Integrand size = 20, antiderivative size = 72 \[ \int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=\frac {A c (e x)^{1+m}}{e (1+m)}+\frac {(B c+A d) x^n (e x)^{1+m}}{e (1+m+n)}+\frac {B d x^{2 n} (e x)^{1+m}}{e (1+m+2 n)} \] Output:
A*c*(e*x)^(1+m)/e/(1+m)+(A*d+B*c)*x^n*(e*x)^(1+m)/e/(1+m+n)+B*d*x^(2*n)*(e *x)^(1+m)/e/(1+m+2*n)
Time = 0.12 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.68 \[ \int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=x (e x)^m \left (\frac {A c}{1+m}+\frac {(B c+A d) x^n}{1+m+n}+\frac {B d x^{2 n}}{1+m+2 n}\right ) \] Input:
Integrate[(e*x)^m*(A + B*x^n)*(c + d*x^n),x]
Output:
x*(e*x)^m*((A*c)/(1 + m) + ((B*c + A*d)*x^n)/(1 + m + n) + (B*d*x^(2*n))/( 1 + m + 2*n))
Time = 0.35 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.92, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {950, 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.
\(\displaystyle \int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx\) |
\(\Big \downarrow \) 950 |
\(\displaystyle \int \left (x^n (e x)^m (A d+B c)+A c (e x)^m+B d x^{2 n} (e x)^m\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {x^{n+1} (e x)^m (A d+B c)}{m+n+1}+\frac {A c (e x)^{m+1}}{e (m+1)}+\frac {B d x^{2 n+1} (e x)^m}{m+2 n+1}\) |
Input:
Int[(e*x)^m*(A + B*x^n)*(c + d*x^n),x]
Output:
((B*c + A*d)*x^(1 + n)*(e*x)^m)/(1 + m + n) + (B*d*x^(1 + 2*n)*(e*x)^m)/(1 + m + 2*n) + (A*c*(e*x)^(1 + m))/(e*(1 + m))
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^ n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && IGt Q[p, 0] && IGtQ[q, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.16 (sec) , antiderivative size = 229, normalized size of antiderivative = 3.18
method | result | size |
risch | \(\frac {x \left (B d \,m^{2} x^{2 n}+B d m n \,x^{2 n}+A d \,m^{2} x^{n}+2 A d m n \,x^{n}+B c \,m^{2} x^{n}+2 B c m n \,x^{n}+2 B \,x^{2 n} d m +B \,x^{2 n} d n +A c \,m^{2}+3 A c m n +2 A c \,n^{2}+2 A \,x^{n} d m +2 A \,x^{n} d n +2 B \,x^{n} c m +2 B \,x^{n} c n +d \,x^{2 n} B +2 A c m +3 A c n +d \,x^{n} A +c B \,x^{n}+A c \right ) e^{m} x^{m} {\mathrm e}^{\frac {i \operatorname {csgn}\left (i e x \right ) \pi m \left (\operatorname {csgn}\left (i e x \right )-\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i e x \right )+\operatorname {csgn}\left (i e \right )\right )}{2}}}{\left (1+m \right ) \left (1+m +n \right ) \left (1+m +2 n \right )}\) | \(229\) |
parallelrisch | \(\frac {B x \,x^{2 n} \left (e x \right )^{m} d \,m^{2}+A x \,x^{n} \left (e x \right )^{m} d \,m^{2}+2 B x \,x^{2 n} \left (e x \right )^{m} d m +B x \,x^{2 n} \left (e x \right )^{m} d n +B x \,x^{n} \left (e x \right )^{m} c \,m^{2}+2 A x \,x^{n} \left (e x \right )^{m} d m +2 A x \,x^{n} \left (e x \right )^{m} d n +3 A x \left (e x \right )^{m} c m n +2 B x \,x^{n} \left (e x \right )^{m} c m +2 B x \,x^{n} \left (e x \right )^{m} c n +B x \,x^{2 n} \left (e x \right )^{m} d m n +2 A x \,x^{n} \left (e x \right )^{m} d m n +2 B x \,x^{n} \left (e x \right )^{m} c m n +A x \left (e x \right )^{m} c \,m^{2}+A x \left (e x \right )^{m} c +A x \,x^{n} \left (e x \right )^{m} d +2 A x \left (e x \right )^{m} c m +3 A x \left (e x \right )^{m} c n +B x \,x^{n} \left (e x \right )^{m} c +B x \,x^{2 n} \left (e x \right )^{m} d +2 A x \left (e x \right )^{m} c \,n^{2}}{\left (1+m \right ) \left (1+m +n \right ) \left (1+m +2 n \right )}\) | \(308\) |
