Integrand size = 22, antiderivative size = 96 \[ \int \frac {d+e x^n}{x^4 \left (a+c x^{2 n}\right )} \, dx=-\frac {d \operatorname {Hypergeometric2F1}\left (1,-\frac {3}{2 n},1-\frac {3}{2 n},-\frac {c x^{2 n}}{a}\right )}{3 a x^3}-\frac {e x^{-3+n} \operatorname {Hypergeometric2F1}\left (1,-\frac {3-n}{2 n},-\frac {3 (1-n)}{2 n},-\frac {c x^{2 n}}{a}\right )}{a (3-n)} \] Output:
-1/3*d*hypergeom([1, -3/2/n],[1-3/2/n],-c*x^(2*n)/a)/a/x^3-e*x^(-3+n)*hype rgeom([1, -1/2*(3-n)/n],[1/2*(-3+3*n)/n],-c*x^(2*n)/a)/a/(3-n)
Time = 0.16 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94 \[ \int \frac {d+e x^n}{x^4 \left (a+c x^{2 n}\right )} \, dx=\frac {-d (-3+n) \operatorname {Hypergeometric2F1}\left (1,-\frac {3}{2 n},1-\frac {3}{2 n},-\frac {c x^{2 n}}{a}\right )+3 e x^n \operatorname {Hypergeometric2F1}\left (1,\frac {-3+n}{2 n},\frac {3 (-1+n)}{2 n},-\frac {c x^{2 n}}{a}\right )}{3 a (-3+n) x^3} \] Input:
Integrate[(d + e*x^n)/(x^4*(a + c*x^(2*n))),x]
Output:
(-(d*(-3 + n)*Hypergeometric2F1[1, -3/(2*n), 1 - 3/(2*n), -((c*x^(2*n))/a) ]) + 3*e*x^n*Hypergeometric2F1[1, (-3 + n)/(2*n), (3*(-1 + n))/(2*n), -((c *x^(2*n))/a)])/(3*a*(-3 + n)*x^3)
Time = 0.26 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1885, 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.
\(\displaystyle \int \frac {d+e x^n}{x^4 \left (a+c x^{2 n}\right )} \, dx\) |
\(\Big \downarrow \) 1885 |
\(\displaystyle \int \left (\frac {d}{x^4 \left (a+c x^{2 n}\right )}+\frac {e x^{n-4}}{a+c x^{2 n}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {d \operatorname {Hypergeometric2F1}\left (1,-\frac {3}{2 n},1-\frac {3}{2 n},-\frac {c x^{2 n}}{a}\right )}{3 a x^3}-\frac {e x^{n-3} \operatorname {Hypergeometric2F1}\left (1,-\frac {3-n}{2 n},-\frac {3 (1-n)}{2 n},-\frac {c x^{2 n}}{a}\right )}{a (3-n)}\) |
Input:
Int[(d + e*x^n)/(x^4*(a + c*x^(2*n))),x]
Output:
-1/3*(d*Hypergeometric2F1[1, -3/(2*n), 1 - 3/(2*n), -((c*x^(2*n))/a)])/(a* x^3) - (e*x^(-3 + n)*Hypergeometric2F1[1, -1/2*(3 - n)/n, (-3*(1 - n))/(2* n), -((c*x^(2*n))/a)])/(a*(3 - n))
Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^ (n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q*(a + c* x^(2*n))^p, x], x] /; FreeQ[{a, c, d, e, f, m, n, p, q}, x] && EqQ[n2, 2*n] && !RationalQ[n] && (IGtQ[p, 0] || IGtQ[q, 0])
\[\int \frac {d +e \,x^{n}}{x^{4} \left (a +c \,x^{2 n}\right )}d x\]
Input:
int((d+e*x^n)/x^4/(a+c*x^(2*n)),x)
Output:
int((d+e*x^n)/x^4/(a+c*x^(2*n)),x)
\[ \int \frac {d+e x^n}{x^4 \left (a+c x^{2 n}\right )} \, dx=\int { \frac {e x^{n} + d}{{\left (c x^{2 \, n} + a\right )} x^{4}} \,d x } \] Input:
integrate((d+e*x^n)/x^4/(a+c*x^(2*n)),x, algorithm="fricas")
Output:
integral((e*x^n + d)/(c*x^4*x^(2*n) + a*x^4), x)
\[ \int \frac {d+e x^n}{x^4 \left (a+c x^{2 n}\right )} \, dx=\int \frac {d + e x^{n}}{x^{4} \left (a + c x^{2 n}\right )}\, dx \] Input:
integrate((d+e*x**n)/x**4/(a+c*x**(2*n)),x)
Output:
Integral((d + e*x**n)/(x**4*(a + c*x**(2*n))), x)
\[ \int \frac {d+e x^n}{x^4 \left (a+c x^{2 n}\right )} \, dx=\int { \frac {e x^{n} + d}{{\left (c x^{2 \, n} + a\right )} x^{4}} \,d x } \] Input:
integrate((d+e*x^n)/x^4/(a+c*x^(2*n)),x, algorithm="maxima")
Output:
integrate((e*x^n + d)/((c*x^(2*n) + a)*x^4), x)
\[ \int \frac {d+e x^n}{x^4 \left (a+c x^{2 n}\right )} \, dx=\int { \frac {e x^{n} + d}{{\left (c x^{2 \, n} + a\right )} x^{4}} \,d x } \] Input:
integrate((d+e*x^n)/x^4/(a+c*x^(2*n)),x, algorithm="giac")
Output:
integrate((e*x^n + d)/((c*x^(2*n) + a)*x^4), x)
Timed out. \[ \int \frac {d+e x^n}{x^4 \left (a+c x^{2 n}\right )} \, dx=\int \frac {d+e\,x^n}{x^4\,\left (a+c\,x^{2\,n}\right )} \,d x \] Input:
int((d + e*x^n)/(x^4*(a + c*x^(2*n))),x)
Output:
int((d + e*x^n)/(x^4*(a + c*x^(2*n))), x)
\[ \int \frac {d+e x^n}{x^4 \left (a+c x^{2 n}\right )} \, dx=\left (\int \frac {x^{n}}{x^{2 n} c \,x^{4}+a \,x^{4}}d x \right ) e +\left (\int \frac {1}{x^{2 n} c \,x^{4}+a \,x^{4}}d x \right ) d \] Input:
int((d+e*x^n)/x^4/(a+c*x^(2*n)),x)
Output:
int(x**n/(x**(2*n)*c*x**4 + a*x**4),x)*e + int(1/(x**(2*n)*c*x**4 + a*x**4 ),x)*d