\(\int \frac {A+B x^n+C x^{2 n}}{(d+e x^n)^2 (a+b x^n+c x^{2 n})} \, dx\) [12]

Optimal result

Integrand size = 40, antiderivative size = 609 \[ \int \frac {A+B x^n+C x^{2 n}}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )} \, dx=\frac {\left (C d^2-e (B d-A e)\right ) x}{d \left (c d^2-b d e+a e^2\right ) n \left (d+e x^n\right )}+\frac {c \left (B c d^2-b C d^2-2 A c d e+2 a C d e+A b e^2-a B e^2+\frac {b^2 \left (C d^2+A e^2\right )-b \left (2 (A c+a C) d e+B \left (c d^2+a e^2\right )\right )+2 \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right )}{\sqrt {b^2-4 a c}}\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^2}+\frac {c \left (B c d^2-b C d^2-2 A c d e+2 a C d e+A b e^2-a B e^2-\frac {b^2 \left (C d^2+A e^2\right )-b \left (2 (A c+a C) d e+B \left (c d^2+a e^2\right )\right )+2 \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right )}{\sqrt {b^2-4 a c}}\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^2}+\frac {\left (e \left (b d \left (C d^2+e (A e (1-2 n)-B d (1-n))\right )+a e \left (e (B d-A e (1-n))-C d^2 (1+n)\right )\right )-c d^2 \left (C d^2 (1-n)+e (A e (1-3 n)-B (d-2 d n))\right )\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right )}{d^2 \left (c d^2-b d e+a e^2\right )^2 n} \] Output:

(C*d^2-e*(-A*e+B*d))*x/d/(a*e^2-b*d*e+c*d^2)/n/(d+e*x^n)+c*(B*c*d^2-b*C*d^ 
2-2*A*c*d*e+2*C*a*d*e+A*b*e^2-B*a*e^2+(b^2*(A*e^2+C*d^2)-b*(2*(A*c+C*a)*d* 
e+B*(a*e^2+c*d^2))+2*A*c*(-a*e^2+c*d^2)+2*a*(a*C*e^2-c*d*(-2*B*e+C*d)))/(- 
4*a*c+b^2)^(1/2))*x*hypergeom([1, 1/n],[1+1/n],-2*c*x^n/(b-(-4*a*c+b^2)^(1 
/2)))/(b-(-4*a*c+b^2)^(1/2))/(a*e^2-b*d*e+c*d^2)^2+c*(B*c*d^2-b*C*d^2-2*A* 
c*d*e+2*C*a*d*e+A*b*e^2-B*a*e^2-(b^2*(A*e^2+C*d^2)-b*(2*(A*c+C*a)*d*e+B*(a 
*e^2+c*d^2))+2*A*c*(-a*e^2+c*d^2)+2*a*(a*C*e^2-c*d*(-2*B*e+C*d)))/(-4*a*c+ 
b^2)^(1/2))*x*hypergeom([1, 1/n],[1+1/n],-2*c*x^n/(b+(-4*a*c+b^2)^(1/2)))/ 
(b+(-4*a*c+b^2)^(1/2))/(a*e^2-b*d*e+c*d^2)^2+(e*(b*d*(C*d^2+e*(A*e*(1-2*n) 
-B*d*(1-n)))+a*e*(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-b*d*e+c*d^2)^2/n
 

Mathematica [A] (verified)

Time = 3.93 (sec) , antiderivative size = 509, normalized size of antiderivative = 0.84 \[ \int \frac {A+B x^n+C x^{2 n}}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )} \, dx=\frac {x \left (\frac {c \left (B c d^2-b C d^2-2 A c d e+2 a C d e+A b e^2-a B e^2+\frac {-2 b (A c+a C) d e+2 A c \left (c d^2-a e^2\right )-b B \left (c d^2+a e^2\right )+b^2 \left (C d^2+A e^2\right )+2 a \left (a C e^2+c d (-C d+2 B e)\right )}{\sqrt {b^2-4 a c}}\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )}{b-\sqrt {b^2-4 a c}}+\frac {c \left (B c d^2-b C d^2-2 A c d e+2 a C d e+A b e^2-a B e^2+\frac {2 b (A c+a C) d e-2 A c \left (c d^2-a e^2\right )+b B \left (c d^2+a e^2\right )-b^2 \left (C d^2+A e^2\right )-2 a \left (a C e^2+c d (-C d+2 B e)\right )}{\sqrt {b^2-4 a c}}\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{b+\sqrt {b^2-4 a c}}+\frac {e \left (-B c d^2+b C d^2+2 A c d e-2 a C d e-A b e^2+a B e^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right )}{d}+\frac {\left (c d^2+e (-b d+a e)\right ) \left (C d^2+e (-B d+A e)\right ) \operatorname {Hypergeometric2F1}\left (2,\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right )}{d^2}\right )}{\left (c d^2+e (-b d+a e)\right )^2} \] Input:

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

Output:

