Optimal. Leaf size=50 \[ \frac{n x \left (a+b x^n\right )^{-1/n}}{a^2 (n+1)}+\frac{x \left (a+b x^n\right )^{-\frac{1}{n}-1}}{a (n+1)} \]
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Rubi [A] time = 0.0405838, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{n x \left (a+b x^n\right )^{-1/n}}{a^2 (n+1)}+\frac{x \left (a+b x^n\right )^{-\frac{1}{n}-1}}{a (n+1)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^n)^(-2 - n^(-1)),x]
[Out]
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Rubi in Sympy [A] time = 3.77428, size = 39, normalized size = 0.78 \[ \frac{x \left (a + b x^{n}\right )^{-1 - \frac{1}{n}}}{a \left (n + 1\right )} + \frac{n x \left (a + b x^{n}\right )^{- \frac{1}{n}}}{a^{2} \left (n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*x**n)**(-2-1/n),x)
[Out]
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Mathematica [C] time = 0.0410468, size = 55, normalized size = 1.1 \[ \frac{x \left (a+b x^n\right )^{-1/n} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}} \, _2F_1\left (2+\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^n)^(-2 - n^(-1)),x]
[Out]
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Maple [F] time = 0.132, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n} \right ) ^{-2-{n}^{-1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*x^n)^(-2-1/n),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{-\frac{1}{n} - 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^(-1/n - 2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.242405, size = 92, normalized size = 1.84 \[ \frac{b^{2} n x x^{2 \, n} +{\left (2 \, a b n + a b\right )} x x^{n} +{\left (a^{2} n + a^{2}\right )} x}{{\left (a^{2} n + a^{2}\right )}{\left (b x^{n} + a\right )}^{\frac{2 \, n + 1}{n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^(-1/n - 2),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*x**n)**(-2-1/n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{-\frac{1}{n} - 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^(-1/n - 2),x, algorithm="giac")
[Out]