3.2470 \(\int \frac{x}{\left (a+b x^n\right )^2} \, dx\)

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)

_______________________________________________________________________________________

Rubi [A]  time = 0.027139, 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 \[ \frac{x^2 \, _2F_1\left (2,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a^2} \]

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)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.29808, size = 22, normalized size = 0.67 \[ \frac{x^{2}{{}_{2}F_{1}\left (\begin{matrix} 2, \frac{2}{n} \\ \frac{n + 2}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(a+b*x**n)**2,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0520846, size = 53, normalized size = 1.61 \[ \frac{x^2 \left ((n-2) \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )+\frac{2 a}{a+b x^n}\right )}{2 a^2 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + b*x^n)^2,x]

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.074, size = 0, normalized size = 0. \[ \int{\frac{x}{ \left ( a+b{x}^{n} \right ) ^{2}}}\, dx \]

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)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x^n + a)^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)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(a+b*x**n)**2,x)

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(b*x^n + a)^2, x)