\(\int \frac {x^{-3 n} (d+e x^n)}{a+b x^n+c x^{2 n}} \, dx\) [155]

Optimal result

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

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(3267\) vs. \(2(184)=368\).

Time = 6.72 (sec) , antiderivative size = 3267, normalized size of antiderivative = 17.76 \[ \int \frac {x^{-3 n} \left (d+e x^n\right )}{a+b x^n+c x^{2 n}} \, dx=\text {Result too large to show} \] Input:

Integrate[(d + e*x^n)/(x^(3*n)*(a + b*x^n + c*x^(2*n))),x]
 

Output:

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

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.

\(\displaystyle \int \frac {x^{-3 n} \left (d+e x^n\right )}{a+b x^n+c x^{2 n}} \, dx\)

\(\Big \downarrow \) 1884

\(\displaystyle \int \left (\frac {x^{-3 n} \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right )}{-\sqrt {b^2-4 a c}+b+2 c x^n}+\frac {x^{-3 n} \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}+b+2 c x^n}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

integrate((d+e*x^n)/(x^(3*n))/(a+b*x^n+c*x^(2*n)),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((d+e*x^n)/(x^(3*n))/(a+b*x^n+c*x^(2*n)),x, algorithm="maxima")
 

Output:

-((2*n^2 - 3*n + 1)*a^2*d*x/x^(3*n) - ((3*n^2 - 4*n + 1)*a*b*d - (3*n^2 - 
4*n + 1)*a^2*e)*x/x^(2*n) + ((6*n^2 - 5*n + 1)*b^2*d - ((6*n^2 - 5*n + 1)* 
c*d + (6*n^2 - 5*n + 1)*b*e)*a)*x/x^n)/((6*n^3 - 11*n^2 + 6*n - 1)*a^3) + 
integrate(-(b^3*d + a^2*c*e - (2*b*c*d + b^2*e)*a + (b^2*c*d - (c^2*d + b* 
c*e)*a)*x^n)/(a^3*c*x^(2*n) + a^3*b*x^n + a^4), x)
 

Giac [F]

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

integrate((d+e*x^n)/(x^(3*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^(3*n)), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^{-3 n} \left (d+e x^n\right )}{a+b x^n+c x^{2 n}} \, dx=\frac {-6 x^{3 n} \left (\int \frac {1}{x^{4 n} c +x^{3 n} b +x^{2 n} a}d x \right ) b d \,n^{2}+5 x^{3 n} \left (\int \frac {1}{x^{4 n} c +x^{3 n} b +x^{2 n} a}d x \right ) b d n -x^{3 n} \left (\int \frac {1}{x^{4 n} c +x^{3 n} b +x^{2 n} a}d x \right ) b d -6 x^{3 n} \left (\int \frac {1}{x^{3 n} c +x^{2 n} b +x^{n} a}d x \right ) b e \,n^{2}+5 x^{3 n} \left (\int \frac {1}{x^{3 n} c +x^{2 n} b +x^{n} a}d x \right ) b e n -x^{3 n} \left (\int \frac {1}{x^{3 n} c +x^{2 n} b +x^{n} a}d x \right ) b e -6 x^{3 n} \left (\int \frac {1}{x^{3 n} c +x^{2 n} b +x^{n} a}d x \right ) c d \,n^{2}+5 x^{3 n} \left (\int \frac {1}{x^{3 n} c +x^{2 n} b +x^{n} a}d x \right ) c d n -x^{3 n} \left (\int \frac {1}{x^{3 n} c +x^{2 n} b +x^{n} a}d x \right ) c d -6 x^{3 n} \left (\int \frac {1}{x^{2 n} c +x^{n} b +a}d x \right ) c e \,n^{2}+5 x^{3 n} \left (\int \frac {1}{x^{2 n} c +x^{n} b +a}d x \right ) c e n -x^{3 n} \left (\int \frac {1}{x^{2 n} c +x^{n} b +a}d x \right ) c e -3 x^{n} e n x +x^{n} e x -2 d n x +d x}{x^{3 n} a \left (6 n^{2}-5 n +1\right )} \] Input:

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

Output:

( - 6*x**(3*n)*int(1/(x**(4*n)*c + x**(3*n)*b + x**(2*n)*a),x)*b*d*n**2 + 
5*x**(3*n)*int(1/(x**(4*n)*c + x**(3*n)*b + x**(2*n)*a),x)*b*d*n - x**(3*n 
)*int(1/(x**(4*n)*c + x**(3*n)*b + x**(2*n)*a),x)*b*d - 6*x**(3*n)*int(1/( 
x**(3*n)*c + x**(2*n)*b + x**n*a),x)*b*e*n**2 + 5*x**(3*n)*int(1/(x**(3*n) 
*c + x**(2*n)*b + x**n*a),x)*b*e*n - x**(3*n)*int(1/(x**(3*n)*c + x**(2*n) 
*b + x**n*a),x)*b*e - 6*x**(3*n)*int(1/(x**(3*n)*c + x**(2*n)*b + x**n*a), 
x)*c*d*n**2 + 5*x**(3*n)*int(1/(x**(3*n)*c + x**(2*n)*b + x**n*a),x)*c*d*n 
 - x**(3*n)*int(1/(x**(3*n)*c + x**(2*n)*b + x**n*a),x)*c*d - 6*x**(3*n)*i 
nt(1/(x**(2*n)*c + x**n*b + a),x)*c*e*n**2 + 5*x**(3*n)*int(1/(x**(2*n)*c 
+ x**n*b + a),x)*c*e*n - x**(3*n)*int(1/(x**(2*n)*c + x**n*b + a),x)*c*e - 
 3*x**n*e*n*x + x**n*e*x - 2*d*n*x + d*x)/(x**(3*n)*a*(6*n**2 - 5*n + 1))