\(\int \frac {d+e x^n}{(a+c x^{2 n})^2} \, dx\) [34]

Optimal result

Integrand size = 19, antiderivative size = 134 \[ \int \frac {d+e x^n}{\left (a+c x^{2 n}\right )^2} \, dx=\frac {x \left (d+e x^n\right )}{2 a n \left (a+c x^{2 n}\right )}-\frac {d (1-2 n) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{2 a^2 n}-\frac {e (1-n) x^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{2 a^2 n (1+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.62 \[ \int \frac {d+e x^n}{\left (a+c x^{2 n}\right )^2} \, dx=\frac {d x \operatorname {Hypergeometric2F1}\left (2,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a^2}+\frac {e x^{1+n} \operatorname {Hypergeometric2F1}\left (2,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a^2 (1+n)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {1761, 1748, 778, 888}

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.

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

\(\Big \downarrow \) 1761

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

\(\Big \downarrow \) 1748

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

\(\Big \downarrow \) 778

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

\(\Big \downarrow \) 888

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 778

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 
1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p 
, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || 
GtQ[a, 0])
 

rule 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 1748

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> Simp[d 
   Int[1/(a + c*x^(2*n)), x], x] + Simp[e   Int[x^n/(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] && ( 
PosQ[a*c] ||  !IntegerQ[n])
 

rule 1761

Int[((d_) + (e_.)*(x_)^(n_))*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> S 
imp[(-x)*(d + e*x^n)*((a + c*x^(2*n))^(p + 1)/(2*a*n*(p + 1))), x] + Simp[1 
/(2*a*n*(p + 1))   Int[(d*(2*n*p + 2*n + 1) + e*(2*n*p + 3*n + 1)*x^n)*(a + 
 c*x^(2*n))^(p + 1), x], x] /; FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && 
 ILtQ[p, -1]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 173.59 (sec) , antiderivative size = 994, normalized size of antiderivative = 7.42 \[ \int \frac {d+e x^n}{\left (a+c x^{2 n}\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

d*(2*a*a**(1/(2*n))*a**(-2 - 1/(2*n))*n*x*lerchphi(c*x**(2*n)*exp_polar(I* 
pi)/a, 1, 1/(2*n))*gamma(1/(2*n))/(8*a*n**3*gamma(1 + 1/(2*n)) + 8*c*n**3* 
x**(2*n)*gamma(1 + 1/(2*n))) + 2*a*a**(1/(2*n))*a**(-2 - 1/(2*n))*n*x*gamm 
a(1/(2*n))/(8*a*n**3*gamma(1 + 1/(2*n)) + 8*c*n**3*x**(2*n)*gamma(1 + 1/(2 
*n))) - a*a**(1/(2*n))*a**(-2 - 1/(2*n))*x*lerchphi(c*x**(2*n)*exp_polar(I 
*pi)/a, 1, 1/(2*n))*gamma(1/(2*n))/(8*a*n**3*gamma(1 + 1/(2*n)) + 8*c*n**3 
*x**(2*n)*gamma(1 + 1/(2*n))) + 2*a**(1/(2*n))*a**(-2 - 1/(2*n))*c*n*x*x** 
(2*n)*lerchphi(c*x**(2*n)*exp_polar(I*pi)/a, 1, 1/(2*n))*gamma(1/(2*n))/(8 
*a*n**3*gamma(1 + 1/(2*n)) + 8*c*n**3*x**(2*n)*gamma(1 + 1/(2*n))) - a**(1 
/(2*n))*a**(-2 - 1/(2*n))*c*x*x**(2*n)*lerchphi(c*x**(2*n)*exp_polar(I*pi) 
/a, 1, 1/(2*n))*gamma(1/(2*n))/(8*a*n**3*gamma(1 + 1/(2*n)) + 8*c*n**3*x** 
(2*n)*gamma(1 + 1/(2*n)))) + e*(a*a**(-5/2 - 1/(2*n))*a**(1/2 + 1/(2*n))*n 
**2*x**(n + 1)*lerchphi(c*x**(2*n)*exp_polar(I*pi)/a, 1, 1/2 + 1/(2*n))*ga 
mma(1/2 + 1/(2*n))/(8*a*n**3*gamma(3/2 + 1/(2*n)) + 8*c*n**3*x**(2*n)*gamm 
a(3/2 + 1/(2*n))) + 2*a*a**(-5/2 - 1/(2*n))*a**(1/2 + 1/(2*n))*n**2*x**(n 
+ 1)*gamma(1/2 + 1/(2*n))/(8*a*n**3*gamma(3/2 + 1/(2*n)) + 8*c*n**3*x**(2* 
n)*gamma(3/2 + 1/(2*n))) + 2*a*a**(-5/2 - 1/(2*n))*a**(1/2 + 1/(2*n))*n*x* 
*(n + 1)*gamma(1/2 + 1/(2*n))/(8*a*n**3*gamma(3/2 + 1/(2*n)) + 8*c*n**3*x* 
*(2*n)*gamma(3/2 + 1/(2*n))) - a*a**(-5/2 - 1/(2*n))*a**(1/2 + 1/(2*n))*x* 
*(n + 1)*lerchphi(c*x**(2*n)*exp_polar(I*pi)/a, 1, 1/2 + 1/(2*n))*gamma...
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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