orering | \(\frac {x \left (3 m^{2}+6 m n +2 n^{2}+3 m +3 n +1\right ) \left (e x \right )^{m} \left (A +B \,x^{n}\right ) \left (c +d \,x^{n}\right )}{\left (m^{2}+2 m n +2 m +2 n +1\right ) \left (1+m +n \right )}-\frac {3 x^{2} \left (m +n \right ) \left (\frac {\left (e x \right )^{m} m \left (A +B \,x^{n}\right ) \left (c +d \,x^{n}\right )}{x}+\frac {\left (e x \right )^{m} B \,x^{n} n \left (c +d \,x^{n}\right )}{x}+\frac {\left (e x \right )^{m} \left (A +B \,x^{n}\right ) d \,x^{n} n}{x}\right )}{\left (m^{2}+2 m n +2 m +2 n +1\right ) \left (1+m +n \right )}+\frac {x^{3} \left (\frac {\left (e x \right )^{m} m^{2} \left (A +B \,x^{n}\right ) \left (c +d \,x^{n}\right )}{x^{2}}-\frac {\left (e x \right )^{m} m \left (A +B \,x^{n}\right ) \left (c +d \,x^{n}\right )}{x^{2}}+\frac {2 \left (e x \right )^{m} m B \,x^{n} n \left (c +d \,x^{n}\right )}{x^{2}}+\frac {2 \left (e x \right )^{m} m \left (A +B \,x^{n}\right ) d \,x^{n} n}{x^{2}}+\frac {\left (e x \right )^{m} B \,x^{n} n^{2} \left (c +d \,x^{n}\right )}{x^{2}}-\frac {\left (e x \right )^{m} B \,x^{n} n \left (c +d \,x^{n}\right )}{x^{2}}+\frac {2 \left (e x \right )^{m} x^{2 n} n^{2} B d}{x^{2}}+\frac {\left (e x \right )^{m} \left (A +B \,x^{n}\right ) d \,x^{n} n^{2}}{x^{2}}-\frac {\left (e x \right )^{m} \left (A +B \,x^{n}\right ) d \,x^{n} n}{x^{2}}\right )}{m^{3}+3 m^{2} n +2 m \,n^{2}+3 m^{2}+6 m n +2 n^{2}+3 m +3 n +1}\) | \(417\) |
Input:
int((e*x)^m*(A+B*x^n)*(c+d*x^n),x,method=_RETURNVERBOSE)
Output:
x*(B*d*m^2*(x^n)^2+B*d*m*n*(x^n)^2+A*d*m^2*x^n+2*A*d*m*n*x^n+B*c*m^2*x^n+2 *B*c*m*n*x^n+2*B*(x^n)^2*d*m+B*(x^n)^2*d*n+A*c*m^2+3*A*c*m*n+2*A*c*n^2+2*A *x^n*d*m+2*A*x^n*d*n+2*B*x^n*c*m+2*B*x^n*c*n+d*(x^n)^2*B+2*A*c*m+3*A*c*n+d *x^n*A+c*B*x^n+A*c)/(1+m)/(1+m+n)/(1+m+2*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. 185 vs. \(2 (72) = 144\).
Time = 0.10 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.57 \[ \int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=\frac {{\left (B d m^{2} + 2 \, B d m + B d + {\left (B d m + B d\right )} n\right )} x x^{2 \, n} e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} + {\left ({\left (B c + A d\right )} m^{2} + B c + A d + 2 \, {\left (B c + A d\right )} m + 2 \, {\left (B c + A d + {\left (B c + A d\right )} m\right )} n\right )} x x^{n} e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} + {\left (A c m^{2} + 2 \, A c n^{2} + 2 \, A c m + A c + 3 \, {\left (A c m + A c\right )} n\right )} x e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )}}{m^{3} + 2 \, {\left (m + 1\right )} n^{2} + 3 \, m^{2} + 3 \, {\left (m^{2} + 2 \, m + 1\right )} n + 3 \, m + 1} \] Input:
integrate((e*x)^m*(A+B*x^n)*(c+d*x^n),x, algorithm="fricas")
Output:
((B*d*m^2 + 2*B*d*m + B*d + (B*d*m + B*d)*n)*x*x^(2*n)*e^(m*log(e) + m*log (x)) + ((B*c + A*d)*m^2 + B*c + A*d + 2*(B*c + A*d)*m + 2*(B*c + A*d + (B* c + A*d)*m)*n)*x*x^n*e^(m*log(e) + m*log(x)) + (A*c*m^2 + 2*A*c*n^2 + 2*A* c*m + A*c + 3*(A*c*m + A*c)*n)*x*e^(m*log(e) + m*log(x)))/(m^3 + 2*(m + 1) *n^2 + 3*m^2 + 3*(m^2 + 2*m + 1)*n + 3*m + 1)
Leaf count of result is larger than twice the leaf count of optimal. 1498 vs. \(2 (61) = 122\).
