\(\int (e x)^m (\frac {c}{a-b x}+\frac {d}{a+b x}) \, dx\) [226]

Optimal result

Integrand size = 26, antiderivative size = 70 \[ \int (e x)^m \left (\frac {c}{a-b x}+\frac {d}{a+b x}\right ) \, dx=\frac {d (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b x}{a}\right )}{a e (1+m)}+\frac {c (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {b x}{a}\right )}{a e (1+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.70 \[ \int (e x)^m \left (\frac {c}{a-b x}+\frac {d}{a+b x}\right ) \, dx=\frac {x (e x)^m \left (d \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b x}{a}\right )+c \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {b x}{a}\right )\right )}{a (1+m)} \] Input:

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

Output:

(x*(e*x)^m*(d*Hypergeometric2F1[1, 1 + m, 2 + m, -((b*x)/a)] + c*Hypergeom 
etric2F1[1, 1 + m, 2 + m, (b*x)/a]))/(a*(1 + m))
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2010, 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.

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

\(\Big \downarrow \) 2010

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 2010

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] 
, x] /; FreeQ[{c, m}, x] && SumQ[u] &&  !LinearQ[u, x] &&  !MatchQ[u, (a_) 
+ (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (51) = 102\).

Time = 2.87 (sec) , antiderivative size = 379, normalized size of antiderivative = 5.41 \[ \int (e x)^m \left (\frac {c}{a-b x}+\frac {d}{a+b x}\right ) \, dx=\frac {c e^{m} m x^{m + 1} \Phi \left (\frac {b^{2} x^{2}}{a^{2}}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {c e^{m} x^{m + 1} \Phi \left (\frac {b^{2} x^{2}}{a^{2}}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {d e^{m} m x^{m + 1} \Phi \left (\frac {b^{2} x^{2}}{a^{2}}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {d e^{m} x^{m + 1} \Phi \left (\frac {b^{2} x^{2}}{a^{2}}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {b c e^{m} m x^{m + 2} \Phi \left (\frac {b^{2} x^{2}}{a^{2}}, 1, \frac {m}{2} + 1\right ) \Gamma \left (\frac {m}{2} + 1\right )}{4 a^{2} \Gamma \left (\frac {m}{2} + 2\right )} + \frac {b c e^{m} x^{m + 2} \Phi \left (\frac {b^{2} x^{2}}{a^{2}}, 1, \frac {m}{2} + 1\right ) \Gamma \left (\frac {m}{2} + 1\right )}{2 a^{2} \Gamma \left (\frac {m}{2} + 2\right )} - \frac {b d e^{m} m x^{m + 2} \Phi \left (\frac {b^{2} x^{2}}{a^{2}}, 1, \frac {m}{2} + 1\right ) \Gamma \left (\frac {m}{2} + 1\right )}{4 a^{2} \Gamma \left (\frac {m}{2} + 2\right )} - \frac {b d e^{m} x^{m + 2} \Phi \left (\frac {b^{2} x^{2}}{a^{2}}, 1, \frac {m}{2} + 1\right ) \Gamma \left (\frac {m}{2} + 1\right )}{2 a^{2} \Gamma \left (\frac {m}{2} + 2\right )} \] Input:

integrate((e*x)**m*(c/(-b*x+a)+d/(b*x+a)),x)
 

Output:

c*e**m*m*x**(m + 1)*lerchphi(b**2*x**2/a**2, 1, m/2 + 1/2)*gamma(m/2 + 1/2 
)/(4*a*gamma(m/2 + 3/2)) + c*e**m*x**(m + 1)*lerchphi(b**2*x**2/a**2, 1, m 
/2 + 1/2)*gamma(m/2 + 1/2)/(4*a*gamma(m/2 + 3/2)) + d*e**m*m*x**(m + 1)*le 
rchphi(b**2*x**2/a**2, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(4*a*gamma(m/2 + 3/2 
)) + d*e**m*x**(m + 1)*lerchphi(b**2*x**2/a**2, 1, m/2 + 1/2)*gamma(m/2 + 
1/2)/(4*a*gamma(m/2 + 3/2)) + b*c*e**m*m*x**(m + 2)*lerchphi(b**2*x**2/a** 
2, 1, m/2 + 1)*gamma(m/2 + 1)/(4*a**2*gamma(m/2 + 2)) + b*c*e**m*x**(m + 2 
)*lerchphi(b**2*x**2/a**2, 1, m/2 + 1)*gamma(m/2 + 1)/(2*a**2*gamma(m/2 + 
2)) - b*d*e**m*m*x**(m + 2)*lerchphi(b**2*x**2/a**2, 1, m/2 + 1)*gamma(m/2 
 + 1)/(4*a**2*gamma(m/2 + 2)) - b*d*e**m*x**(m + 2)*lerchphi(b**2*x**2/a** 
2, 1, m/2 + 1)*gamma(m/2 + 1)/(2*a**2*gamma(m/2 + 2))
 

Maxima [F]

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

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

Output:

-integrate((e*x)^m*(c/(b*x - a) - d/(b*x + a)), x)
 

Giac [F]

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

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

Output:

integrate(-(e*x)^m*(c/(b*x - a) - d/(b*x + a)), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (e x)^m \left (\frac {c}{a-b x}+\frac {d}{a+b x}\right ) \, dx=\frac {e^{m} \left (-x^{m} c +x^{m} d +\left (\int \frac {x^{m}}{-b^{2} x^{2}+a^{2}}d x \right ) a b c m +\left (\int \frac {x^{m}}{-b^{2} x^{2}+a^{2}}d x \right ) a b d m +\left (\int \frac {x^{m}}{-b^{2} x^{3}+a^{2} x}d x \right ) a^{2} c m -\left (\int \frac {x^{m}}{-b^{2} x^{3}+a^{2} x}d x \right ) a^{2} d m \right )}{b m} \] Input:

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

Output:

(e**m*( - x**m*c + x**m*d + int(x**m/(a**2 - b**2*x**2),x)*a*b*c*m + int(x 
**m/(a**2 - b**2*x**2),x)*a*b*d*m + int(x**m/(a**2*x - b**2*x**3),x)*a**2* 
c*m - int(x**m/(a**2*x - b**2*x**3),x)*a**2*d*m))/(b*m)