3.2708 \(\int (a+b x^n)^{-1-\frac{1}{n}} \, dx\)

Optimal. Leaf size=18 \[ \frac{x \left (a+b x^n\right )^{-1/n}}{a} \]

[Out]

x/(a*(a + b*x^n)^n^(-1))

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Rubi [A]  time = 0.002757, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {191} \[ \frac{x \left (a+b x^n\right )^{-1/n}}{a} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)^(-1 - n^(-1)),x]

[Out]

x/(a*(a + b*x^n)^n^(-1))

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

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

Mathematica [A]  time = 0.0247102, size = 18, normalized size = 1. \[ \frac{x \left (a+b x^n\right )^{-1/n}}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(a*(a + b*x^n)^n^(-1))

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Maple [B]  time = 0.021, size = 53, normalized size = 2.9 \begin{align*} x{{\rm e}^{ \left ( -1-{n}^{-1} \right ) \ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }}+{\frac{bx{{\rm e}^{n\ln \left ( x \right ) }}}{a}{{\rm e}^{ \left ( -1-{n}^{-1} \right ) \ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*exp((-1-1/n)*ln(a+b*exp(n*ln(x))))+x*b/a*exp(n*ln(x))*exp((-1-1/n)*ln(a+b*exp(n*ln(x))))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{n} + a\right )}^{-\frac{1}{n} - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^n + a)^(-1/n - 1), x)

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Fricas [A]  time = 1.33565, size = 61, normalized size = 3.39 \begin{align*} \frac{b x x^{n} + a x}{{\left (b x^{n} + a\right )}^{\frac{n + 1}{n}} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x*x^n + a*x)/((b*x^n + a)^((n + 1)/n)*a)

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Sympy [A]  time = 37.2195, size = 211, normalized size = 11.72 \begin{align*} \begin{cases} - \frac{b^{- \frac{1}{n}} x x^{- n} \left (x^{n}\right )^{- \frac{1}{n}}}{b n} & \text{for}\: a = 0 \\0^{-1 - \frac{1}{n}} x & \text{for}\: a = - b x^{n} \\x \left (0^{n}\right )^{-1 - \frac{1}{n}} & \text{for}\: a = 0^{n} - b x^{n} \\\frac{a^{2} x}{a^{3} \left (a + b x^{n}\right )^{\frac{1}{n}} + 2 a^{2} b x^{n} \left (a + b x^{n}\right )^{\frac{1}{n}} + a b^{2} x^{2 n} \left (a + b x^{n}\right )^{\frac{1}{n}}} + \frac{a b x x^{n}}{a^{3} \left (a + b x^{n}\right )^{\frac{1}{n}} + 2 a^{2} b x^{n} \left (a + b x^{n}\right )^{\frac{1}{n}} + a b^{2} x^{2 n} \left (a + b x^{n}\right )^{\frac{1}{n}}} + \frac{b x x^{n}}{a^{2} \left (a + b x^{n}\right )^{\frac{1}{n}} + a b x^{n} \left (a + b x^{n}\right )^{\frac{1}{n}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-b**(-1/n)*x*x**(-n)*(x**n)**(-1/n)/(b*n), Eq(a, 0)), (0**(-1 - 1/n)*x, Eq(a, -b*x**n)), (x*(0**n)*
*(-1 - 1/n), Eq(a, 0**n - b*x**n)), (a**2*x/(a**3*(a + b*x**n)**(1/n) + 2*a**2*b*x**n*(a + b*x**n)**(1/n) + a*
b**2*x**(2*n)*(a + b*x**n)**(1/n)) + a*b*x*x**n/(a**3*(a + b*x**n)**(1/n) + 2*a**2*b*x**n*(a + b*x**n)**(1/n)
+ a*b**2*x**(2*n)*(a + b*x**n)**(1/n)) + b*x*x**n/(a**2*(a + b*x**n)**(1/n) + a*b*x**n*(a + b*x**n)**(1/n)), T
rue))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{n} + a\right )}^{-\frac{1}{n} - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^n + a)^(-1/n - 1), x)