\(\int \frac {(e x)^m (a+b x^n) (A+B x^n)}{c+d x^n} \, dx\) [42]

Optimal result

Integrand size = 29, antiderivative size = 125 \[ \int \frac {(e x)^m \left (a+b x^n\right ) \left (A+B x^n\right )}{c+d x^n} \, dx=-\frac {(b B c-A b d-a B d) (e x)^{1+m}}{d^2 e (1+m)}+\frac {b B x^n (e x)^{1+m}}{d e (1+m+n)}+\frac {(b c-a d) (B c-A d) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )}{c d^2 e (1+m)} \] Output:

-(-A*b*d-B*a*d+B*b*c)*(e*x)^(1+m)/d^2/e/(1+m)+b*B*x^n*(e*x)^(1+m)/d/e/(1+m 
+n)+(-a*d+b*c)*(-A*d+B*c)*(e*x)^(1+m)*hypergeom([1, (1+m)/n],[(1+m+n)/n],- 
d*x^n/c)/c/d^2/e/(1+m)
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.76 \[ \int \frac {(e x)^m \left (a+b x^n\right ) \left (A+B x^n\right )}{c+d x^n} \, dx=\frac {x (e x)^m \left (\frac {-b B c+A b d+a B d}{1+m}+\frac {b B d x^n}{1+m+n}+\frac {(b c-a d) (B c-A d) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )}{c (1+m)}\right )}{d^2} \] Input:

Integrate[((e*x)^m*(a + b*x^n)*(A + B*x^n))/(c + d*x^n),x]
 

Output:

(x*(e*x)^m*((-(b*B*c) + A*b*d + a*B*d)/(1 + m) + (b*B*d*x^n)/(1 + m + n) + 
 ((b*c - a*d)*(B*c - A*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, - 
((d*x^n)/c)])/(c*(1 + m))))/d^2
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, 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.

\(\displaystyle \int \frac {(e x)^m \left (a+b x^n\right ) \left (A+B x^n\right )}{c+d x^n} \, dx\)

\(\Big \downarrow \) 1040

\(\displaystyle \int \left (\frac {(e x)^m (a d-b c) (A d-B c)}{d^2 \left (c+d x^n\right )}+\frac {(e x)^m (a B d+A b d-b B c)}{d^2}+\frac {b B x^n (e x)^m}{d}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(e x)^{m+1} (b c-a d) (B c-A d) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{n},\frac {m+n+1}{n},-\frac {d x^n}{c}\right )}{c d^2 e (m+1)}-\frac {(e x)^{m+1} (-a B d-A b d+b B c)}{d^2 e (m+1)}+\frac {b B x^{n+1} (e x)^m}{d (m+n+1)}\)

Input:

Int[((e*x)^m*(a + b*x^n)*(A + B*x^n))/(c + d*x^n),x]
 

Output:

(b*B*x^(1 + n)*(e*x)^m)/(d*(1 + m + n)) - ((b*B*c - A*b*d - a*B*d)*(e*x)^( 
1 + m))/(d^2*e*(1 + m)) + ((b*c - a*d)*(B*c - A*d)*(e*x)^(1 + m)*Hypergeom 
etric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(c*d^2*e*(1 + m))
 

Defintions of rubi rules used

rule 1040

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

\[\int \frac {\left (e x \right )^{m} \left (a +b \,x^{n}\right ) \left (A +B \,x^{n}\right )}{c +d \,x^{n}}d x\]

Input:

int((e*x)^m*(a+b*x^n)*(A+B*x^n)/(c+d*x^n),x)
 

Output:

int((e*x)^m*(a+b*x^n)*(A+B*x^n)/(c+d*x^n),x)
 

Fricas [F]

\[ \int \frac {(e x)^m \left (a+b x^n\right ) \left (A+B x^n\right )}{c+d x^n} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )} \left (e x\right )^{m}}{d x^{n} + c} \,d x } \] Input:

integrate((e*x)^m*(a+b*x^n)*(A+B*x^n)/(c+d*x^n),x, algorithm="fricas")
 

