3.2733 \(\int (a+b x^n)^{-\frac{1+n}{n}} \, dx\)

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 = 0.0029852, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, 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)/n)),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 )^{-\frac{1+n}{n}} \, dx &=\frac{x \left (a+b x^n\right )^{-1/n}}{a}\\ \end{align*}

Mathematica [A]  time = 0.0205476, 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)/n)),x]

[Out]

x/(a*(a + b*x^n)^n^(-1))

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Maple [A]  time = 0.019, size = 35, normalized size = 1.9 \begin{align*}{ \left ( x+{\frac{bx{{\rm e}^{n\ln \left ( x \right ) }}}{a}} \right ) \left ({{\rm e}^{{\frac{ \left ( 1+n \right ) \ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{n}}}} \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.37482, 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(1/((a+b*x^n)^((1+n)/n)),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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