\(\int \frac {d+e x^n}{a-c x^{2 n}} \, dx\) [47]

Optimal result

Integrand size = 20, antiderivative size = 81 \[ \int \frac {d+e x^n}{a-c x^{2 n}} \, dx=\frac {d x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),\frac {c x^{2 n}}{a}\right )}{a}+\frac {e x^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),\frac {c x^{2 n}}{a}\right )}{a (1+n)} \]

[Out]

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)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1432, 251, 371} \[ \int \frac {d+e x^n}{a-c x^{2 n}} \, dx=\frac {d x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),\frac {c x^{2 n}}{a}\right )}{a}+\frac {e x^{n+1} \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),\frac {c x^{2 n}}{a}\right )}{a (n+1)} \]

[In]

Int[(d + e*x^n)/(a - c*x^(2*n)),x]

[Out]

(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))

Rule 251

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-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])

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 1432

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> Dist[d, Int[1/(a + c*x^(2*n)), x], x] + D
ist[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])

Rubi steps \begin{align*} \text {integral}& = d \int \frac {1}{a-c x^{2 n}} \, dx+e \int \frac {x^n}{a-c x^{2 n}} \, dx \\ & = \frac {d x \, _2F_1\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} \, _2F_1\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)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00 \[ \int \frac {d+e x^n}{a-c x^{2 n}} \, dx=\frac {d x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),\frac {c x^{2 n}}{a}\right )}{a}+\frac {e x^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),\frac {c x^{2 n}}{a}\right )}{a (1+n)} \]

[In]

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

[Out]

(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))

Maple [F]

\[\int \frac {d +e \,x^{n}}{a -c \,x^{2 n}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {d+e x^n}{a-c x^{2 n}} \, dx=\int { -\frac {e x^{n} + d}{c x^{2 \, n} - a} \,d x } \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.25 (sec) , antiderivative size = 214, normalized size of antiderivative = 2.64 \[ \int \frac {d+e x^n}{a-c x^{2 n}} \, dx=\frac {a^{\frac {1}{2 n}} a^{-1 - \frac {1}{2 n}} d x \Phi \left (\frac {c x^{2 n} e^{2 i \pi }}{a}, 1, \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2 n}\right )}{4 n^{2} \Gamma \left (1 + \frac {1}{2 n}\right )} + \frac {a^{- \frac {3}{2} - \frac {1}{2 n}} a^{\frac {1}{2} + \frac {1}{2 n}} e x^{n + 1} \Phi \left (\frac {c x^{2 n} e^{2 i \pi }}{a}, 1, \frac {1}{2} + \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2} + \frac {1}{2 n}\right )}{4 n \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )} + \frac {a^{- \frac {3}{2} - \frac {1}{2 n}} a^{\frac {1}{2} + \frac {1}{2 n}} e x^{n + 1} \Phi \left (\frac {c x^{2 n} e^{2 i \pi }}{a}, 1, \frac {1}{2} + \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2} + \frac {1}{2 n}\right )}{4 n^{2} \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )} \]

[In]

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

[Out]

a**(1/(2*n))*a**(-1 - 1/(2*n))*d*x*lerchphi(c*x**(2*n)*exp_polar(2*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(2*I*p
i)/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_polar(2*I*pi)/a, 1, 1/2 + 1/(2*n))*gamma(1/2 + 1/(2*n))/(4*n**2*gamma(
3/2 + 1/(2*n)))

Maxima [F]

\[ \int \frac {d+e x^n}{a-c x^{2 n}} \, dx=\int { -\frac {e x^{n} + d}{c x^{2 \, n} - a} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {d+e x^n}{a-c x^{2 n}} \, dx=\int { -\frac {e x^{n} + d}{c x^{2 \, n} - a} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {d+e x^n}{a-c x^{2 n}} \, dx=\int \frac {d+e\,x^n}{a-c\,x^{2\,n}} \,d x \]

[In]

int((d + e*x^n)/(a - c*x^(2*n)),x)

[Out]

int((d + e*x^n)/(a - c*x^(2*n)), x)