\(\int \frac {x^{-2 n} (d+e x^n)}{a+b x^n+c x^{2 n}} \, dx\) [154]

Optimal result

Integrand size = 29, antiderivative size = 184 \[ \int \frac {x^{-2 n} \left (d+e x^n\right )}{a+b x^n+c x^{2 n}} \, dx=\frac {\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) x^{1-2 n} \operatorname {Hypergeometric2F1}\left (1,-2+\frac {1}{n},-1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) (1-2 n)}+\frac {\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) x^{1-2 n} \operatorname {Hypergeometric2F1}\left (1,-2+\frac {1}{n},-1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) (1-2 n)} \] Output:

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1254\) vs. \(2(184)=368\).

Time = 6.63 (sec) , antiderivative size = 1254, normalized size of antiderivative = 6.82 \[ \int \frac {x^{-2 n} \left (d+e x^n\right )}{a+b x^n+c x^{2 n}} \, dx =\text {Too large to display} \] Input:

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

Output:

(x*(b^2*d - a*c*d - a*b*e + (a^2*d)/((1 - 2*n)*x^(2*n)) + (a^2*e)/((1 - n) 
*x^n) + (a*b*d)/((-1 + n)*x^n) - (2^(1 + n^(-1))*a*c*(-(b^2*d) + a*c*d + a 
*b*e)*x^n*(a/(2*a + (b - Sqrt[b^2 - 4*a*c])*x^n))^n^(-1)*Hypergeometric2F1 
[1 + n^(-1), 1 + n^(-1), 2 + n^(-1), ((-b + Sqrt[b^2 - 4*a*c])*x^n)/(-2*a 
+ (-b + Sqrt[b^2 - 4*a*c])*x^n)])/(Sqrt[b^2 - 4*a*c]*(1 + n)*(-2*a + (-b + 
 Sqrt[b^2 - 4*a*c])*x^n)) - (2^(1 + n^(-1))*c*(-(b^2*d) + a*c*d + a*b*e)*x 
^n*(a/(2*a + (b + Sqrt[b^2 - 4*a*c])*x^n))^(1 + n^(-1))*Hypergeometric2F1[ 
1 + n^(-1), 1 + n^(-1), 2 + n^(-1), ((b + Sqrt[b^2 - 4*a*c])*x^n)/(2*a + ( 
b + Sqrt[b^2 - 4*a*c])*x^n)])/(Sqrt[b^2 - 4*a*c]*(1 + n)) - (2^n^(-1)*b^3* 
d*(a/(2*a + (b - Sqrt[b^2 - 4*a*c])*x^n))^n^(-1)*Hypergeometric2F1[n^(-1), 
 n^(-1), 1 + n^(-1), ((-b + Sqrt[b^2 - 4*a*c])*x^n)/(-2*a + (-b + Sqrt[b^2 
 - 4*a*c])*x^n)])/Sqrt[b^2 - 4*a*c] + (2^(1 + n^(-1))*a*b*c*d*(a/(2*a + (b 
 - Sqrt[b^2 - 4*a*c])*x^n))^n^(-1)*Hypergeometric2F1[n^(-1), n^(-1), 1 + n 
^(-1), ((-b + Sqrt[b^2 - 4*a*c])*x^n)/(-2*a + (-b + Sqrt[b^2 - 4*a*c])*x^n 
)])/Sqrt[b^2 - 4*a*c] + (2^n^(-1)*a*b^2*e*(a/(2*a + (b - Sqrt[b^2 - 4*a*c] 
)*x^n))^n^(-1)*Hypergeometric2F1[n^(-1), n^(-1), 1 + n^(-1), ((-b + Sqrt[b 
^2 - 4*a*c])*x^n)/(-2*a + (-b + Sqrt[b^2 - 4*a*c])*x^n)])/Sqrt[b^2 - 4*a*c 
] - (2^n^(-1)*a^2*c*e*(a/(2*a + (b - Sqrt[b^2 - 4*a*c])*x^n))^n^(-1)*Hyper 
geometric2F1[n^(-1), n^(-1), 1 + n^(-1), ((-b + Sqrt[b^2 - 4*a*c])*x^n)/(- 
2*a + (-b + Sqrt[b^2 - 4*a*c])*x^n)])/Sqrt[b^2 - 4*a*c] + (2^n^(-1)*b^3...
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {1884, 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.

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

\(\Big \downarrow \) 1884

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 1884

Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*( 
(d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d 
+ e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, 
 n, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] &&  !RationalQ[n] && ( 
IGtQ[p, 0] || IGtQ[q, 0])
 

rule 2009

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-2)]

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

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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