Time = 2.19 (sec) , antiderivative size = 1498, normalized size of antiderivative = 20.81 \[ \int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=\text {Too large to display} \] Input:
integrate((e*x)**m*(A+B*x**n)*(c+d*x**n),x)
Output:
Piecewise(((A + B)*(c + d)*log(x)/e, Eq(m, -1) & Eq(n, 0)), ((A*c*log(x) + A*d*x**n/n + B*c*x**n/n + B*d*x**(2*n)/(2*n))/e, Eq(m, -1)), (A*c*Piecewi se((0**(-2*n - 1)*x, Eq(e, 0)), (Piecewise((-1/(2*n*(e*x)**(2*n)), Ne(n, 0 )), (log(e*x), True))/e, True)) + A*d*Piecewise((-x*x**n*(e*x)**(-2*n - 1) /n, Ne(n, 0)), (x*x**n*(e*x)**(-2*n - 1)*log(x), True)) + 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), T rue)) + B*d*x*x**(2*n)*(e*x)**(-2*n - 1)*log(x), Eq(m, -2*n - 1)), (A*c*Pi ecewise((0**(-n - 1)*x, Eq(e, 0)), (Piecewise((-1/(n*(e*x)**n), Ne(n, 0)), (log(e*x), True))/e, True)) + A*d*x*x**n*(e*x)**(-n - 1)*log(x) + B*c*x*x **n*(e*x)**(-n - 1)*log(x) + B*d*Piecewise((x*x**(2*n)*(e*x)**(-n - 1)/n, Ne(n, 0)), (x*x**(2*n)*(e*x)**(-n - 1)*log(x), True)), Eq(m, -n - 1)), (A* c*m**2*x*(e*x)**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n **2 + 3*n + 1) + 3*A*c*m*n*x*(e*x)**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + 2*A*c*m*x*(e*x)**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + 2*A*c*n**2*x*(e*x)* *m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + 3*A*c*n*x*(e*x)**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + A*c*x*(e*x)**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + A*d*m**2*x*x**n*(e*x)**m/(m**3 + 3*m**2* n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + 2*A*d*m*n*x*x...
Time = 0.04 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.26 \[ \int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=\frac {B d e^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {B c e^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 1} + \frac {A d e^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 1} + \frac {\left (e x\right )^{m + 1} A c}{e {\left (m + 1\right )}} \] Input:
integrate((e*x)^m*(A+B*x^n)*(c+d*x^n),x, algorithm="maxima")
Output:
B*d*e^m*x*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1) + B*c*e^m*x*e^(m*log(x) + n*log(x))/(m + n + 1) + A*d*e^m*x*e^(m*log(x) + n*log(x))/(m + n + 1) + (e*x)^(m + 1)*A*c/(e*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 763 vs. \(2 (72) = 144\).
