Optimal. Leaf size=51 \[ \frac{n x \left (a+b x^n\right )^{-1/n}}{a^2 (n+1)}+\frac{x \left (a+b x^n\right )^{-\frac{n+1}{n}}}{a (n+1)} \]
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Rubi [A] time = 0.0489196, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{n x \left (a+b x^n\right )^{-1/n}}{a^2 (n+1)}+\frac{x \left (a+b x^n\right )^{-\frac{n+1}{n}}}{a (n+1)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^n)^(-((1 + 2*n)/n)),x]
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Rubi in Sympy [A] time = 4.63327, size = 46, normalized size = 0.9 \[ x \left (1 + \frac{b x^{n}}{a}\right )^{2 + \frac{1}{n}} \left (a + b x^{n}\right )^{-2 - \frac{1}{n}}{{}_{2}F_{1}\left (\begin{matrix} 2 + \frac{1}{n}, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/((a+b*x**n)**((1+2*n)/n)),x)
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Mathematica [C] time = 0.0366189, size = 55, normalized size = 1.08 \[ \frac{x \left (a+b x^n\right )^{-1/n} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}} \, _2F_1\left (2+\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^n)^(-((1 + 2*n)/n)),x]
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Maple [F] time = 0.128, size = 0, normalized size = 0. \[ \int \left ( \left ( a+b{x}^{n} \right ) ^{{\frac{1+2\,n}{n}}} \right ) ^{-1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/((a+b*x^n)^((1+2*n)/n)),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{-\frac{2 \, n + 1}{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)^((2*n + 1)/n)),x, algorithm="maxima")
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Fricas [A] time = 0.240196, size = 92, normalized size = 1.8 \[ \frac{b^{2} n x x^{2 \, n} +{\left (2 \, a b n + a b\right )} x x^{n} +{\left (a^{2} n + a^{2}\right )} x}{{\left (a^{2} n + a^{2}\right )}{\left (b x^{n} + a\right )}^{\frac{2 \, n + 1}{n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)^((2*n + 1)/n)),x, algorithm="fricas")
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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(1/((a+b*x**n)**((1+2*n)/n)),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{n} + a\right )}^{\frac{2 \, n + 1}{n}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)^((2*n + 1)/n)),x, algorithm="giac")
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