Optimal. Leaf size=65 \[ -\frac{\left (a+b x^n\right ) \, _2F_1\left (1,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a x \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]
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Rubi [A] time = 0.0740505, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{\left (a+b x^n\right ) \, _2F_1\left (1,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a x \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)]),x]
[Out]
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Rubi in Sympy [A] time = 14.8806, size = 56, normalized size = 0.86 \[ - \frac{b \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}{{}_{2}F_{1}\left (\begin{matrix} 1, - \frac{1}{n} \\ \frac{n - 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a x \left (a b + b^{2} x^{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(a**2+2*a*b*x**n+b**2*x**(2*n))**(1/2),x)
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Mathematica [A] time = 0.0291856, size = 51, normalized size = 0.78 \[ -\frac{\left (a+b x^n\right ) \, _2F_1\left (1,-\frac{1}{n};1-\frac{1}{n};-\frac{b x^n}{a}\right )}{a x \sqrt{\left (a+b x^n\right )^2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)]),x]
[Out]
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Maple [F] time = 0.03, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}}{\frac{1}{\sqrt{{a}^{2}+2\,ab{x}^{n}+{b}^{2}{x}^{2\,n}}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(a^2+2*a*b*x^n+b^2*x^(2*n))^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b^2*x^(2*n) + 2*a*b*x^n + a^2)*x^2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b^2*x^(2*n) + 2*a*b*x^n + a^2)*x^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \sqrt{\left (a + b x^{n}\right )^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(a**2+2*a*b*x**n+b**2*x**(2*n))**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b^2*x^(2*n) + 2*a*b*x^n + a^2)*x^2),x, algorithm="giac")
[Out]