Output:

integral((B*b*x^(2*n) + A*a + (B*a + A*b)*x^n)*(e*x)^m/(d*x^n + c), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 6.52 (sec) , antiderivative size = 872, normalized size of antiderivative = 6.98 \[ \int \frac {(e x)^m \left (a+b x^n\right ) \left (A+B x^n\right )}{c+d x^n} \, dx=\text {Too large to display} \] Input:

integrate((e*x)**m*(a+b*x**n)*(A+B*x**n)/(c+d*x**n),x)
 

Output:

A*a*c**(m/n + 1/n)*c**(-m/n - 1 - 1/n)*e**m*m*x**(m + 1)*lerchphi(d*x**n*e 
xp_polar(I*pi)/c, 1, m/n + 1/n)*gamma(m/n + 1/n)/(n**2*gamma(m/n + 1 + 1/n 
)) + A*a*c**(m/n + 1/n)*c**(-m/n - 1 - 1/n)*e**m*x**(m + 1)*lerchphi(d*x** 
n*exp_polar(I*pi)/c, 1, m/n + 1/n)*gamma(m/n + 1/n)/(n**2*gamma(m/n + 1 + 
1/n)) + A*b*c**(-m/n - 2 - 1/n)*c**(m/n + 1 + 1/n)*e**m*m*x**(m + n + 1)*l 
erchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/( 
n**2*gamma(m/n + 2 + 1/n)) + A*b*c**(-m/n - 2 - 1/n)*c**(m/n + 1 + 1/n)*e* 
*m*x**(m + n + 1)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gam 
ma(m/n + 1 + 1/n)/(n*gamma(m/n + 2 + 1/n)) + A*b*c**(-m/n - 2 - 1/n)*c**(m 
/n + 1 + 1/n)*e**m*x**(m + n + 1)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/ 
n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(n**2*gamma(m/n + 2 + 1/n)) + B*a*c**(-m 
/n - 2 - 1/n)*c**(m/n + 1 + 1/n)*e**m*m*x**(m + n + 1)*lerchphi(d*x**n*exp 
_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(n**2*gamma(m/n + 2 
 + 1/n)) + B*a*c**(-m/n - 2 - 1/n)*c**(m/n + 1 + 1/n)*e**m*x**(m + n + 1)* 
lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/ 
(n*gamma(m/n + 2 + 1/n)) + B*a*c**(-m/n - 2 - 1/n)*c**(m/n + 1 + 1/n)*e**m 
*x**(m + n + 1)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma 
(m/n + 1 + 1/n)/(n**2*gamma(m/n + 2 + 1/n)) + B*b*c**(-m/n - 3 - 1/n)*c**( 
m/n + 2 + 1/n)*e**m*m*x**(m + 2*n + 1)*lerchphi(d*x**n*exp_polar(I*pi)/c, 
1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(n**2*gamma(m/n + 3 + 1/n)) + 2*...
 

Maxima [F]

\[ \int \frac {(e x)^m \left (a+b x^n\right ) \left (A+B x^n\right )}{c+d x^n} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )} \left (e x\right )^{m}}{d x^{n} + c} \,d x } \] Input:

integrate((e*x)^m*(a+b*x^n)*(A+B*x^n)/(c+d*x^n),x, algorithm="maxima")
 

Output:

-((b*c*d*e^m - a*d^2*e^m)*A - (b*c^2*e^m - a*c*d*e^m)*B)*integrate(x^m/(d^ 
3*x^n + c*d^2), x) + (B*b*d*e^m*(m + 1)*x*e^(m*log(x) + n*log(x)) + (A*b*d 
*e^m*(m + n + 1) - (b*c*e^m*(m + n + 1) - a*d*e^m*(m + n + 1))*B)*x*x^m)/( 
(m^2 + m*(n + 2) + n + 1)*d^2)
 

Giac [F]

\[ \int \frac {(e x)^m \left (a+b x^n\right ) \left (A+B x^n\right )}{c+d x^n} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )} \left (e x\right )^{m}}{d x^{n} + c} \,d x } \] Input:

integrate((e*x)^m*(a+b*x^n)*(A+B*x^n)/(c+d*x^n),x, algorithm="giac")
 

Output:

integrate((B*x^n + A)*(b*x^n + a)*(e*x)^m/(d*x^n + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^m \left (a+b x^n\right ) \left (A+B x^n\right )}{c+d x^n} \, dx=\int \frac {{\left (e\,x\right )}^m\,\left (A+B\,x^n\right )\,\left (a+b\,x^n\right )}{c+d\,x^n} \,d x \] Input:

int(((e*x)^m*(A + B*x^n)*(a + b*x^n))/(c + d*x^n),x)
 

