Integrand size = 24, antiderivative size = 63 \[ \int (e x)^m (c+d x) \left (a x^n+b x^{1+n}\right ) \, dx=\frac {a c x^{1+n} (e x)^m}{1+m+n}+\frac {(b c+a d) x^{2+n} (e x)^m}{2+m+n}+\frac {b d x^{3+n} (e x)^m}{3+m+n} \] Output:
a*c*x^(1+n)*(e*x)^m/(1+m+n)+(a*d+b*c)*x^(2+n)*(e*x)^m/(2+m+n)+b*d*x^(3+n)* (e*x)^m/(3+m+n)
Time = 0.09 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.17 \[ \int (e x)^m (c+d x) \left (a x^n+b x^{1+n}\right ) \, dx=\frac {x^{1+n} (e x)^m \left (b (c+d x)^2-\frac {(b c (1+m+n)-a d (3+m+n)) (c (2+m+n)+d (1+m+n) x)}{(1+m+n) (2+m+n)}\right )}{d (3+m+n)} \] Input:
Integrate[(e*x)^m*(c + d*x)*(a*x^n + b*x^(1 + n)),x]
Output:
(x^(1 + n)*(e*x)^m*(b*(c + d*x)^2 - ((b*c*(1 + m + n) - a*d*(3 + m + n))*( c*(2 + m + n) + d*(1 + m + n)*x))/((1 + m + n)*(2 + m + n))))/(d*(3 + m + n))
Time = 0.38 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2027, 30, 85, 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.
\(\displaystyle \int (c+d x) (e x)^m \left (a x^n+b x^{n+1}\right ) \, dx\) |
\(\Big \downarrow \) 2027 |
\(\displaystyle \int x^n (a+b x) (c+d x) (e x)^mdx\) |
\(\Big \downarrow \) 30 |
\(\displaystyle x^{-m} (e x)^m \int x^{m+n} (a+b x) (c+d x)dx\) |
\(\Big \downarrow \) 85 |
\(\displaystyle x^{-m} (e x)^m \int \left (a c x^{m+n}+(b c+a d) x^{m+n+1}+b d x^{m+n+2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^{-m} (e x)^m \left (\frac {x^{m+n+2} (a d+b c)}{m+n+2}+\frac {a c x^{m+n+1}}{m+n+1}+\frac {b d x^{m+n+3}}{m+n+3}\right )\) |
Input:
Int[(e*x)^m*(c + d*x)*(a*x^n + b*x^(1 + n)),x]
Output:
((e*x)^m*((a*c*x^(1 + m + n))/(1 + m + n) + ((b*c + a*d)*x^(2 + m + n))/(2 + m + n) + (b*d*x^(3 + m + n))/(3 + m + n)))/x^m
Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[b^I ntPart[p]*((b*x^i)^FracPart[p]/(a^(i*IntPart[p])*(a*x)^(i*FracPart[p]))) Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p}, x] && IntegerQ[i] & & !IntegerQ[p]
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : > Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && NeQ[b*e + a* f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
Leaf count of result is larger than twice the leaf count of optimal. \(204\) vs. \(2(63)=126\).
