\(\int \frac {(c x)^{4+n}}{a+b x^n} \, dx\) [2772]

Optimal result

Integrand size = 17, antiderivative size = 44 \[ \int \frac {(c x)^{4+n}}{a+b x^n} \, dx=\frac {(c x)^{5+n} \operatorname {Hypergeometric2F1}\left (1,\frac {5+n}{n},2+\frac {5}{n},-\frac {b x^n}{a}\right )}{a c (5+n)} \]

[Out]

(c*x)^(5+n)*hypergeom([1, (5+n)/n],[2+5/n],-b*x^n/a)/a/c/(5+n)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {371} \[ \int \frac {(c x)^{4+n}}{a+b x^n} \, dx=\frac {(c x)^{n+5} \operatorname {Hypergeometric2F1}\left (1,\frac {n+5}{n},2+\frac {5}{n},-\frac {b x^n}{a}\right )}{a c (n+5)} \]

[In]

Int[(c*x)^(4 + n)/(a + b*x^n),x]

[Out]

((c*x)^(5 + n)*Hypergeometric2F1[1, (5 + n)/n, 2 + 5/n, -((b*x^n)/a)])/(a*c*(5 + 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)^{5+n} \, _2F_1\left (1,\frac {5+n}{n};2+\frac {5}{n};-\frac {b x^n}{a}\right )}{a c (5+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \frac {(c x)^{4+n}}{a+b x^n} \, dx=\frac {x (c x)^{4+n} \operatorname {Hypergeometric2F1}\left (1,\frac {5+n}{n},1+\frac {5+n}{n},-\frac {b x^n}{a}\right )}{a (5+n)} \]

[In]

Integrate[(c*x)^(4 + n)/(a + b*x^n),x]

[Out]

(x*(c*x)^(4 + n)*Hypergeometric2F1[1, (5 + n)/n, 1 + (5 + n)/n, -((b*x^n)/a)])/(a*(5 + n))

Maple [F]

\[\int \frac {\left (c x \right )^{4+n}}{a +b \,x^{n}}d x\]

[In]

int((c*x)^(4+n)/(a+b*x^n),x)

[Out]

int((c*x)^(4+n)/(a+b*x^n),x)

Fricas [F]

\[ \int \frac {(c x)^{4+n}}{a+b x^n} \, dx=\int { \frac {\left (c x\right )^{n + 4}}{b x^{n} + a} \,d x } \]

[In]

integrate((c*x)^(4+n)/(a+b*x^n),x, algorithm="fricas")

[Out]

integral((c*x)^(n + 4)/(b*x^n + a), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.56 (sec) , antiderivative size = 117, normalized size of antiderivative = 2.66 \[ \int \frac {(c x)^{4+n}}{a+b x^n} \, dx=\frac {a^{-2 - \frac {5}{n}} a^{1 + \frac {5}{n}} c^{n + 4} x^{n + 5} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, 1 + \frac {5}{n}\right ) \Gamma \left (1 + \frac {5}{n}\right )}{n \Gamma \left (2 + \frac {5}{n}\right )} + \frac {5 a^{-2 - \frac {5}{n}} a^{1 + \frac {5}{n}} c^{n + 4} x^{n + 5} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, 1 + \frac {5}{n}\right ) \Gamma \left (1 + \frac {5}{n}\right )}{n^{2} \Gamma \left (2 + \frac {5}{n}\right )} \]

[In]

integrate((c*x)**(4+n)/(a+b*x**n),x)

[Out]

a**(-2 - 5/n)*a**(1 + 5/n)*c**(n + 4)*x**(n + 5)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1 + 5/n)*gamma(1 + 5/n)
/(n*gamma(2 + 5/n)) + 5*a**(-2 - 5/n)*a**(1 + 5/n)*c**(n + 4)*x**(n + 5)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1,
1 + 5/n)*gamma(1 + 5/n)/(n**2*gamma(2 + 5/n))

Maxima [F]

\[ \int \frac {(c x)^{4+n}}{a+b x^n} \, dx=\int { \frac {\left (c x\right )^{n + 4}}{b x^{n} + a} \,d x } \]

[In]

integrate((c*x)^(4+n)/(a+b*x^n),x, algorithm="maxima")

[Out]

1/5*c^(n + 4)*x^5/b - a*c^(n + 4)*integrate(x^4/(b^2*x^n + a*b), x)

Giac [F]

\[ \int \frac {(c x)^{4+n}}{a+b x^n} \, dx=\int { \frac {\left (c x\right )^{n + 4}}{b x^{n} + a} \,d x } \]

[In]

integrate((c*x)^(4+n)/(a+b*x^n),x, algorithm="giac")

[Out]

integrate((c*x)^(n + 4)/(b*x^n + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(c x)^{4+n}}{a+b x^n} \, dx=\int \frac {{\left (c\,x\right )}^{n+4}}{a+b\,x^n} \,d x \]

[In]

int((c*x)^(n + 4)/(a + b*x^n),x)

[Out]

int((c*x)^(n + 4)/(a + b*x^n), x)