\(\int \frac {x^{2 n} (d+e x^n)}{a+b x^n+c x^{2 n}} \, dx\) [150]

Optimal result

Integrand size = 29, antiderivative size = 184 \[ \int \frac {x^{2 n} \left (d+e x^n\right )}{a+b x^n+c x^{2 n}} \, dx=\frac {\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) x^{1+2 n} \operatorname {Hypergeometric2F1}\left (1,2+\frac {1}{n},3+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) (1+2 n)}+\frac {\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) x^{1+2 n} \operatorname {Hypergeometric2F1}\left (1,2+\frac {1}{n},3+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) (1+2 n)} \] Output:

(e+(-b*e+2*c*d)/(-4*a*c+b^2)^(1/2))*x^(1+2*n)*hypergeom([1, 2+1/n],[3+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],[3+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 [A] (warning: unable to verify)

Time = 2.14 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.84 \[ \int \frac {x^{2 n} \left (d+e x^n\right )}{a+b x^n+c x^{2 n}} \, dx=-\frac {2^{-\frac {1+n}{n}} x \left (-\frac {2^{1+\frac {1}{n}} c \sqrt {b^2-4 a c} e x^n}{1+n}+\left (b c d-c \sqrt {b^2-4 a c} d-b^2 e+2 a c e+b \sqrt {b^2-4 a c} e\right ) \left (\frac {c x^n}{b-\sqrt {b^2-4 a c}+2 c x^n}\right )^{-1/n} \operatorname {Hypergeometric2F1}\left (-\frac {1}{n},-\frac {1}{n},\frac {-1+n}{n},\frac {b-\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}+2 c x^n}\right )+\left (-b c d-c \sqrt {b^2-4 a c} d+b^2 e-2 a c e+b \sqrt {b^2-4 a c} e\right ) \left (\frac {c x^n}{b+\sqrt {b^2-4 a c}+2 c x^n}\right )^{-1/n} \operatorname {Hypergeometric2F1}\left (-\frac {1}{n},-\frac {1}{n},\frac {-1+n}{n},\frac {b+\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}+2 c x^n}\right )\right )}{c^2 \sqrt {b^2-4 a c}} \] Input:

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

Output:

-((x*(-((2^(1 + n^(-1))*c*Sqrt[b^2 - 4*a*c]*e*x^n)/(1 + n)) + ((b*c*d - c* 
Sqrt[b^2 - 4*a*c]*d - b^2*e + 2*a*c*e + b*Sqrt[b^2 - 4*a*c]*e)*Hypergeomet 
ric2F1[-n^(-1), -n^(-1), (-1 + n)/n, (b - Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 
 - 4*a*c] + 2*c*x^n)])/((c*x^n)/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1) 
+ ((-(b*c*d) - c*Sqrt[b^2 - 4*a*c]*d + b^2*e - 2*a*c*e + b*Sqrt[b^2 - 4*a* 
c]*e)*Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, (b + Sqrt[b^2 - 4*a* 
c])/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)])/((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 
2*c*x^n))^n^(-1)))/(2^((1 + n)/n)*c^2*Sqrt[b^2 - 4*a*c]))
 

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

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

Input:

Int[(x^(2*n)*(d + e*x^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), 3 + 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), 3 + 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(x**n*b*e*x + int(x**(2*n)/(x**(2*n)*c + x**n*b + a),x)*a*c*e*n + int(x**( 
2*n)/(x**(2*n)*c + x**n*b + a),x)*a*c*e - int(x**(2*n)/(x**(2*n)*c + x**n* 
b + a),x)*b**2*e*n - int(x**(2*n)/(x**(2*n)*c + x**n*b + a),x)*b**2*e + in 
t(x**(2*n)/(x**(2*n)*c + x**n*b + a),x)*b*c*d*n + int(x**(2*n)/(x**(2*n)*c 
 + x**n*b + a),x)*b*c*d + int(1/(x**(2*n)*c + x**n*b + a),x)*a**2*e*n + in 
t(1/(x**(2*n)*c + x**n*b + a),x)*a**2*e - a*e*n*x - a*e*x)/(b*c*(n + 1))