Time = 0.41 (sec) , antiderivative size = 205, normalized size of antiderivative = 3.25
method | result | size |
orering | \(\frac {\left (b d \,m^{2} x^{2}+2 b d m n \,x^{2}+b d \,n^{2} x^{2}+a d \,m^{2} x +2 a d m n x +a d \,n^{2} x +b c \,m^{2} x +2 b c m n x +b c \,n^{2} x +3 b d m \,x^{2}+3 b d n \,x^{2}+a c \,m^{2}+2 a c m n +a c \,n^{2}+4 a d m x +4 a d n x +4 b c m x +4 n b c x +2 b d \,x^{2}+5 a c m +5 a c n +3 a d x +3 c b x +6 a c \right ) x \left (e x \right )^{m} \left (a \,x^{n}+b \,x^{1+n}\right )}{\left (1+m +n \right ) \left (2+m +n \right ) \left (3+m +n \right ) \left (b x +a \right )}\) | \(205\) |
risch | \(\frac {x \left (b d \,m^{2} x^{2}+2 b d m n \,x^{2}+b d \,n^{2} x^{2}+a d \,m^{2} x +2 a d m n x +a d \,n^{2} x +b c \,m^{2} x +2 b c m n x +b c \,n^{2} x +3 b d m \,x^{2}+3 b d n \,x^{2}+a c \,m^{2}+2 a c m n +a c \,n^{2}+4 a d m x +4 a d n x +4 b c m x +4 n b c x +2 b d \,x^{2}+5 a c m +5 a c n +3 a d x +3 c b x +6 a c \right ) x^{n} x^{m} e^{m} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i e x \right ) m \left (\operatorname {csgn}\left (i e x \right )-\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i e x \right )+\operatorname {csgn}\left (i e \right )\right )}{2}}}{\left (1+m +n \right ) \left (2+m +n \right ) \left (3+m +n \right )}\) | \(229\) |
parallelrisch | \(\frac {2 x \,x^{1+n} \left (e x \right )^{m} b c m n +2 x^{2} x^{1+n} \left (e x \right )^{m} b d m n +2 x \,x^{n} \left (e x \right )^{m} a c m n +2 x^{2} x^{n} \left (e x \right )^{m} a d m n +x^{2} x^{n} \left (e x \right )^{m} a d \,m^{2}+x^{2} x^{n} \left (e x \right )^{m} a d \,n^{2}+x^{2} x^{1+n} \left (e x \right )^{m} b d \,m^{2}+x^{2} x^{1+n} \left (e x \right )^{m} b d \,n^{2}+4 x^{2} x^{n} \left (e x \right )^{m} a d m +4 x^{2} x^{n} \left (e x \right )^{m} a d n +3 x^{2} x^{1+n} \left (e x \right )^{m} b d m +3 x^{2} x^{1+n} \left (e x \right )^{m} b d n +x \,x^{n} \left (e x \right )^{m} a c \,m^{2}+x \,x^{n} \left (e x \right )^{m} a c \,n^{2}+x \,x^{1+n} \left (e x \right )^{m} b c \,m^{2}+x \,x^{1+n} \left (e x \right )^{m} b c \,n^{2}+5 x \,x^{n} \left (e x \right )^{m} a c m +5 x \,x^{n} \left (e x \right )^{m} a c n +4 x \,x^{1+n} \left (e x \right )^{m} b c m +4 x \,x^{1+n} \left (e x \right )^{m} b c n +3 x^{2} x^{n} \left (e x \right )^{m} a d +2 x^{2} x^{1+n} \left (e x \right )^{m} b d +6 x \,x^{n} \left (e x \right )^{m} a c +3 x \,x^{1+n} \left (e x \right )^{m} b c}{\left (1+m +n \right ) \left (2+m +n \right ) \left (3+m +n \right )}\) | \(413\) |
Input:
int((e*x)^m*(d*x+c)*(a*x^n+b*x^(1+n)),x,method=_RETURNVERBOSE)
Output:
(b*d*m^2*x^2+2*b*d*m*n*x^2+b*d*n^2*x^2+a*d*m^2*x+2*a*d*m*n*x+a*d*n^2*x+b*c *m^2*x+2*b*c*m*n*x+b*c*n^2*x+3*b*d*m*x^2+3*b*d*n*x^2+a*c*m^2+2*a*c*m*n+a*c *n^2+4*a*d*m*x+4*a*d*n*x+4*b*c*m*x+4*b*c*n*x+2*b*d*x^2+5*a*c*m+5*a*c*n+3*a *d*x+3*b*c*x+6*a*c)/(1+m+n)/(2+m+n)/(3+m+n)/(b*x+a)*x*(e*x)^m*(a*x^n+b*x^( 1+n))
Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (63) = 126\).
