m*log(x)) + 6*A*b*c*m*n^2 *x*x^n*e^(m*log(e) + m*log(x)) + 3*B*b*c*m*n^2*x*x^n*e^(m*log(e) + m*lo...
Time = 4.83 (sec) , antiderivative size = 271, normalized size of antiderivative = 2.32 \[ \int (e x)^m \left (a+b x^n\right ) \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=\frac {A\,a\,c\,x\,{\left (e\,x\right )}^m}{m+1}+\frac {x\,x^{2\,n}\,{\left (e\,x\right )}^m\,\left (A\,b\,d+B\,a\,d+B\,b\,c\right )\,\left (m^2+4\,m\,n+2\,m+3\,n^2+4\,n+1\right )}{m^3+6\,m^2\,n+3\,m^2+11\,m\,n^2+12\,m\,n+3\,m+6\,n^3+11\,n^2+6\,n+1}+\frac {x\,x^n\,{\left (e\,x\right )}^m\,\left (A\,a\,d+A\,b\,c+B\,a\,c\right )\,\left (m^2+5\,m\,n+2\,m+6\,n^2+5\,n+1\right )}{m^3+6\,m^2\,n+3\,m^2+11\,m\,n^2+12\,m\,n+3\,m+6\,n^3+11\,n^2+6\,n+1}+\frac {B\,b\,d\,x\,x^{3\,n}\,{\left (e\,x\right )}^m\,\left (m^2+3\,m\,n+2\,m+2\,n^2+3\,n+1\right )}{m^3+6\,m^2\,n+3\,m^2+11\,m\,n^2+12\,m\,n+3\,m+6\,n^3+11\,n^2+6\,n+1} \] Input:
int((e*x)^m*(A + B*x^n)*(a + b*x^n)*(c + d*x^n),x)
Output:
(A*a*c*x*(e*x)^m)/(m + 1) + (x*x^(2*n)*(e*x)^m*(A*b*d + B*a*d + B*b*c)*(2* m + 4*n + 4*m*n + m^2 + 3*n^2 + 1))/(3*m + 6*n + 12*m*n + 11*m*n^2 + 6*m^2 *n + 3*m^2 + m^3 + 11*n^2 + 6*n^3 + 1) + (x*x^n*(e*x)^m*(A*a*d + A*b*c + B *a*c)*(2*m + 5*n + 5*m*n + m^2 + 6*n^2 + 1))/(3*m + 6*n + 12*m*n + 11*m*n^ 2 + 6*m^2*n + 3*m^2 + m^3 + 11*n^2 + 6*n^3 + 1) + (B*b*d*x*x^(3*n)*(e*x)^m *(2*m + 3*n + 3*m*n + m^2 + 2*n^2 + 1))/(3*m + 6*n + 12*m*n + 11*m*n^2 + 6 *m^2*n + 3*m^2 + m^3 + 11*n^2 + 6*n^3 + 1)
Time = 0.30 (sec) , antiderivative size = 700, normalized size of antiderivative = 5.98 \[ \int (e x)^m \left (a+b x^n\right ) \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=\frac {x^{m} e^{m} x \left (6 x^{2 n} a b d m +6 x^{2 n} a b d \,n^{2}+8 x^{2 n} a b d n +4 x^{2 n} b^{2} c \,m^{2} n +3 x^{2 n} b^{2} c m \,n^{2}+8 x^{2 n} b^{2} c m n +5 x^{n} a^{2} d \,m^{2} n +6 x^{n} a^{2} d m \,n^{2}+10 x^{n} a^{2} d m n +2 x^{n} a b c \,m^{3}+6 x^{n} a b c \,m^{2}+6 x^{n} a b c m +12 x^{n} a b c \,n^{2}+10 x^{n} a b c n +3 a^{2} c \,m^{2}+3 a^{2} c m +6 a^{2} c \,n^{3}+11 a^{2} c \,n^{2}+6 a^{2} c n +3 x^{3 n} b^{2} d \,m^{2}+x^{3 n} b^{2} d \,m^{3}+x^{2 n} b^{2} c \,m^{3}+x^{n} a^{2} d \,m^{3}+a^{2} c \,m^{3}+3 x^{3 n} b^{2} d m +2 x^{3 n} b^{2} d \,n^{2}+3 x^{3 n} b^{2} d n +3 x^{2 n} b^{2} c \,m^{2}+3 x^{2 n} b^{2} c m +3 x^{2 n} b^{2} c \,n^{2}+4 x^{2 n} b^{2} c n +3 x^{n} a^{2} d \,m^{2}+3 x^{n} a^{2} d m +6 x^{n} a^{2} d \,n^{2}+5 x^{n} a^{2} d n +6 a^{2} c \,m^{2} n +11 a^{2} c m \,n^{2}+12 a^{2} c m n +8 x^{2 n} a b d \,m^{2} n +6 x^{2 n} a b d m \,n^{2}+16 x^{2 n} a b d m n +10 x^{n} a b c \,m^{2} n +12 x^{n} a b c m \,n^{2}+20 x^{n} a b c m n +3 x^{3 n} b^{2} d \,m^{2} n +2 x^{3 n} b^{2} d m \,n^{2}+6 x^{3 n} b^{2} d m n +2 x^{2 n} a b d \,m^{3}+6 x^{2 n} a b d \,m^{2}+2 x^{n} a b c +x^{3 n} b^{2} d +x^{2 n} b^{2} c +2 x^{2 n} a b d +x^{n} a^{2} d +a^{2} c \right )}{m^{4}+6 m^{3} n +11 m^{2} n^{2}+6 m \,n^{3}+4 m^{3}+18 m^{2} n +22 m \,n^{2}+6 n^{3}+6 m^{2}+18 m n +11 n^{2}+4 m +6 n +1} \] Input:
int((e*x)^m*(a+b*x^n)*(A+B*x^n)*(c+d*x^n),x)
Output:
(x**m*e**m*x*(x**(3*n)*b**2*d*m**3 + 3*x**(3*n)*b**2*d*m**2*n + 3*x**(3*n) *b**2*d*m**2 + 2*x**(3*n)*b**2*d*m*n**2 + 6*x**(3*n)*b**2*d*m*n + 3*x**(3* n)*b**2*d*m + 2*x**(3*n)*b**2*d*n**2 + 3*x**(3*n)*b**2*d*n + x**(3*n)*b**2 *d + 2*x**(2*n)*a*b*d*m**3 + 8*x**(2*n)*a*b*d*m**2*n + 6*x**(2*n)*a*b*d*m* *2 + 6*x**(2*n)*a*b*d*m*n**2 + 16*x**(2*n)*a*b*d*m*n + 6*x**(2*n)*a*b*d*m + 6*x**(2*n)*a*b*d*n**2 + 8*x**(2*n)*a*b*d*n + 2*x**(2*n)*a*b*d + x**(2*n) *b**2*c*m**3 + 4*x**(2*n)*b**2*c*m**2*n + 3*x**(2*n)*b**2*c*m**2 + 3*x**(2 *n)*b**2*c*m*n**2 + 8*x**(2*n)*b**2*c*m*n + 3*x**(2*n)*b**2*c*m + 3*x**(2* n)*b**2*c*n**2 + 4*x**(2*n)*b**2*c*n + x**(2*n)*b**2*c + x**n*a**2*d*m**3 + 5*x**n*a**2*d*m**2*n + 3*x**n*a**2*d*m**2 + 6*x**n*a**2*d*m*n**2 + 10*x* *n*a**2*d*m*n + 3*x**n*a**2*d*m + 6*x**n*a**2*d*n**2 + 5*x**n*a**2*d*n + x **n*a**2*d + 2*x**n*a*b*c*m**3 + 10*x**n*a*b*c*m**2*n + 6*x**n*a*b*c*m**2 + 12*x**n*a*b*c*m*n**2 + 20*x**n*a*b*c*m*n + 6*x**n*a*b*c*m + 12*x**n*a*b* c*n**2 + 10*x**n*a*b*c*n + 2*x**n*a*b*c + a**2*c*m**3 + 6*a**2*c*m**2*n + 3*a**2*c*m**2 + 11*a**2*c*m*n**2 + 12*a**2*c*m*n + 3*a**2*c*m + 6*a**2*c*n **3 + 11*a**2*c*n**2 + 6*a**2*c*n + a**2*c))/(m**4 + 6*m**3*n + 4*m**3 + 1 1*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6 *n**3 + 11*n**2 + 6*n + 1)