Integrand size = 19, antiderivative size = 83 \[ \int \frac {d+e x^n}{a+c x^{2 n}} \, dx=\frac {d x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a}+\frac {e x^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a (1+n)} \] Output:
d*x*hypergeom([1, 1/2/n],[1+1/2/n],-c*x^(2*n)/a)/a+e*x^(1+n)*hypergeom([1, 1/2*(1+n)/n],[3/2+1/2/n],-c*x^(2*n)/a)/a/(1+n)
Time = 0.02 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00 \[ \int \frac {d+e x^n}{a+c x^{2 n}} \, dx=\frac {d x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a}+\frac {e x^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a (1+n)} \] Input:
Integrate[(d + e*x^n)/(a + c*x^(2*n)),x]
Output:
(d*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/a + (e*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2 *n))/a)])/(a*(1 + n))
Time = 0.22 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {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.
\(\displaystyle \int \frac {d+e x^n}{a+c x^{2 n}} \, dx\) |
\(\Big \downarrow \) 1748 |
\(\displaystyle d \int \frac {1}{c x^{2 n}+a}dx+e \int \frac {x^n}{c x^{2 n}+a}dx\) |
\(\Big \downarrow \) 778 |
\(\displaystyle e \int \frac {x^n}{c x^{2 n}+a}dx+\frac {d x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {d x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a}+\frac {e x^{n+1} \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a (n+1)}\) |
Input:
Int[(d + e*x^n)/(a + c*x^(2*n)),x]
Output:
(d*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/a + (e*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2 *n))/a)])/(a*(1 + n))
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])
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])
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])
\[\int \frac {d +e \,x^{n}}{a +c \,x^{2 n}}d x\]
Input:
int((d+e*x^n)/(a+c*x^(2*n)),x)
Output:
int((d+e*x^n)/(a+c*x^(2*n)),x)
\[ \int \frac {d+e x^n}{a+c x^{2 n}} \, dx=\int { \frac {e x^{n} + d}{c x^{2 \, n} + a} \,d x } \] Input:
integrate((d+e*x^n)/(a+c*x^(2*n)),x, algorithm="fricas")
Output:
integral((e*x^n + d)/(c*x^(2*n) + a), x)
Result contains complex when optimal does not.
Time = 2.22 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.52 \[ \int \frac {d+e x^n}{a+c x^{2 n}} \, dx=\frac {a^{\frac {1}{2 n}} a^{-1 - \frac {1}{2 n}} d x \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2 n}\right )}{4 n^{2} \Gamma \left (1 + \frac {1}{2 n}\right )} + \frac {a^{- \frac {3}{2} - \frac {1}{2 n}} a^{\frac {1}{2} + \frac {1}{2 n}} e x^{n + 1} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2} + \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2} + \frac {1}{2 n}\right )}{4 n \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )} + \frac {a^{- \frac {3}{2} - \frac {1}{2 n}} a^{\frac {1}{2} + \frac {1}{2 n}} e x^{n + 1} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2} + \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2} + \frac {1}{2 n}\right )}{4 n^{2} \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )} \] Input:
integrate((d+e*x**n)/(a+c*x**(2*n)),x)
Output:
a**(1/(2*n))*a**(-1 - 1/(2*n))*d*x*lerchphi(c*x**(2*n)*exp_polar(I*pi)/a, 1, 1/(2*n))*gamma(1/(2*n))/(4*n**2*gamma(1 + 1/(2*n))) + a**(-3/2 - 1/(2*n ))*a**(1/2 + 1/(2*n))*e*x**(n + 1)*lerchphi(c*x**(2*n)*exp_polar(I*pi)/a, 1, 1/2 + 1/(2*n))*gamma(1/2 + 1/(2*n))/(4*n*gamma(3/2 + 1/(2*n))) + a**(-3 /2 - 1/(2*n))*a**(1/2 + 1/(2*n))*e*x**(n + 1)*lerchphi(c*x**(2*n)*exp_pola r(I*pi)/a, 1, 1/2 + 1/(2*n))*gamma(1/2 + 1/(2*n))/(4*n**2*gamma(3/2 + 1/(2 *n)))
\[ \int \frac {d+e x^n}{a+c x^{2 n}} \, dx=\int { \frac {e x^{n} + d}{c x^{2 \, n} + a} \,d x } \] Input:
integrate((d+e*x^n)/(a+c*x^(2*n)),x, algorithm="maxima")
Output:
integrate((e*x^n + d)/(c*x^(2*n) + a), x)
\[ \int \frac {d+e x^n}{a+c x^{2 n}} \, dx=\int { \frac {e x^{n} + d}{c x^{2 \, n} + a} \,d x } \] Input:
integrate((d+e*x^n)/(a+c*x^(2*n)),x, algorithm="giac")
Output:
integrate((e*x^n + d)/(c*x^(2*n) + a), x)
Timed out. \[ \int \frac {d+e x^n}{a+c x^{2 n}} \, dx=\int \frac {d+e\,x^n}{a+c\,x^{2\,n}} \,d x \] Input:
int((d + e*x^n)/(a + c*x^(2*n)),x)
Output:
int((d + e*x^n)/(a + c*x^(2*n)), x)
\[ \int \frac {d+e x^n}{a+c x^{2 n}} \, dx=\left (\int \frac {x^{n}}{x^{2 n} c +a}d x \right ) e +\left (\int \frac {1}{x^{2 n} c +a}d x \right ) d \] Input:
int((d+e*x^n)/(a+c*x^(2*n)),x)
Output:
int(x**n/(x**(2*n)*c + a),x)*e + int(1/(x**(2*n)*c + a),x)*d