Optimal. Leaf size=56 \[ x \left (a+b x^n\right )^{2-\frac{1}{n}} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}-2} \, _2F_1\left (\frac{1}{n}-2,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]
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Rubi [A] time = 0.0398779, antiderivative size = 56, 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 \[ x \left (a+b x^n\right )^{2-\frac{1}{n}} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}-2} \, _2F_1\left (\frac{1}{n}-2,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^n)^(-((1 - 2*n)/n)),x]
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Rubi in Sympy [A] time = 4.74937, size = 44, normalized size = 0.79 \[ x \left (1 + \frac{b x^{n}}{a}\right )^{-2 + \frac{1}{n}} \left (a + b x^{n}\right )^{2 - \frac{1}{n}}{{}_{2}F_{1}\left (\begin{matrix} -2 + \frac{1}{n}, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )} \]
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 [A] time = 0.046043, size = 55, normalized size = 0.98 \[ a^2 x \left (a+b x^n\right )^{-1/n} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}} \, _2F_1\left (\frac{1}{n}-2,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^n)^(-((1 - 2*n)/n)),x]
[Out]
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Maple [F] time = 0.116, 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)
[Out]
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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")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{n} + a\right )}^{\frac{2 \, n - 1}{n}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^((2*n - 1)/n),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(1/((a+b*x**n)**((1-2*n)/n)),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{\frac{2 \, n - 1}{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^((2*n - 1)/n),x, algorithm="giac")
[Out]