Optimal. Leaf size=59 \[ x \left (a+b x^n\right )^{-\frac{1-n}{n}} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}-1} \, _2F_1\left (\frac{1}{n}-1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]
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Rubi [A] time = 0.0384988, antiderivative size = 59, 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 \[ x \left (a+b x^n\right )^{-\frac{1-n}{n}} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}-1} \, _2F_1\left (\frac{1}{n}-1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^n)^(-((1 - n)/n)),x]
[Out]
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Rubi in Sympy [A] time = 4.93949, size = 46, normalized size = 0.78 \[ x \left (1 + \frac{b x^{n}}{a}\right )^{- \frac{n - 1}{n}} \left (a + b x^{n}\right )^{\frac{n - 1}{n}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{n - 1}{n}, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/((a+b*x**n)**((1-n)/n)),x)
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Mathematica [A] time = 0.0358157, size = 53, normalized size = 0.9 \[ a x \left (a+b x^n\right )^{-1/n} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}} \, _2F_1\left (\frac{1}{n}-1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^n)^(-((1 - n)/n)),x]
[Out]
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Maple [F] time = 0.103, size = 0, normalized size = 0. \[ \int \left ( \left ( a+b{x}^{n} \right ) ^{{\frac{1-n}{n}}} \right ) ^{-1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/((a+b*x^n)^((1-n)/n)),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{\frac{n - 1}{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)^(-(n - 1)/n)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{n} + a\right )}^{\frac{n - 1}{n}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^((n - 1)/n),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)**((1-n)/n)),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{\frac{n - 1}{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^((n - 1)/n),x, algorithm="giac")
[Out]