\(\int \frac {(c x)^n}{a+b x^n} \, dx\) [2776]

Optimal result

Integrand size = 15, antiderivative size = 40 \[ \int \frac {(c x)^n}{a+b x^n} \, dx=\frac {(c x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+\frac {1}{n},2+\frac {1}{n},-\frac {b x^n}{a}\right )}{a c (1+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {371} \[ \int \frac {(c x)^n}{a+b x^n} \, dx=\frac {(c x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,1+\frac {1}{n},2+\frac {1}{n},-\frac {b x^n}{a}\right )}{a c (n+1)} \]

[In]

Int[(c*x)^n/(a + b*x^n),x]

[Out]

((c*x)^(1 + n)*Hypergeometric2F1[1, 1 + n^(-1), 2 + n^(-1), -((b*x^n)/a)])/(a*c*(1 + n))

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])

Rubi steps \begin{align*} \text {integral}& = \frac {(c x)^{1+n} \, _2F_1\left (1,1+\frac {1}{n};2+\frac {1}{n};-\frac {b x^n}{a}\right )}{a c (1+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.90 \[ \int \frac {(c x)^n}{a+b x^n} \, dx=\frac {x (c x)^n \operatorname {Hypergeometric2F1}\left (1,1+\frac {1}{n},2+\frac {1}{n},-\frac {b x^n}{a}\right )}{a (1+n)} \]

[In]

Integrate[(c*x)^n/(a + b*x^n),x]

[Out]

(x*(c*x)^n*Hypergeometric2F1[1, 1 + n^(-1), 2 + n^(-1), -((b*x^n)/a)])/(a*(1 + n))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((c*x)^n/(b*x^n + a), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.53 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.65 \[ \int \frac {(c x)^n}{a+b x^n} \, dx=- \frac {a^{- \frac {1}{n}} a^{1 + \frac {1}{n}} b^{\frac {1}{n}} b^{-1 - \frac {1}{n}} c^{n} x \Phi \left (\frac {a x^{- n} e^{i \pi }}{b}, 1, \frac {e^{i \pi }}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a n^{2} \Gamma \left (1 + \frac {1}{n}\right )} \]

[In]

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

[Out]

-a**(1 + 1/n)*b**(1/n)*b**(-1 - 1/n)*c**n*x*lerchphi(a*exp_polar(I*pi)/(b*x**n), 1, exp_polar(I*pi)/n)*gamma(1
/n)/(a*a**(1/n)*n**2*gamma(1 + 1/n))

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate((c*x)^n/(b*x^n + a), x)

Mupad [F(-1)]

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

[In]

int((c*x)^n/(a + b*x^n),x)

[Out]

int((c*x)^n/(a + b*x^n), x)