3.2472 \(\int \frac{x}{a+b x^n} \, dx\)

Optimal. Leaf size=33 \[ \frac{x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.0077799, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {364} \[ \frac{x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x^n),x]

[Out]

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

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{x}{a+b x^n} \, dx &=\frac{x^2 \, _2F_1\left (1,\frac{2}{n};\frac{2+n}{n};-\frac{b x^n}{a}\right )}{2 a}\\ \end{align*}

Mathematica [A]  time = 0.002836, size = 33, normalized size = 1. \[ \frac{x^2 \, _2F_1\left (1,\frac{2}{n};1+\frac{2}{n};-\frac{b x^n}{a}\right )}{2 a} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x^n),x]

[Out]

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

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Maple [F]  time = 0.048, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{a+b{x}^{n}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/(b*x^n + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x/(b*x^n + a), x)

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Sympy [C]  time = 0.920911, size = 36, normalized size = 1.09 \begin{align*} \frac{2 x^{2} \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{2}{n}\right ) \Gamma \left (\frac{2}{n}\right )}{a n^{2} \Gamma \left (1 + \frac{2}{n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/(b*x^n + a), x)