\(\int \frac {1}{(d+e x^n)^2 (a+c x^{2 n})} \, dx\) [18]

Optimal result

Integrand size = 21, antiderivative size = 224 \[ \int \frac {1}{\left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )} \, dx=\frac {e^2 x}{d \left (c d^2+a e^2\right ) n \left (d+e x^n\right )}+\frac {c \left (c d^2-a e^2\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a \left (c d^2+a e^2\right )^2}-\frac {e^2 \left (c d^2 (1-3 n)+a e^2 (1-n)\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right )}{d^2 \left (c d^2+a e^2\right )^2 n}-\frac {2 c^2 d e x^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a \left (c d^2+a e^2\right )^2 (1+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )} \, dx=\frac {x \left (c d^2 \left (c d^2-a e^2\right ) (1+n) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )+e \left (2 a c d^2 e (1+n) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right )-2 c^2 d^3 x^n \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )+a e \left (c d^2+a e^2\right ) (1+n) \operatorname {Hypergeometric2F1}\left (2,\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right )\right )\right )}{a \left (c d^3+a d e^2\right )^2 (1+n)} \] Input:

Integrate[1/((d + e*x^n)^2*(a + c*x^(2*n))),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.92, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1755, 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.

\(\displaystyle \int \frac {1}{\left (a+c x^{2 n}\right ) \left (d+e x^n\right )^2} \, dx\)

\(\Big \downarrow \) 1755

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

\(\Big \downarrow \) 2009

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

Input:

Int[1/((d + e*x^n)^2*(a + c*x^(2*n))),x]
 

Output:

(c*(c*d^2 - a*e^2)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^ 
(2*n))/a)])/(a*(c*d^2 + a*e^2)^2) + (2*c*e^2*x*Hypergeometric2F1[1, n^(-1) 
, 1 + n^(-1), -((e*x^n)/d)])/(c*d^2 + a*e^2)^2 - (2*c^2*d*e*x^(1 + n)*Hype 
rgeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(c*d 
^2 + a*e^2)^2*(1 + n)) + (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 1755

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> I 
nt[ExpandIntegrand[(d + e*x^n)^q/(a + c*x^(2*n)), x], x] /; FreeQ[{a, c, d, 
 e, n}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[q]
 

rule 2009

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

Maple [F]

\[\int \frac {1}{\left (d +e \,x^{n}\right )^{2} \left (a +c \,x^{2 n}\right )}d x\]

Input:

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

Output:

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

Fricas [F]

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

integrate(1/(d+e*x^n)^2/(a+c*x^(2*n)),x, algorithm="fricas")
 

Output:

integral(1/(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. \[ \int \frac {1}{\left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate(1/(d+e*x**n)**2/(a+c*x**(2*n)),x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

integrate(1/(d+e*x^n)^2/(a+c*x^(2*n)),x, algorithm="maxima")
 

Output:

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) + (c*d^2*e^2*( 
3*n - 1) + a*e^4*(n - 1))*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) - integr 
ate((2*c^2*d*e*x^n - c^2*d^2 + a*c*e^2)/(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]

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

integrate(1/(d+e*x^n)^2/(a+c*x^(2*n)),x, algorithm="giac")
 

Output:

integrate(1/((c*x^(2*n) + a)*(e*x^n + d)^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )} \, dx=\int \frac {1}{\left (a+c\,x^{2\,n}\right )\,{\left (d+e\,x^n\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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)