\(\int \frac {x}{(a+b x^n)^2} \, dx\) [2477]

Optimal result

Integrand size = 11, antiderivative size = 33 \[ \int \frac {x}{\left (a+b x^n\right )^2} \, dx=\frac {x^2 \operatorname {Hypergeometric2F1}\left (2,\frac {2}{n},\frac {2+n}{n},-\frac {b x^n}{a}\right )}{2 a^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {371} \[ \int \frac {x}{\left (a+b x^n\right )^2} \, dx=\frac {x^2 \operatorname {Hypergeometric2F1}\left (2,\frac {2}{n},\frac {n+2}{n},-\frac {b x^n}{a}\right )}{2 a^2} \]

[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 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 {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] (verified)

Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\left (a+b x^n\right )^2} \, dx=\frac {x^2 \operatorname {Hypergeometric2F1}\left (2,\frac {2}{n},1+\frac {2}{n},-\frac {b x^n}{a}\right )}{2 a^2} \]

[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)

Maple [F]

\[\int \frac {x}{\left (a +b \,x^{n}\right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {x}{\left (a+b x^n\right )^2} \, dx=\int { \frac {x}{{\left (b x^{n} + a\right )}^{2}} \,d x } \]

[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)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.64 (sec) , antiderivative size = 335, normalized size of antiderivative = 10.15 \[ \int \frac {x}{\left (a+b x^n\right )^2} \, dx=\frac {2 a a^{\frac {2}{n}} a^{-2 - \frac {2}{n}} n x^{2} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {2}{n}\right ) \Gamma \left (\frac {2}{n}\right )}{a n^{3} \Gamma \left (1 + \frac {2}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {2}{n}\right )} + \frac {2 a a^{\frac {2}{n}} a^{-2 - \frac {2}{n}} n x^{2} \Gamma \left (\frac {2}{n}\right )}{a n^{3} \Gamma \left (1 + \frac {2}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {2}{n}\right )} - \frac {4 a a^{\frac {2}{n}} a^{-2 - \frac {2}{n}} x^{2} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {2}{n}\right ) \Gamma \left (\frac {2}{n}\right )}{a n^{3} \Gamma \left (1 + \frac {2}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {2}{n}\right )} + \frac {2 a^{\frac {2}{n}} a^{-2 - \frac {2}{n}} b n x^{2} x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {2}{n}\right ) \Gamma \left (\frac {2}{n}\right )}{a n^{3} \Gamma \left (1 + \frac {2}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {2}{n}\right )} - \frac {4 a^{\frac {2}{n}} a^{-2 - \frac {2}{n}} b x^{2} x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {2}{n}\right ) \Gamma \left (\frac {2}{n}\right )}{a n^{3} \Gamma \left (1 + \frac {2}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {2}{n}\right )} \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {x}{\left (a+b x^n\right )^2} \, dx=\int { \frac {x}{{\left (b x^{n} + a\right )}^{2}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {x}{\left (a+b x^n\right )^2} \, dx=\int { \frac {x}{{\left (b x^{n} + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

integrate(x/(b*x^n + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{\left (a+b x^n\right )^2} \, dx=\int \frac {x}{{\left (a+b\,x^n\right )}^2} \,d x \]

[In]

int(x/(a + b*x^n)^2,x)

[Out]

int(x/(a + b*x^n)^2, x)