Time = 0.09 (sec) , antiderivative size = 190, normalized size of antiderivative = 3.02 \[ \int (e x)^m (c+d x) \left (a x^n+b x^{1+n}\right ) \, dx=\frac {{\left (a c m^{2} + a c n^{2} + 5 \, a c m + {\left (b d m^{2} + b d n^{2} + 3 \, b d m + 2 \, b d + {\left (2 \, b d m + 3 \, b d\right )} n\right )} x^{2} + 6 \, a c + {\left (2 \, a c m + 5 \, a c\right )} n + {\left ({\left (b c + a d\right )} m^{2} + {\left (b c + a d\right )} n^{2} + 3 \, b c + 3 \, a d + 4 \, {\left (b c + a d\right )} m + 2 \, {\left (2 \, b c + 2 \, a d + {\left (b c + a d\right )} m\right )} n\right )} x\right )} x^{n + 1} e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )}}{m^{3} + 3 \, {\left (m + 2\right )} n^{2} + n^{3} + 6 \, m^{2} + {\left (3 \, m^{2} + 12 \, m + 11\right )} n + 11 \, m + 6} \] Input:
integrate((e*x)^m*(d*x+c)*(a*x^n+b*x^(1+n)),x, algorithm="fricas")
Output:
(a*c*m^2 + a*c*n^2 + 5*a*c*m + (b*d*m^2 + b*d*n^2 + 3*b*d*m + 2*b*d + (2*b *d*m + 3*b*d)*n)*x^2 + 6*a*c + (2*a*c*m + 5*a*c)*n + ((b*c + a*d)*m^2 + (b *c + a*d)*n^2 + 3*b*c + 3*a*d + 4*(b*c + a*d)*m + 2*(2*b*c + 2*a*d + (b*c + a*d)*m)*n)*x)*x^(n + 1)*e^(m*log(e) + m*log(x))/(m^3 + 3*(m + 2)*n^2 + n ^3 + 6*m^2 + (3*m^2 + 12*m + 11)*n + 11*m + 6)
Leaf count of result is larger than twice the leaf count of optimal. 1756 vs. \(2 (58) = 116\).
Time = 11.99 (sec) , antiderivative size = 1756, normalized size of antiderivative = 27.87 \[ \int (e x)^m (c+d x) \left (a x^n+b x^{1+n}\right ) \, dx=\text {Too large to display} \] Input:
integrate((e*x)**m*(d*x+c)*(a*x**n+b*x**(1+n)),x)
Output:
Piecewise((-a*c*x*x**n*(e*x)**(-n - 3)/2 - a*d*x**2*x**n*(e*x)**(-n - 3) - b*c*x*x**(n + 1)*(e*x)**(-n - 3) + b*d*x**2*x**(n + 1)*(e*x)**(-n - 3)*lo g(x), Eq(m, -n - 3)), (-a*c*x*x**n*(e*x)**(-n - 2) + a*d*x**2*x**n*(e*x)** (-n - 2)*log(x) + b*c*x*x**(n + 1)*(e*x)**(-n - 2)*log(x) + b*d*x**2*x**(n + 1)*(e*x)**(-n - 2), Eq(m, -n - 2)), (a*c*x*x**n*(e*x)**(-n - 1)*log(x) + a*d*x**2*x**n*(e*x)**(-n - 1) + b*c*x*x**(n + 1)*(e*x)**(-n - 1) + b*d*x **2*x**(n + 1)*(e*x)**(-n - 1)/2, Eq(m, -n - 1)), (a*c*m**2*x*x**n*(e*x)** m/(m**3 + 3*m**2*n + 6*m**2 + 3*m*n**2 + 12*m*n + 11*m + n**3 + 6*n**2 + 1 1*n + 6) + 2*a*c*m*n*x*x**n*(e*x)**m/(m**3 + 3*m**2*n + 6*m**2 + 3*m*n**2 + 12*m*n + 11*m + n**3 + 6*n**2 + 11*n + 6) + 5*a*c*m*x*x**n*(e*x)**m/(m** 3 + 3*m**2*n + 6*m**2 + 3*m*n**2 + 12*m*n + 11*m + n**3 + 6*n**2 + 11*n + 6) + a*c*n**2*x*x**n*(e*x)**m/(m**3 + 3*m**2*n + 6*m**2 + 3*m*n**2 + 12*m* n + 11*m + n**3 + 6*n**2 + 11*n + 6) + 5*a*c*n*x*x**n*(e*x)**m/(m**3 + 3*m **2*n + 6*m**2 + 3*m*n**2 + 12*m*n + 11*m + n**3 + 6*n**2 + 11*n + 6) + 6* a*c*x*x**n*(e*x)**m/(m**3 + 3*m**2*n + 6*m**2 + 3*m*n**2 + 12*m*n + 11*m + n**3 + 6*n**2 + 11*n + 6) + a*d*m**2*x**2*x**n*(e*x)**m/(m**3 + 3*m**2*n + 6*m**2 + 3*m*n**2 + 12*m*n + 11*m + n**3 + 6*n**2 + 11*n + 6) + 2*a*d*m* n*x**2*x**n*(e*x)**m/(m**3 + 3*m**2*n + 6*m**2 + 3*m*n**2 + 12*m*n + 11*m + n**3 + 6*n**2 + 11*n + 6) + 4*a*d*m*x**2*x**n*(e*x)**m/(m**3 + 3*m**2*n + 6*m**2 + 3*m*n**2 + 12*m*n + 11*m + n**3 + 6*n**2 + 11*n + 6) + a*d*n...