(x*((c*(B*c*d^2 - b*C*d^2 - 2*A*c*d*e + 2*a*C*d*e + A*b*e^2 - a*B*e^2 + (- 
2*b*(A*c + a*C)*d*e + 2*A*c*(c*d^2 - a*e^2) - b*B*(c*d^2 + a*e^2) + b^2*(C 
*d^2 + A*e^2) + 2*a*(a*C*e^2 + c*d*(-(C*d) + 2*B*e)))/Sqrt[b^2 - 4*a*c])*H 
ypergeometric2F1[1, n^(-1), 1 + n^(-1), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c]) 
])/(b - Sqrt[b^2 - 4*a*c]) + (c*(B*c*d^2 - b*C*d^2 - 2*A*c*d*e + 2*a*C*d*e 
 + A*b*e^2 - a*B*e^2 + (2*b*(A*c + a*C)*d*e - 2*A*c*(c*d^2 - a*e^2) + b*B* 
(c*d^2 + a*e^2) - b^2*(C*d^2 + A*e^2) - 2*a*(a*C*e^2 + c*d*(-(C*d) + 2*B*e 
)))/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n) 
/(b + Sqrt[b^2 - 4*a*c])])/(b + Sqrt[b^2 - 4*a*c]) + (e*(-(B*c*d^2) + b*C* 
d^2 + 2*A*c*d*e - 2*a*C*d*e - A*b*e^2 + a*B*e^2)*Hypergeometric2F1[1, n^(- 
1), 1 + n^(-1), -((e*x^n)/d)])/d + ((c*d^2 + e*(-(b*d) + a*e))*(C*d^2 + e* 
(-(B*d) + A*e))*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/d^ 
2))/(c*d^2 + e*(-(b*d) + a*e))^2
 

Rubi [A] (verified)

Time = 1.93 (sec) , antiderivative size = 552, normalized size of antiderivative = 0.91, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, 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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {c x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right ) \left (-\frac {2 b d e (a C+A c)-2 A c \left (c d^2-a e^2\right )+b B \left (a e^2+c d^2\right )-2 a \left (a C e^2-c d (C d-2 B e)\right )-\left (b^2 \left (A e^2+C d^2\right )\right )}{\sqrt {b^2-4 a c}}-a B e^2+2 a C d e+A b e^2-2 A c d e-b C d^2+B c d^2\right )}{\left (b-\sqrt {b^2-4 a c}\right ) \left (a e^2-b d e+c d^2\right )^2}+\frac {c x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right ) \left (\frac {2 b d e (a C+A c)-2 A c \left (c d^2-a e^2\right )+b B \left (a e^2+c d^2\right )-2 a \left (a C e^2-c d (C d-2 B e)\right )-\left (b^2 \left (A e^2+C d^2\right )\right )}{\sqrt {b^2-4 a c}}-a B e^2+2 a C d e+A b e^2-2 A c d e-b C d^2+B c d^2\right )}{\left (\sqrt {b^2-4 a c}+b\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac {e x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right ) \left (-a B e^2+2 a C d e+A b e^2-2 A c d e-b C d^2+B c d^2\right )}{d \left (a e^2-b d e+c d^2\right )^2}+\frac {x \left (C d^2-e (B d-A e)\right ) \operatorname {Hypergeometric2F1}\left (2,\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right )}{d^2 \left (a e^2-b d e+c d^2\right )}\)

Input:

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

Output:

(c*(B*c*d^2 - b*C*d^2 - 2*A*c*d*e + 2*a*C*d*e + A*b*e^2 - a*B*e^2 - (2*b*( 
A*c + a*C)*d*e - 2*A*c*(c*d^2 - a*e^2) + b*B*(c*d^2 + a*e^2) - b^2*(C*d^2 
+ A*e^2) - 2*a*(a*C*e^2 - c*d*(C*d - 2*B*e)))/Sqrt[b^2 - 4*a*c])*x*Hyperge 
ometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/((b 
 - Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)^2) + (c*(B*c*d^2 - b*C*d^2 - 
 2*A*c*d*e + 2*a*C*d*e + A*b*e^2 - a*B*e^2 + (2*b*(A*c + a*C)*d*e - 2*A*c* 
(c*d^2 - a*e^2) + b*B*(c*d^2 + a*e^2) - b^2*(C*d^2 + A*e^2) - 2*a*(a*C*e^2 
 - c*d*(C*d - 2*B*e)))/Sqrt[b^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 
 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/((b + Sqrt[b^2 - 4*a*c])*( 
c*d^2 - b*d*e + a*e^2)^2) - (e*(B*c*d^2 - b*C*d^2 - 2*A*c*d*e + 2*a*C*d*e 
+ A*b*e^2 - a*B*e^2)*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/ 
d)])/(d*(c*d^2 - b*d*e + a*e^2)^2) + ((C*d^2 - e*(B*d - A*e))*x*Hypergeome 
tric2F1[2, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d^2*(c*d^2 - b*d*e + 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((C*x^(2*n) + B*x^n + A)/(b*e^2*x^(3*n) + a*d^2 + (c*e^2*x^(2*n) + 
 2*c*d*e*x^n + c*d^2)*x^(2*n) + (2*b*d*e + a*e^2)*x^(2*n) + (b*d^2 + 2*a*d 
*e)*x^n), x)
 

Sympy [F(-2)]

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

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(1/(x**(2*n)*e**2 + 2*x**n*d*e + d**2),x)