3.2477 \(\int \frac{x}{(a+b x^n)^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0060406, 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} \[ \frac{x^2 \, _2F_1\left (2,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a^2} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x^n)^2,x]

[Out]

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

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

Mathematica [A]  time = 0.0030431, size = 33, normalized size = 1. \[ \frac{x^2 \, _2F_1\left (2,\frac{2}{n};1+\frac{2}{n};-\frac{b x^n}{a}\right )}{2 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x^n)^2,x]

[Out]

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

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{ \left ( a+b{x}^{n} \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a+b*x^n)^2,x)

[Out]

int(x/(a+b*x^n)^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*x^n)^2,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x}{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*x^n)^2,x, algorithm="fricas")

[Out]

integral(x/(b^2*x^(2*n) + 2*a*b*x^n + a^2), x)

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Sympy [C]  time = 1.32171, size = 274, normalized size = 8.3 \begin{align*} \frac{2 n x^{2} \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{2}{n}\right ) \Gamma \left (\frac{2}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac{2}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac{2}{n}\right )\right )} + \frac{2 n x^{2} \Gamma \left (\frac{2}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac{2}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac{2}{n}\right )\right )} - \frac{4 x^{2} \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{2}{n}\right ) \Gamma \left (\frac{2}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac{2}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac{2}{n}\right )\right )} + \frac{2 b n x^{2} x^{n} \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{2}{n}\right ) \Gamma \left (\frac{2}{n}\right )}{a^{2} \left (a n^{3} \Gamma \left (1 + \frac{2}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac{2}{n}\right )\right )} - \frac{4 b x^{2} x^{n} \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{2}{n}\right ) \Gamma \left (\frac{2}{n}\right )}{a^{2} \left (a n^{3} \Gamma \left (1 + \frac{2}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac{2}{n}\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*x**n)**2,x)

[Out]

2*n*x**2*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 2/n)*gamma(2/n)/(a*(a*n**3*gamma(1 + 2/n) + b*n**3*x**n*gamma(1
+ 2/n))) + 2*n*x**2*gamma(2/n)/(a*(a*n**3*gamma(1 + 2/n) + b*n**3*x**n*gamma(1 + 2/n))) - 4*x**2*lerchphi(b*x
**n*exp_polar(I*pi)/a, 1, 2/n)*gamma(2/n)/(a*(a*n**3*gamma(1 + 2/n) + b*n**3*x**n*gamma(1 + 2/n))) + 2*b*n*x**
2*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 2/n)*gamma(2/n)/(a**2*(a*n**3*gamma(1 + 2/n) + b*n**3*x**n*gamma(
1 + 2/n))) - 4*b*x**2*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 2/n)*gamma(2/n)/(a**2*(a*n**3*gamma(1 + 2/n)
+ b*n**3*x**n*gamma(1 + 2/n)))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*x^n)^2,x, algorithm="giac")

[Out]

integrate(x/(b*x^n + a)^2, x)