Time = 0.13 (sec) , antiderivative size = 763, normalized size of antiderivative = 10.60 \[ \int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=\text {Too large to display} \] Input:
integrate((e*x)^m*(A+B*x^n)*(c+d*x^n),x, algorithm="giac")
Output:
(B*d*m^2*x*x^(2*n)*e^(m*log(e) + m*log(x)) + B*d*m*n*x*x^(2*n)*e^(m*log(e) + m*log(x)) + B*c*m^2*x*x^n*e^(m*log(e) + m*log(x)) + A*d*m^2*x*x^n*e^(m* log(e) + m*log(x)) + B*d*m^2*x*x^n*e^(m*log(e) + m*log(x)) + 2*B*c*m*n*x*x ^n*e^(m*log(e) + m*log(x)) + 2*A*d*m*n*x*x^n*e^(m*log(e) + m*log(x)) + B*d *m*n*x*x^n*e^(m*log(e) + m*log(x)) + A*c*m^2*x*e^(m*log(e) + m*log(x)) + B *c*m^2*x*e^(m*log(e) + m*log(x)) + A*d*m^2*x*e^(m*log(e) + m*log(x)) + B*d *m^2*x*e^(m*log(e) + m*log(x)) + 3*A*c*m*n*x*e^(m*log(e) + m*log(x)) + 2*B *c*m*n*x*e^(m*log(e) + m*log(x)) + 2*A*d*m*n*x*e^(m*log(e) + m*log(x)) + B *d*m*n*x*e^(m*log(e) + m*log(x)) + 2*A*c*n^2*x*e^(m*log(e) + m*log(x)) + 2 *B*d*m*x*x^(2*n)*e^(m*log(e) + m*log(x)) + B*d*n*x*x^(2*n)*e^(m*log(e) + m *log(x)) + 2*B*c*m*x*x^n*e^(m*log(e) + m*log(x)) + 2*A*d*m*x*x^n*e^(m*log( e) + m*log(x)) + 2*B*d*m*x*x^n*e^(m*log(e) + m*log(x)) + 2*B*c*n*x*x^n*e^( m*log(e) + m*log(x)) + 2*A*d*n*x*x^n*e^(m*log(e) + m*log(x)) + B*d*n*x*x^n *e^(m*log(e) + m*log(x)) + 2*A*c*m*x*e^(m*log(e) + m*log(x)) + 2*B*c*m*x*e ^(m*log(e) + m*log(x)) + 2*A*d*m*x*e^(m*log(e) + m*log(x)) + 2*B*d*m*x*e^( m*log(e) + m*log(x)) + 3*A*c*n*x*e^(m*log(e) + m*log(x)) + 2*B*c*n*x*e^(m* log(e) + m*log(x)) + 2*A*d*n*x*e^(m*log(e) + m*log(x)) + B*d*n*x*e^(m*log( e) + m*log(x)) + B*d*x*x^(2*n)*e^(m*log(e) + m*log(x)) + B*c*x*x^n*e^(m*lo g(e) + m*log(x)) + A*d*x*x^n*e^(m*log(e) + m*log(x)) + B*d*x*x^n*e^(m*log( e) + m*log(x)) + A*c*x*e^(m*log(e) + m*log(x)) + B*c*x*e^(m*log(e) + m*...
Time = 4.76 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.26 \[ \int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx={\left (e\,x\right )}^m\,\left (\frac {A\,c\,x}{m+1}+\frac {x\,x^n\,\left (A\,d+B\,c\right )\,\left (m+2\,n+1\right )}{m^2+3\,m\,n+2\,m+2\,n^2+3\,n+1}+\frac {B\,d\,x\,x^{2\,n}\,\left (m+n+1\right )}{m^2+3\,m\,n+2\,m+2\,n^2+3\,n+1}\right ) \] Input:
int((e*x)^m*(A + B*x^n)*(c + d*x^n),x)
Output:
(e*x)^m*((A*c*x)/(m + 1) + (x*x^n*(A*d + B*c)*(m + 2*n + 1))/(2*m + 3*n + 3*m*n + m^2 + 2*n^2 + 1) + (B*d*x*x^(2*n)*(m + n + 1))/(2*m + 3*n + 3*m*n + m^2 + 2*n^2 + 1))
Time = 0.24 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.89 \[ \int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=\frac {x^{m} e^{m} x \left (x^{2 n} b d \,m^{2}+x^{2 n} b d m n +2 x^{2 n} b d m +x^{2 n} b d n +x^{2 n} b d +x^{n} a d \,m^{2}+2 x^{n} a d m n +2 x^{n} a d m +2 x^{n} a d n +x^{n} a d +x^{n} b c \,m^{2}+2 x^{n} b c m n +2 x^{n} b c m +2 x^{n} b c n +x^{n} b c +a c \,m^{2}+3 a c m n +2 a c m +2 a c \,n^{2}+3 a c n +a c \right )}{m^{3}+3 m^{2} n +2 m \,n^{2}+3 m^{2}+6 m n +2 n^{2}+3 m +3 n +1} \] Input:
int((e*x)^m*(A+B*x^n)*(c+d*x^n),x)
Output:
(x**m*e**m*x*(x**(2*n)*b*d*m**2 + x**(2*n)*b*d*m*n + 2*x**(2*n)*b*d*m + x* *(2*n)*b*d*n + x**(2*n)*b*d + x**n*a*d*m**2 + 2*x**n*a*d*m*n + 2*x**n*a*d* m + 2*x**n*a*d*n + x**n*a*d + x**n*b*c*m**2 + 2*x**n*b*c*m*n + 2*x**n*b*c* m + 2*x**n*b*c*n + x**n*b*c + a*c*m**2 + 3*a*c*m*n + 2*a*c*m + 2*a*c*n**2 + 3*a*c*n + a*c))/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n **2 + 3*n + 1)