3.2776 \(\int \frac{(c x)^n}{a+b x^n} \, dx\)

Optimal. Leaf size=40 \[ \frac{(c x)^{n+1} \, _2F_1\left (1,1+\frac{1}{n};2+\frac{1}{n};-\frac{b x^n}{a}\right )}{a c (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.012494, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {364} \[ \frac{(c x)^{n+1} \, _2F_1\left (1,1+\frac{1}{n};2+\frac{1}{n};-\frac{b x^n}{a}\right )}{a c (n+1)} \]

Antiderivative was successfully verified.

[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 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \frac{(c x)^n}{a+b x^n} \, dx &=\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]  time = 0.0083094, size = 36, normalized size = 0.9 \[ \frac{x (c x)^n \, _2F_1\left (1,1+\frac{1}{n};2+\frac{1}{n};-\frac{b x^n}{a}\right )}{a (n+1)} \]

Antiderivative was successfully verified.

[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))

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Maple [F]  time = 0.067, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( cx \right ) ^{n}}{a+b{x}^{n}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -a c^{n} \int \frac{1}{b^{2} x^{n} + a b}\,{d x} + \frac{c^{n} x}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\left (c x\right )^{n}}{b x^{n} + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((c*x)^n/(b*x^n + a), x)

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Sympy [C]  time = 0.81903, size = 41, normalized size = 1.02 \begin{align*} - \frac{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 )}{b n^{2} \Gamma \left (1 + \frac{1}{n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x\right )^{n}}{b x^{n} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*x)^n/(b*x^n + a), x)