3.2704 \(\int \left (a+b x^n\right )^{2-\frac{1}{n}} \, dx\)

Optimal. Leaf size=55 \[ a^2 x \left (a+b x^n\right )^{-1/n} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}} \, _2F_1\left (\frac{1}{n}-2,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]

[Out]

(a^2*x*(1 + (b*x^n)/a)^n^(-1)*Hypergeometric2F1[-2 + n^(-1), n^(-1), 1 + n^(-1),
-((b*x^n)/a)])/(a + b*x^n)^n^(-1)

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Rubi [A]  time = 0.0359431, antiderivative size = 55, 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 \[ a^2 x \left (a+b x^n\right )^{-1/n} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}} \, _2F_1\left (\frac{1}{n}-2,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^n)^(2 - n^(-1)),x]

[Out]

(a^2*x*(1 + (b*x^n)/a)^n^(-1)*Hypergeometric2F1[-2 + n^(-1), n^(-1), 1 + n^(-1),
-((b*x^n)/a)])/(a + b*x^n)^n^(-1)

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Rubi in Sympy [A]  time = 4.5856, size = 44, normalized size = 0.8 \[ a^{2} x \left (1 + \frac{b x^{n}}{a}\right )^{\frac{1}{n}} \left (a + b x^{n}\right )^{- \frac{1}{n}}{{}_{2}F_{1}\left (\begin{matrix} -2 + \frac{1}{n}, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**n)**(2-1/n),x)

[Out]

a**2*x*(1 + b*x**n/a)**(1/n)*(a + b*x**n)**(-1/n)*hyper((-2 + 1/n, 1/n), (1 + 1/
n,), -b*x**n/a)

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Mathematica [A]  time = 0.0492444, size = 55, normalized size = 1. \[ a^2 x \left (a+b x^n\right )^{-1/n} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}} \, _2F_1\left (\frac{1}{n}-2,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^n)^(2 - n^(-1)),x]

[Out]

(a^2*x*(1 + (b*x^n)/a)^n^(-1)*Hypergeometric2F1[-2 + n^(-1), n^(-1), 1 + n^(-1),
-((b*x^n)/a)])/(a + b*x^n)^n^(-1)

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Maple [F]  time = 0.108, 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]

int((a+b*x^n)^(2-1/n),x)

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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]

integrate((b*x^n + a)^(-1/n + 2), x)

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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)^(-1/n + 2),x, algorithm="fricas")

[Out]

integral((b*x^n + a)^((2*n - 1)/n), x)

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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]

Timed 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]

integrate((b*x^n + a)^(-1/n + 2), x)