Output:

int(((e*x)^m*(A + B*x^n)*(a + b*x^n))/(c + d*x^n), x)
 

Reduce [F]

\[ \int \frac {(e x)^m \left (a+b x^n\right ) \left (A+B x^n\right )}{c+d x^n} \, dx=\frac {e^{m} \left (x^{m +n} b^{2} d m x +x^{m +n} b^{2} d x +2 x^{m} a b d m x +2 x^{m} a b d n x +2 x^{m} a b d x -x^{m} b^{2} c m x -x^{m} b^{2} c n x -x^{m} b^{2} c x +\left (\int \frac {x^{m}}{x^{n} d +c}d x \right ) a^{2} d^{2} m^{2}+\left (\int \frac {x^{m}}{x^{n} d +c}d x \right ) a^{2} d^{2} m n +2 \left (\int \frac {x^{m}}{x^{n} d +c}d x \right ) a^{2} d^{2} m +\left (\int \frac {x^{m}}{x^{n} d +c}d x \right ) a^{2} d^{2} n +\left (\int \frac {x^{m}}{x^{n} d +c}d x \right ) a^{2} d^{2}-2 \left (\int \frac {x^{m}}{x^{n} d +c}d x \right ) a b c d \,m^{2}-2 \left (\int \frac {x^{m}}{x^{n} d +c}d x \right ) a b c d m n -4 \left (\int \frac {x^{m}}{x^{n} d +c}d x \right ) a b c d m -2 \left (\int \frac {x^{m}}{x^{n} d +c}d x \right ) a b c d n -2 \left (\int \frac {x^{m}}{x^{n} d +c}d x \right ) a b c d +\left (\int \frac {x^{m}}{x^{n} d +c}d x \right ) b^{2} c^{2} m^{2}+\left (\int \frac {x^{m}}{x^{n} d +c}d x \right ) b^{2} c^{2} m n +2 \left (\int \frac {x^{m}}{x^{n} d +c}d x \right ) b^{2} c^{2} m +\left (\int \frac {x^{m}}{x^{n} d +c}d x \right ) b^{2} c^{2} n +\left (\int \frac {x^{m}}{x^{n} d +c}d x \right ) b^{2} c^{2}\right )}{d^{2} \left (m^{2}+m n +2 m +n +1\right )} \] Input:

int((e*x)^m*(a+b*x^n)*(A+B*x^n)/(c+d*x^n),x)
 

Output:

(e**m*(x**(m + n)*b**2*d*m*x + x**(m + n)*b**2*d*x + 2*x**m*a*b*d*m*x + 2* 
x**m*a*b*d*n*x + 2*x**m*a*b*d*x - x**m*b**2*c*m*x - x**m*b**2*c*n*x - x**m 
*b**2*c*x + int(x**m/(x**n*d + c),x)*a**2*d**2*m**2 + int(x**m/(x**n*d + c 
),x)*a**2*d**2*m*n + 2*int(x**m/(x**n*d + c),x)*a**2*d**2*m + int(x**m/(x* 
*n*d + c),x)*a**2*d**2*n + int(x**m/(x**n*d + c),x)*a**2*d**2 - 2*int(x**m 
/(x**n*d + c),x)*a*b*c*d*m**2 - 2*int(x**m/(x**n*d + c),x)*a*b*c*d*m*n - 4 
*int(x**m/(x**n*d + c),x)*a*b*c*d*m - 2*int(x**m/(x**n*d + c),x)*a*b*c*d*n 
 - 2*int(x**m/(x**n*d + c),x)*a*b*c*d + int(x**m/(x**n*d + c),x)*b**2*c**2 
*m**2 + int(x**m/(x**n*d + c),x)*b**2*c**2*m*n + 2*int(x**m/(x**n*d + c),x 
)*b**2*c**2*m + int(x**m/(x**n*d + c),x)*b**2*c**2*n + int(x**m/(x**n*d + 
c),x)*b**2*c**2))/(d**2*(m**2 + m*n + 2*m + n + 1))