\(\int \frac {1}{(d+e x^n) (a+c x^{2 n})^2} \, dx\) [35]

Optimal result

Integrand size = 21, antiderivative size = 249 \[ \int \frac {1}{\left (d+e x^n\right ) \left (a+c x^{2 n}\right )^2} \, dx=\frac {c x \left (d-e x^n\right )}{2 a \left (c d^2+a e^2\right ) n \left (a+c x^{2 n}\right )}-\frac {c d \left (a e^2 (1-4 n)+c d^2 (1-2 n)\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{2 a^2 \left (c d^2+a e^2\right )^2 n}+\frac {e^4 x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right )}{d \left (c d^2+a e^2\right )^2}+\frac {c e \left (a e^2 (1-3 n)+c d^2 (1-n)\right ) x^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{2 a^2 \left (c d^2+a e^2\right )^2 n (1+n)} \] Output:

1/2*c*x*(d-e*x^n)/a/(a*e^2+c*d^2)/n/(a+c*x^(2*n))-1/2*c*d*(a*e^2*(1-4*n)+c 
*d^2*(1-2*n))*x*hypergeom([1, 1/2/n],[1+1/2/n],-c*x^(2*n)/a)/a^2/(a*e^2+c* 
d^2)^2/n+e^4*x*hypergeom([1, 1/n],[1+1/n],-e*x^n/d)/d/(a*e^2+c*d^2)^2+1/2* 
c*e*(a*e^2*(1-3*n)+c*d^2*(1-n))*x^(1+n)*hypergeom([1, 1/2*(1+n)/n],[3/2+1/ 
2/n],-c*x^(2*n)/a)/a^2/(a*e^2+c*d^2)^2/n/(1+n)
 

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (d+e x^n\right ) \left (a+c x^{2 n}\right )^2} \, dx=\frac {x \left (a c d^2 e^2 (1+n) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )+a^2 e^4 (1+n) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right )+c d \left (-a e^3 x^n \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )+\left (c d^2+a e^2\right ) \left (d (1+n) \operatorname {Hypergeometric2F1}\left (2,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )-e x^n \operatorname {Hypergeometric2F1}\left (2,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )\right )\right )\right )}{a^2 d \left (c d^2+a e^2\right )^2 (1+n)} \] Input:

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

Output:

(x*(a*c*d^2*e^2*(1 + n)*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c 
*x^(2*n))/a)] + a^2*e^4*(1 + n)*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), - 
((e*x^n)/d)] + c*d*(-(a*e^3*x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n 
^(-1))/2, -((c*x^(2*n))/a)]) + (c*d^2 + a*e^2)*(d*(1 + n)*Hypergeometric2F 
1[2, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)] - e*x^n*Hypergeometric2F1[ 
2, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)]))))/(a^2*d*(c*d^2 + a* 
e^2)^2*(1 + n))
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.34, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1767, 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.

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

\(\Big \downarrow \) 1767

\(\displaystyle \int \left (-\frac {c e^2 \left (e x^n-d\right )}{\left (a e^2+c d^2\right )^2 \left (a+c x^{2 n}\right )}-\frac {c \left (e x^n-d\right )}{\left (a e^2+c d^2\right ) \left (a+c x^{2 n}\right )^2}+\frac {e^4}{\left (a e^2+c d^2\right )^2 \left (d+e x^n\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {c e (1-n) x^{n+1} \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{2 a^2 n (n+1) \left (a e^2+c d^2\right )}-\frac {c d (1-2 n) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{2 a^2 n \left (a e^2+c d^2\right )}+\frac {c d e^2 x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a \left (a e^2+c d^2\right )^2}+\frac {c x \left (d-e x^n\right )}{2 a n \left (a e^2+c d^2\right ) \left (a+c x^{2 n}\right )}+\frac {e^4 x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right )}{d \left (a e^2+c d^2\right )^2}-\frac {c e^3 x^{n+1} \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a (n+1) \left (a e^2+c d^2\right )^2}\)

Input:

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

Output:

(c*x*(d - e*x^n))/(2*a*(c*d^2 + a*e^2)*n*(a + c*x^(2*n))) + (c*d*e^2*x*Hyp 
ergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(c*d^2 + 
a*e^2)^2) - (c*d*(1 - 2*n)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, 
 -((c*x^(2*n))/a)])/(2*a^2*(c*d^2 + a*e^2)*n) + (e^4*x*Hypergeometric2F1[1 
, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d*(c*d^2 + a*e^2)^2) - (c*e^3*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*e*(1 - n)*x^(1 + n)*Hypergeometric2F1[ 
1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(2*a^2*(c*d^2 + a*e^2 
)*n*(1 + n))
 

Defintions of rubi rules used

rule 1767

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

rule 2009

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(1/(a^2*e*x^n + a^2*d + (c^2*e*x^n + c^2*d)*x^(4*n) + 2*(a*c*e*x^n 
 + a*c*d)*x^(2*n)), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (d+e x^n\right ) \left (a+c x^{2 n}\right )^2} \, dx=\int \frac {1}{x^{5 n} c^{2} e +x^{4 n} c^{2} d +2 x^{3 n} a c e +2 x^{2 n} a c d +x^{n} a^{2} e +a^{2} d}d x \] Input:

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

Output:

int(1/(x**(5*n)*c**2*e + x**(4*n)*c**2*d + 2*x**(3*n)*a*c*e + 2*x**(2*n)*a 
*c*d + x**n*a**2*e + a**2*d),x)