3.2471 \(\int \frac{1}{\left (a+b x^n\right )^2} \, dx\)

Optimal. Leaf size=24 \[ \frac{x \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2} \]

[Out]

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

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Rubi [A]  time = 0.0188835, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{x \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^n)^(-2),x]

[Out]

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

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Rubi in Sympy [A]  time = 1.83814, size = 19, normalized size = 0.79 \[ \frac{x{{}_{2}F_{1}\left (\begin{matrix} 2, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b*x**n)**2,x)

[Out]

x*hyper((2, 1/n), (1 + 1/n,), -b*x**n/a)/a**2

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Mathematica [B]  time = 0.0437842, size = 49, normalized size = 2.04 \[ \frac{x \left ((n-1) \left (a+b x^n\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )+a\right )}{a^2 n \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^n)^(-2),x]

[Out]

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

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Maple [F]  time = 0.063, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n} \right ) ^{-2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*x^n)^2,x)

[Out]

int(1/(a+b*x^n)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[{\left (n - 1\right )} \int \frac{1}{a b n x^{n} + a^{2} n}\,{d x} + \frac{x}{a b n x^{n} + a^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(n - 1)*integrate(1/(a*b*n*x^n + a^2*n), x) + x/(a*b*n*x^n + a^2*n)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/(b^2*x^(2*n) + 2*a*b*x^n + a^2), x)

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

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^n + a)^(-2), x)