Time = 0.04 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.57 \[ \int (e x)^m (c+d x) \left (a x^n+b x^{1+n}\right ) \, dx=\frac {b d e^{m} x^{3} e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 3} + \frac {b c e^{m} x^{2} e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 2} + \frac {a d e^{m} x^{2} e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 2} + \frac {a c e^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 1} \] Input:
integrate((e*x)^m*(d*x+c)*(a*x^n+b*x^(1+n)),x, algorithm="maxima")
Output:
b*d*e^m*x^3*e^(m*log(x) + n*log(x))/(m + n + 3) + b*c*e^m*x^2*e^(m*log(x) + n*log(x))/(m + n + 2) + a*d*e^m*x^2*e^(m*log(x) + n*log(x))/(m + n + 2) + a*c*e^m*x*e^(m*log(x) + n*log(x))/(m + n + 1)
Leaf count of result is larger than twice the leaf count of optimal. 544 vs. \(2 (63) = 126\).
Time = 0.29 (sec) , antiderivative size = 544, normalized size of antiderivative = 8.63 \[ \int (e x)^m (c+d x) \left (a x^n+b x^{1+n}\right ) \, dx =\text {Too large to display} \] Input:
integrate((e*x)^m*(d*x+c)*(a*x^n+b*x^(1+n)),x, algorithm="giac")
Output:
(b*d*m^2*x^3*x^n*e^(m*log(e) + m*log(x)) + 2*b*d*m*n*x^3*x^n*e^(m*log(e) + m*log(x)) + b*d*n^2*x^3*x^n*e^(m*log(e) + m*log(x)) + b*c*m^2*x^2*x^n*e^( m*log(e) + m*log(x)) + a*d*m^2*x^2*x^n*e^(m*log(e) + m*log(x)) + 2*b*c*m*n *x^2*x^n*e^(m*log(e) + m*log(x)) + 2*a*d*m*n*x^2*x^n*e^(m*log(e) + m*log(x )) + b*c*n^2*x^2*x^n*e^(m*log(e) + m*log(x)) + a*d*n^2*x^2*x^n*e^(m*log(e) + m*log(x)) + 3*b*d*m*x^3*x^n*e^(m*log(e) + m*log(x)) + 3*b*d*n*x^3*x^n*e ^(m*log(e) + m*log(x)) + a*c*m^2*x*x^n*e^(m*log(e) + m*log(x)) + 2*a*c*m*n *x*x^n*e^(m*log(e) + m*log(x)) + a*c*n^2*x*x^n*e^(m*log(e) + m*log(x)) + 4 *b*c*m*x^2*x^n*e^(m*log(e) + m*log(x)) + 4*a*d*m*x^2*x^n*e^(m*log(e) + m*l og(x)) + 4*b*c*n*x^2*x^n*e^(m*log(e) + m*log(x)) + 4*a*d*n*x^2*x^n*e^(m*lo g(e) + m*log(x)) + 2*b*d*x^3*x^n*e^(m*log(e) + m*log(x)) + 5*a*c*m*x*x^n*e ^(m*log(e) + m*log(x)) + 5*a*c*n*x*x^n*e^(m*log(e) + m*log(x)) + 3*b*c*x^2 *x^n*e^(m*log(e) + m*log(x)) + 3*a*d*x^2*x^n*e^(m*log(e) + m*log(x)) + 6*a *c*x*x^n*e^(m*log(e) + m*log(x)))/(m^3 + 3*m^2*n + 3*m*n^2 + n^3 + 6*m^2 + 12*m*n + 6*n^2 + 11*m + 11*n + 6)
