Integrand size = 26, antiderivative size = 99 \[ \int \frac {A+B x^n+C x^{2 n}}{a+c x^{2 n}} \, dx=\frac {C x}{c}+\frac {(A c-a C) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a c}+\frac {B x^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a (1+n)} \] Output:
C*x/c+(A*c-C*a)*x*hypergeom([1, 1/2/n],[1+1/2/n],-c*x^(2*n)/a)/a/c+B*x^(1+ n)*hypergeom([1, 1/2*(1+n)/n],[3/2+1/2/n],-c*x^(2*n)/a)/a/(1+n)
Time = 0.30 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x^n+C x^{2 n}}{a+c x^{2 n}} \, dx=\frac {C x}{c}+\frac {(A c-a C) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a c}+\frac {B x^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a (1+n)} \] Input:
Integrate[(A + B*x^n + C*x^(2*n))/(a + c*x^(2*n)),x]
Output:
(C*x)/c + ((A*c - a*C)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -(( c*x^(2*n))/a)])/(a*c) + (B*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), ( 3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(1 + n))
Time = 0.36 (sec) , antiderivative size = 99, 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 = {7293, 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.
\(\displaystyle \int \frac {A+B x^n+C x^{2 n}}{a+c x^{2 n}} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {-a C+A c+B c x^n}{c \left (a+c x^{2 n}\right )}+\frac {C}{c}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {x (A c-a C) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a c}+\frac {B x^{n+1} \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a (n+1)}+\frac {C x}{c}\) |
Input:
Int[(A + B*x^n + C*x^(2*n))/(a + c*x^(2*n)),x]
Output:
(C*x)/c + ((A*c - a*C)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -(( c*x^(2*n))/a)])/(a*c) + (B*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), ( 3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(1 + n))
\[\int \frac {A +B \,x^{n}+C \,x^{2 n}}{a +c \,x^{2 n}}d x\]
Input:
int((A+B*x^n+C*x^(2*n))/(a+c*x^(2*n)),x)
Output:
int((A+B*x^n+C*x^(2*n))/(a+c*x^(2*n)),x)
\[ \int \frac {A+B x^n+C x^{2 n}}{a+c x^{2 n}} \, dx=\int { \frac {C x^{2 \, n} + B x^{n} + A}{c x^{2 \, n} + a} \,d x } \] Input:
integrate((A+B*x^n+C*x^(2*n))/(a+c*x^(2*n)),x, algorithm="fricas")
Output:
integral((C*x^(2*n) + B*x^n + A)/(c*x^(2*n) + a), x)
Result contains complex when optimal does not.
Time = 2.71 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.92 \[ \int \frac {A+B x^n+C x^{2 n}}{a+c x^{2 n}} \, dx=\frac {A a^{\frac {1}{2 n}} a^{-1 - \frac {1}{2 n}} x \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2 n}\right )}{4 n^{2} \Gamma \left (1 + \frac {1}{2 n}\right )} + \frac {B a^{- \frac {3}{2} - \frac {1}{2 n}} a^{\frac {1}{2} + \frac {1}{2 n}} x^{n + 1} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2} + \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2} + \frac {1}{2 n}\right )}{4 n \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )} + \frac {B a^{- \frac {3}{2} - \frac {1}{2 n}} a^{\frac {1}{2} + \frac {1}{2 n}} x^{n + 1} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2} + \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2} + \frac {1}{2 n}\right )}{4 n^{2} \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )} - \frac {C a^{- \frac {1}{2 n}} a^{1 + \frac {1}{2 n}} c^{\frac {1}{2 n}} c^{-1 - \frac {1}{2 n}} x \Phi \left (\frac {a x^{- 2 n} e^{i \pi }}{c}, 1, \frac {e^{i \pi }}{2 n}\right ) \Gamma \left (\frac {1}{2 n}\right )}{4 a n^{2} \Gamma \left (1 + \frac {1}{2 n}\right )} \] Input:
integrate((A+B*x**n+C*x**(2*n))/(a+c*x**(2*n)),x)
Output:
A*a**(1/(2*n))*a**(-1 - 1/(2*n))*x*lerchphi(c*x**(2*n)*exp_polar(I*pi)/a, 1, 1/(2*n))*gamma(1/(2*n))/(4*n**2*gamma(1 + 1/(2*n))) + B*a**(-3/2 - 1/(2 *n))*a**(1/2 + 1/(2*n))*x**(n + 1)*lerchphi(c*x**(2*n)*exp_polar(I*pi)/a, 1, 1/2 + 1/(2*n))*gamma(1/2 + 1/(2*n))/(4*n*gamma(3/2 + 1/(2*n))) + B*a**( -3/2 - 1/(2*n))*a**(1/2 + 1/(2*n))*x**(n + 1)*lerchphi(c*x**(2*n)*exp_pola r(I*pi)/a, 1, 1/2 + 1/(2*n))*gamma(1/2 + 1/(2*n))/(4*n**2*gamma(3/2 + 1/(2 *n))) - C*a**(1 + 1/(2*n))*c**(1/(2*n))*c**(-1 - 1/(2*n))*x*lerchphi(a*exp _polar(I*pi)/(c*x**(2*n)), 1, exp_polar(I*pi)/(2*n))*gamma(1/(2*n))/(4*a*a **(1/(2*n))*n**2*gamma(1 + 1/(2*n)))
\[ \int \frac {A+B x^n+C x^{2 n}}{a+c x^{2 n}} \, dx=\int { \frac {C x^{2 \, n} + B x^{n} + A}{c x^{2 \, n} + a} \,d x } \] Input:
integrate((A+B*x^n+C*x^(2*n))/(a+c*x^(2*n)),x, algorithm="maxima")
Output:
C*x/c + integrate((B*c*x^n - C*a + A*c)/(c^2*x^(2*n) + a*c), x)
\[ \int \frac {A+B x^n+C x^{2 n}}{a+c x^{2 n}} \, dx=\int { \frac {C x^{2 \, n} + B x^{n} + A}{c x^{2 \, n} + a} \,d x } \] Input:
integrate((A+B*x^n+C*x^(2*n))/(a+c*x^(2*n)),x, algorithm="giac")
Output:
integrate((C*x^(2*n) + B*x^n + A)/(c*x^(2*n) + a), x)
Timed out. \[ \int \frac {A+B x^n+C x^{2 n}}{a+c x^{2 n}} \, dx=\int \frac {A+C\,x^{2\,n}+B\,x^n}{a+c\,x^{2\,n}} \,d x \] Input:
int((A + C*x^(2*n) + B*x^n)/(a + c*x^(2*n)),x)
Output:
int((A + C*x^(2*n) + B*x^n)/(a + c*x^(2*n)), x)
\[ \int \frac {A+B x^n+C x^{2 n}}{a+c x^{2 n}} \, dx=\left (\int \frac {x^{2 n}}{x^{2 n} c +a}d x \right ) c +\left (\int \frac {x^{n}}{x^{2 n} c +a}d x \right ) b +\left (\int \frac {1}{x^{2 n} c +a}d x \right ) a \] Input:
int((A+B*x^n+C*x^(2*n))/(a+c*x^(2*n)),x)
Output:
int(x**(2*n)/(x**(2*n)*c + a),x)*c + int(x**n/(x**(2*n)*c + a),x)*b + int( 1/(x**(2*n)*c + a),x)*a