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