Integrand size = 30, antiderivative size = 70 \[ \int \left (\frac {c (e x)^m}{a-b x}+\frac {d (e x)^m}{a+b x}\right ) \, dx=\frac {d (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b x}{a}\right )}{a e (1+m)}+\frac {c (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {b x}{a}\right )}{a e (1+m)} \] 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)
Time = 0.11 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.70 \[ \int \left (\frac {c (e x)^m}{a-b x}+\frac {d (e x)^m}{a+b x}\right ) \, dx=\frac {x (e x)^m \left (d \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b x}{a}\right )+c \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {b x}{a}\right )\right )}{a (1+m)} \] Input:
Integrate[(c*(e*x)^m)/(a - b*x) + (d*(e*x)^m)/(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))
Time = 0.21 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {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.
\(\displaystyle \int \left (\frac {c (e x)^m}{a-b x}+\frac {d (e x)^m}{a+b x}\right ) \, dx\) |
\(\Big \downarrow \) 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[(c*(e*x)^m)/(a - b*x) + (d*(e*x)^m)/(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))
\[\int \left (\frac {c \left (e x \right )^{m}}{-b x +a}+\frac {d \left (e x \right )^{m}}{b x +a}\right )d x\]
Input:
int(c*(e*x)^m/(-b*x+a)+d*(e*x)^m/(b*x+a),x)
Output:
int(c*(e*x)^m/(-b*x+a)+d*(e*x)^m/(b*x+a),x)
\[ \int \left (\frac {c (e x)^m}{a-b x}+\frac {d (e x)^m}{a+b x}\right ) \, dx=\int { -\frac {\left (e x\right )^{m} c}{b x - a} + \frac {\left (e x\right )^{m} d}{b x + a} \,d x } \] Input:
integrate(c*(e*x)^m/(-b*x+a)+d*(e*x)^m/(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)
Result contains complex when optimal does not.
Time = 0.52 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.06 \[ \int \left (\frac {c (e x)^m}{a-b x}+\frac {d (e x)^m}{a+b x}\right ) \, dx=c \left (\frac {e^{m} m x^{m + 1} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a \Gamma \left (m + 2\right )} + \frac {e^{m} x^{m + 1} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a \Gamma \left (m + 2\right )}\right ) + d \left (\frac {e^{m} m x^{m + 1} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a \Gamma \left (m + 2\right )} + \frac {e^{m} x^{m + 1} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a \Gamma \left (m + 2\right )}\right ) \] Input:
integrate(c*(e*x)**m/(-b*x+a)+d*(e*x)**m/(b*x+a),x)
Output:
c*(e**m*m*x**(m + 1)*lerchphi(b*x*exp_polar(2*I*pi)/a, 1, m + 1)*gamma(m + 1)/(a*gamma(m + 2)) + e**m*x**(m + 1)*lerchphi(b*x*exp_polar(2*I*pi)/a, 1 , m + 1)*gamma(m + 1)/(a*gamma(m + 2))) + d*(e**m*m*x**(m + 1)*lerchphi(b* x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(a*gamma(m + 2)) + e**m*x**(m + 1)*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(a*gamma(m + 2 )))
\[ \int \left (\frac {c (e x)^m}{a-b x}+\frac {d (e x)^m}{a+b x}\right ) \, dx=\int { -\frac {\left (e x\right )^{m} c}{b x - a} + \frac {\left (e x\right )^{m} d}{b x + a} \,d x } \] Input:
integrate(c*(e*x)^m/(-b*x+a)+d*(e*x)^m/(b*x+a),x, algorithm="maxima")
Output:
integrate(-(e*x)^m*c/(b*x - a) + (e*x)^m*d/(b*x + a), x)
\[ \int \left (\frac {c (e x)^m}{a-b x}+\frac {d (e x)^m}{a+b x}\right ) \, dx=\int { -\frac {\left (e x\right )^{m} c}{b x - a} + \frac {\left (e x\right )^{m} d}{b x + a} \,d x } \] Input:
integrate(c*(e*x)^m/(-b*x+a)+d*(e*x)^m/(b*x+a),x, algorithm="giac")
Output:
integrate(-(e*x)^m*c/(b*x - a) + (e*x)^m*d/(b*x + a), x)
Timed out. \[ \int \left (\frac {c (e x)^m}{a-b x}+\frac {d (e x)^m}{a+b x}\right ) \, dx=\int \frac {c\,{\left (e\,x\right )}^m}{a-b\,x}+\frac {d\,{\left (e\,x\right )}^m}{a+b\,x} \,d x \] Input:
int((c*(e*x)^m)/(a - b*x) + (d*(e*x)^m)/(a + b*x),x)
Output:
int((c*(e*x)^m)/(a - b*x) + (d*(e*x)^m)/(a + b*x), x)
\[ \int \left (\frac {c (e x)^m}{a-b x}+\frac {d (e x)^m}{a+b x}\right ) \, dx=\frac {e^{m} \left (-x^{m} c +x^{m} d +\left (\int \frac {x^{m}}{-b^{2} x^{2}+a^{2}}d x \right ) a b c m +\left (\int \frac {x^{m}}{-b^{2} x^{2}+a^{2}}d x \right ) a b d m +\left (\int \frac {x^{m}}{-b^{2} x^{3}+a^{2} x}d x \right ) a^{2} c m -\left (\int \frac {x^{m}}{-b^{2} x^{3}+a^{2} x}d x \right ) a^{2} d m \right )}{b m} \] Input:
int(c*(e*x)^m/(-b*x+a)+d*(e*x)^m/(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)