Time = 5.36 (sec) , antiderivative size = 297, normalized size of antiderivative = 4.71 \[ \int (e x)^m (c+d x) \left (a x^n+b x^{1+n}\right ) \, dx=\frac {b\,d\,x^{n+1}\,x^2\,{\left (e\,x\right )}^m\,\left (m^2+2\,m\,n+3\,m+n^2+3\,n+2\right )}{m^3+3\,m^2\,n+6\,m^2+3\,m\,n^2+12\,m\,n+11\,m+n^3+6\,n^2+11\,n+6}+\frac {a\,c\,x\,x^n\,{\left (e\,x\right )}^m\,\left (m^2+2\,m\,n+5\,m+n^2+5\,n+6\right )}{m^3+3\,m^2\,n+6\,m^2+3\,m\,n^2+12\,m\,n+11\,m+n^3+6\,n^2+11\,n+6}+\frac {b\,c\,x\,x^{n+1}\,{\left (e\,x\right )}^m\,\left (m^2+2\,m\,n+4\,m+n^2+4\,n+3\right )}{m^3+3\,m^2\,n+6\,m^2+3\,m\,n^2+12\,m\,n+11\,m+n^3+6\,n^2+11\,n+6}+\frac {a\,d\,x^n\,x^2\,{\left (e\,x\right )}^m\,\left (m^2+2\,m\,n+4\,m+n^2+4\,n+3\right )}{m^3+3\,m^2\,n+6\,m^2+3\,m\,n^2+12\,m\,n+11\,m+n^3+6\,n^2+11\,n+6} \] Input:
int((e*x)^m*(a*x^n + b*x^(n + 1))*(c + d*x),x)
Output:
(b*d*x^(n + 1)*x^2*(e*x)^m*(3*m + 3*n + 2*m*n + m^2 + n^2 + 2))/(11*m + 11 *n + 12*m*n + 3*m*n^2 + 3*m^2*n + 6*m^2 + m^3 + 6*n^2 + n^3 + 6) + (a*c*x* x^n*(e*x)^m*(5*m + 5*n + 2*m*n + m^2 + n^2 + 6))/(11*m + 11*n + 12*m*n + 3 *m*n^2 + 3*m^2*n + 6*m^2 + m^3 + 6*n^2 + n^3 + 6) + (b*c*x*x^(n + 1)*(e*x) ^m*(4*m + 4*n + 2*m*n + m^2 + n^2 + 3))/(11*m + 11*n + 12*m*n + 3*m*n^2 + 3*m^2*n + 6*m^2 + m^3 + 6*n^2 + n^3 + 6) + (a*d*x^n*x^2*(e*x)^m*(4*m + 4*n + 2*m*n + m^2 + n^2 + 3))/(11*m + 11*n + 12*m*n + 3*m*n^2 + 3*m^2*n + 6*m ^2 + m^3 + 6*n^2 + n^3 + 6)
Time = 0.28 (sec) , antiderivative size = 211, normalized size of antiderivative = 3.35 \[ \int (e x)^m (c+d x) \left (a x^n+b x^{1+n}\right ) \, dx=\frac {x^{m +n} e^{m} x \left (b d \,m^{2} x^{2}+2 b d m n \,x^{2}+b d \,n^{2} x^{2}+a d \,m^{2} x +2 a d m n x +a d \,n^{2} x +b c \,m^{2} x +2 b c m n x +b c \,n^{2} x +3 b d m \,x^{2}+3 b d n \,x^{2}+a c \,m^{2}+2 a c m n +a c \,n^{2}+4 a d m x +4 a d n x +4 b c m x +4 b c n x +2 b d \,x^{2}+5 a c m +5 a c n +3 a d x +3 b c x +6 a c \right )}{m^{3}+3 m^{2} n +3 m \,n^{2}+n^{3}+6 m^{2}+12 m n +6 n^{2}+11 m +11 n +6} \] Input:
int((e*x)^m*(d*x+c)*(a*x^n+b*x^(1+n)),x)
Output:
(x**(m + n)*e**m*x*(a*c*m**2 + 2*a*c*m*n + 5*a*c*m + a*c*n**2 + 5*a*c*n + 6*a*c + a*d*m**2*x + 2*a*d*m*n*x + 4*a*d*m*x + a*d*n**2*x + 4*a*d*n*x + 3* a*d*x + b*c*m**2*x + 2*b*c*m*n*x + 4*b*c*m*x + b*c*n**2*x + 4*b*c*n*x + 3* b*c*x + b*d*m**2*x**2 + 2*b*d*m*n*x**2 + 3*b*d*m*x**2 + b*d*n**2*x**2 + 3* b*d*n*x**2 + 2*b*d*x**2))/(m**3 + 3*m**2*n + 6*m**2 + 3*m*n**2 + 12*m*n + 11*m + n**3 + 6*n**2 + 11*n + 6)