\(\int \frac {A+B x^n+C x^{2 n}}{(d+e x^n) (a+c x^{2 n})} \, dx\) [5]

Optimal result

Integrand size = 35, antiderivative size = 188 \[ \int \frac {A+B x^n+C x^{2 n}}{\left (d+e x^n\right ) \left (a+c x^{2 n}\right )} \, dx=\frac {(A c d-a C d+a B e) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a \left (c d^2+a e^2\right )}+\frac {\left (C d^2-B d e+A e^2\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right )}{d \left (c d^2+a e^2\right )}+\frac {(B c d-A c e+a C e) x^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a \left (c d^2+a e^2\right ) (1+n)} \] Output:

(A*c*d+B*a*e-C*a*d)*x*hypergeom([1, 1/2/n],[1+1/2/n],-c*x^(2*n)/a)/a/(a*e^ 
2+c*d^2)+(A*e^2-B*d*e+C*d^2)*x*hypergeom([1, 1/n],[1+1/n],-e*x^n/d)/d/(a*e 
^2+c*d^2)+(-A*c*e+B*c*d+C*a*e)*x^(1+n)*hypergeom([1, 1/2*(1+n)/n],[3/2+1/2 
/n],-c*x^(2*n)/a)/a/(a*e^2+c*d^2)/(1+n)
 

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.85 \[ \int \frac {A+B x^n+C x^{2 n}}{\left (d+e x^n\right ) \left (a+c x^{2 n}\right )} \, dx=\frac {x \left (\frac {(A c d-a C d+a B e) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a}+\frac {\left (C d^2+e (-B d+A e)\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right )}{d}+\frac {(B c d-A c e+a C e) x^n \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a (1+n)}\right )}{c d^2+a e^2} \] Input:

Integrate[(A + B*x^n + C*x^(2*n))/((d + e*x^n)*(a + c*x^(2*n))),x]
 

Output:

(x*(((A*c*d - a*C*d + a*B*e)*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, 
 -((c*x^(2*n))/a)])/a + ((C*d^2 + e*(-(B*d) + A*e))*Hypergeometric2F1[1, n 
^(-1), 1 + n^(-1), -((e*x^n)/d)])/d + ((B*c*d - A*c*e + a*C*e)*x^n*Hyperge 
ometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(1 + n) 
)))/(c*d^2 + a*e^2)
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, 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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x^{n+1} \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right ) (a C e-A c e+B c d)}{a (n+1) \left (a e^2+c d^2\right )}+\frac {x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right ) (a B e-a C d+A c d)}{a \left (a e^2+c d^2\right )}+\frac {x \left (A e^2-B d e+C d^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right )}{d \left (a e^2+c d^2\right )}\)

Input:

Int[(A + B*x^n + C*x^(2*n))/((d + e*x^n)*(a + c*x^(2*n))),x]
 

Output:

((A*c*d - a*C*d + a*B*e)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, - 
((c*x^(2*n))/a)])/(a*(c*d^2 + a*e^2)) + ((C*d^2 - B*d*e + A*e^2)*x*Hyperge 
ometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d*(c*d^2 + a*e^2)) + ((B 
*c*d - A*c*e + a*C*e)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n 
^(-1))/2, -((c*x^(2*n))/a)])/(a*(c*d^2 + a*e^2)*(1 + n))
 

Defintions of rubi rules used

rule 2009

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

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [F]

\[\int \frac {A +B \,x^{n}+C \,x^{2 n}}{\left (d +e \,x^{n}\right ) \left (a +c \,x^{2 n}\right )}d x\]

Input:

int((A+B*x^n+C*x^(2*n))/(d+e*x^n)/(a+c*x^(2*n)),x)
 

Output:

int((A+B*x^n+C*x^(2*n))/(d+e*x^n)/(a+c*x^(2*n)),x)
 

Fricas [F]

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

integrate((A+B*x^n+C*x^(2*n))/(d+e*x^n)/(a+c*x^(2*n)),x, algorithm="fricas 
")
 

Output:

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

Sympy [F(-2)]

Exception generated. \[ \int \frac {A+B x^n+C x^{2 n}}{\left (d+e x^n\right ) \left (a+c x^{2 n}\right )} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate((A+B*x**n+C*x**(2*n))/(d+e*x**n)/(a+c*x**(2*n)),x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

integrate((A+B*x^n+C*x^(2*n))/(d+e*x^n)/(a+c*x^(2*n)),x, algorithm="maxima 
")
 

Output:

integrate((C*x^(2*n) + B*x^n + A)/((c*x^(2*n) + a)*(e*x^n + d)), x)
 

Giac [F]

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

integrate((A+B*x^n+C*x^(2*n))/(d+e*x^n)/(a+c*x^(2*n)),x, algorithm="giac")
 

Output:

integrate((C*x^(2*n) + B*x^n + A)/((c*x^(2*n) + a)*(e*x^n + d)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^n+C x^{2 n}}{\left (d+e x^n\right ) \left (a+c x^{2 n}\right )} \, dx=\int \frac {A+C\,x^{2\,n}+B\,x^n}{\left (a+c\,x^{2\,n}\right )\,\left (d+e\,x^n\right )} \,d x \] Input:

int((A + C*x^(2*n) + B*x^n)/((a + c*x^(2*n))*(d + e*x^n)),x)
 

Output:

int((A + C*x^(2*n) + B*x^n)/((a + c*x^(2*n))*(d + e*x^n)), x)
 

Reduce [F]

\[ \int \frac {A+B x^n+C x^{2 n}}{\left (d+e x^n\right ) \left (a+c x^{2 n}\right )} \, dx=\left (\int \frac {x^{2 n}}{x^{3 n} c e +x^{2 n} c d +x^{n} a e +a d}d x \right ) c +\left (\int \frac {x^{n}}{x^{3 n} c e +x^{2 n} c d +x^{n} a e +a d}d x \right ) b +\left (\int \frac {1}{x^{3 n} c e +x^{2 n} c d +x^{n} a e +a d}d x \right ) a \] Input:

int((A+B*x^n+C*x^(2*n))/(d+e*x^n)/(a+c*x^(2*n)),x)
 

Output:

int(x**(2*n)/(x**(3*n)*c*e + x**(2*n)*c*d + x**n*a*e + a*d),x)*c + int(x** 
n/(x**(3*n)*c*e + x**(2*n)*c*d + x**n*a*e + a*d),x)*b + int(1/(x**(3*n)*c* 
e + x**(2*n)*c*d + x**n*a*e + a*d),x)*a