\(\int \frac {(e x)^m (A+B x^n)}{c+d x^n} \, dx\) [43]

Optimal result

Integrand size = 22, antiderivative size = 78 \[ \int \frac {(e x)^m \left (A+B x^n\right )}{c+d x^n} \, dx=\frac {B (e x)^{1+m}}{d e (1+m)}-\frac {(B c-A d) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )}{c d e (1+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.73 \[ \int \frac {(e x)^m \left (A+B x^n\right )}{c+d x^n} \, dx=\frac {x (e x)^m \left (B c+(-B c+A d) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )\right )}{c d (1+m)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {959, 888}

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.

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

\(\Big \downarrow \) 959

\(\displaystyle \frac {B (e x)^{m+1}}{d e (m+1)}-\frac {(B c-A d) \int \frac {(e x)^m}{d x^n+c}dx}{d}\)

\(\Big \downarrow \) 888

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

Input:

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

Output:

(B*(e*x)^(1 + m))/(d*e*(1 + m)) - ((B*c - A*d)*(e*x)^(1 + m)*Hypergeometri 
c2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(c*d*e*(1 + m))
 

Defintions of rubi rules used

rule 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 959

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p 
+ 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p 
 + 1) + 1))   Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, 
 n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.19 (sec) , antiderivative size = 377, normalized size of antiderivative = 4.83 \[ \int \frac {(e x)^m \left (A+B x^n\right )}{c+d x^n} \, dx=\frac {A c^{\frac {m}{n} + \frac {1}{n}} c^{- \frac {m}{n} - 1 - \frac {1}{n}} e^{m} m x^{m + 1} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, \frac {m}{n} + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + \frac {1}{n}\right )}{n^{2} \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )} + \frac {A c^{\frac {m}{n} + \frac {1}{n}} c^{- \frac {m}{n} - 1 - \frac {1}{n}} e^{m} x^{m + 1} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, \frac {m}{n} + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + \frac {1}{n}\right )}{n^{2} \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )} + \frac {B c^{- \frac {m}{n} - 2 - \frac {1}{n}} c^{\frac {m}{n} + 1 + \frac {1}{n}} e^{m} m x^{m + n + 1} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, \frac {m}{n} + 1 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )}{n^{2} \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )} + \frac {B c^{- \frac {m}{n} - 2 - \frac {1}{n}} c^{\frac {m}{n} + 1 + \frac {1}{n}} e^{m} x^{m + n + 1} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, \frac {m}{n} + 1 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )}{n \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )} + \frac {B c^{- \frac {m}{n} - 2 - \frac {1}{n}} c^{\frac {m}{n} + 1 + \frac {1}{n}} e^{m} x^{m + n + 1} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, \frac {m}{n} + 1 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )}{n^{2} \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )} \] Input:

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

Output:

A*c**(m/n + 1/n)*c**(-m/n - 1 - 1/n)*e**m*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*c**(m/n + 1/n)*c**(-m/n - 1 - 1/n)*e**m*x**(m + 1)*lerchphi(d*x**n*ex 
p_polar(I*pi)/c, 1, m/n + 1/n)*gamma(m/n + 1/n)/(n**2*gamma(m/n + 1 + 1/n) 
) + B*c**(-m/n - 2 - 1/n)*c**(m/n + 1 + 1/n)*e**m*m*x**(m + n + 1)*lerchph 
i(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(n**2*g 
amma(m/n + 2 + 1/n)) + 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*gamma(m/n + 2 + 1/n)) + 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))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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