3.2711 \(\int (a+b x^n)^{2-\frac {1}{n}} \, dx\)

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

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Rubi [A]  time = 0.01, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {246, 245} \[ a^2 x \left (a+b x^n\right )^{-1/n} \left (\frac {b x^n}{a}+1\right )^{\frac {1}{n}} \, _2F_1\left (\frac {1}{n}-2,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right ) \]

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)

Rule 245

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, -((b*x^n)/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^Fr
acPart[p], Int[(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])

Rubi steps

\begin {align*} \int \left (a+b x^n\right )^{2-\frac {1}{n}} \, dx &=\left (a^2 \left (a+b x^n\right )^{-1/n} \left (1+\frac {b x^n}{a}\right )^{\frac {1}{n}}\right ) \int \left (1+\frac {b x^n}{a}\right )^{2-\frac {1}{n}} \, dx\\ &=a^2 x \left (a+b x^n\right )^{-1/n} \left (1+\frac {b x^n}{a}\right )^{\frac {1}{n}} \, _2F_1\left (-2+\frac {1}{n},\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 55, normalized size = 1.00 \[ a^2 x \left (a+b x^n\right )^{-1/n} \left (\frac {b x^n}{a}+1\right )^{\frac {1}{n}} \, _2F_1\left (\frac {1}{n}-2,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right ) \]

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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fricas [F]  time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{n} + a\right )}^{\frac {2 \, n - 1}{n}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^(2-1/n),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{n} + a\right )}^{-\frac {1}{n} + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^(2-1/n),x, algorithm="giac")

[Out]

integrate((b*x^n + a)^(-1/n + 2), x)

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maple [F]  time = 0.29, size = 0, normalized size = 0.00 \[ \int \left (b \,x^{n}+a \right )^{-\frac {1}{n}+2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{n} + a\right )}^{-\frac {1}{n} + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^(2-1/n),x, algorithm="maxima")

[Out]

integrate((b*x^n + a)^(-1/n + 2), x)

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mupad [B]  time = 1.31, size = 56, normalized size = 1.02 \[ \frac {a^2\,x\,{\left (\frac {b\,x^n}{a}+1\right )}^{1/n}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{n}-2,\frac {1}{n};\ \frac {1}{n}+1;\ -\frac {b\,x^n}{a}\right )}{{\left (a+b\,x^n\right )}^{1/n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^n)^(2 - 1/n),x)

[Out]

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

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sympy [C]  time = 90.92, size = 44, normalized size = 0.80 \[ \frac {a^{2} a^{- \frac {1}{n}} x \Gamma \left (\frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{n}, -2 + \frac {1}{n} \\ 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (1 + \frac {1}{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*a**(-1/n)*x*gamma(1/n)*hyper((1/n, -2 + 1/n), (1 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(1 + 1/n))

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