\(\int \frac {A+B x^n+C x^{2 n}}{(d+e x^n)^2 (a+c x^{2 n})} \, dx\) [6]

Optimal result

Integrand size = 35, antiderivative size = 325 \[ \int \frac {A+B x^n+C x^{2 n}}{\left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )} \, dx=\frac {\left (C d^2-B d e+A e^2\right ) x}{d \left (c d^2+a e^2\right ) n \left (d+e x^n\right )}+\frac {\left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a \left (c d^2+a e^2\right )^2}+\frac {\left (a e^2 \left (e (B d-A e (1-n))-C d^2 (1+n)\right )-c d^2 \left (C d^2 (1-n)+e (A e (1-3 n)-B (d-2 d n))\right )\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right )}{d^2 \left (c d^2+a e^2\right )^2 n}+\frac {c \left (B c d^2-2 A c d e+2 a C d e-a B e^2\right ) x^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a \left (c d^2+a e^2\right )^2 (1+n)} \] Output:

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

Mathematica [A] (verified)

Time = 1.61 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.79 \[ \int \frac {A+B x^n+C x^{2 n}}{\left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )} \, dx=\frac {x \left (\frac {\left (A c \left (c d^2-a e^2\right )+a \left (a C e^2+c d (-C d+2 B e)\right )\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a}+\frac {e \left (-B c d^2+2 A c d e-2 a C d e+a B e^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right )}{d}+\frac {c \left (B c d^2-2 A c d e+2 a C d e-a B e^2\right ) x^n \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a (1+n)}+\frac {\left (c d^2+a e^2\right ) \left (C d^2+e (-B d+A e)\right ) \operatorname {Hypergeometric2F1}\left (2,\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right )}{d^2}\right )}{\left (c d^2+a e^2\right )^2} \] Input:

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

Output:

(x*(((A*c*(c*d^2 - a*e^2) + a*(a*C*e^2 + c*d*(-(C*d) + 2*B*e)))*Hypergeome 
tric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/a + (e*(-(B*c*d^2) 
+ 2*A*c*d*e - 2*a*C*d*e + a*B*e^2)*Hypergeometric2F1[1, n^(-1), 1 + n^(-1) 
, -((e*x^n)/d)])/d + (c*(B*c*d^2 - 2*A*c*d*e + 2*a*C*d*e - a*B*e^2)*x^n*Hy 
pergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(1 
 + n)) + ((c*d^2 + a*e^2)*(C*d^2 + e*(-(B*d) + A*e))*Hypergeometric2F1[2, 
n^(-1), 1 + n^(-1), -((e*x^n)/d)])/d^2))/(c*d^2 + a*e^2)^2
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.89, 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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((A*c*(c*d^2 - a*e^2) + a*(a*C*e^2 - c*d*(C*d - 2*B*e)))*x*Hypergeometric2 
F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(c*d^2 + a*e^2)^2) - 
(e*(B*c*d^2 - 2*A*c*d*e + 2*a*C*d*e - a*B*e^2)*x*Hypergeometric2F1[1, n^(- 
1), 1 + n^(-1), -((e*x^n)/d)])/(d*(c*d^2 + a*e^2)^2) + (c*(B*c*d^2 - 2*A*c 
*d*e + 2*a*C*d*e - a*B*e^2)*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)^2*(1 + n)) + ((C*d^2 
 - B*d*e + A*e^2)*x*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((e*x^n)/d)] 
)/(d^2*(c*d^2 + a*e^2))
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-2)]

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

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

((c*d^2*e^2*(3*n - 1) + a*e^4*(n - 1))*A - (c*d^3*e*(2*n - 1) - a*d*e^3)*B 
 - (a*d^2*e^2*(n + 1) - c*d^4*(n - 1))*C)*integrate(1/(c^2*d^6*n + 2*a*c*d 
^4*e^2*n + a^2*d^2*e^4*n + (c^2*d^5*e*n + 2*a*c*d^3*e^3*n + a^2*d*e^5*n)*x 
^n), x) + (C*d^2 - B*d*e + A*e^2)*x/(c*d^4*n + a*d^2*e^2*n + (c*d^3*e*n + 
a*d*e^3*n)*x^n) - integrate(-(2*B*a*c*d*e + (c^2*d^2 - a*c*e^2)*A - (a*c*d 
^2 - a^2*e^2)*C + (2*C*a*c*d*e - 2*A*c^2*d*e + (c^2*d^2 - a*c*e^2)*B)*x^n) 
/(a*c^2*d^4 + 2*a^2*c*d^2*e^2 + a^3*e^4 + (c^3*d^4 + 2*a*c^2*d^2*e^2 + a^2 
*c*e^4)*x^(2*n)), x)
 

Giac [F]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {A+B x^n+C x^{2 n}}{\left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )} \, dx=\left (\int \frac {x^{2 n}}{x^{4 n} c \,e^{2}+2 x^{3 n} c d e +x^{2 n} a \,e^{2}+x^{2 n} c \,d^{2}+2 x^{n} a d e +a \,d^{2}}d x \right ) c +\left (\int \frac {x^{n}}{x^{4 n} c \,e^{2}+2 x^{3 n} c d e +x^{2 n} a \,e^{2}+x^{2 n} c \,d^{2}+2 x^{n} a d e +a \,d^{2}}d x \right ) b +\left (\int \frac {1}{x^{4 n} c \,e^{2}+2 x^{3 n} c d e +x^{2 n} a \,e^{2}+x^{2 n} c \,d^{2}+2 x^{n} a d e +a \,d^{2}}d x \right ) a \] Input:

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

Output:

int(x**(2*n)/(x**(4*n)*c*e**2 + 2*x**(3*n)*c*d*e + x**(2*n)*a*e**2 + x**(2 
*n)*c*d**2 + 2*x**n*a*d*e + a*d**2),x)*c + int(x**n/(x**(4*n)*c*e**2 + 2*x 
**(3*n)*c*d*e + x**(2*n)*a*e**2 + x**(2*n)*c*d**2 + 2*x**n*a*d*e + a*d**2) 
,x)*b + int(1/(x**(4*n)*c*e**2 + 2*x**(3*n)*c*d*e + x**(2*n)*a*e**2 + x**( 
2*n)*c*d**2 + 2*x**n*a*d*e + a*d**2),x)*a