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)^(1/n))

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Rubi [A]  time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.04, size = 18, normalized size = 1.00 \[ \frac {x \left (a+b x^n\right )^{-1/n}}{a} \]

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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fricas [A]  time = 0.69, size = 31, normalized size = 1.72 \[ \frac {b x x^{n} + a x}{{\left (b x^{n} + a\right )}^{\frac {n + 1}{n}} a} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 35, normalized size = 1.94 \[ \left (\frac {b x \,{\mathrm e}^{n \ln \relax (x )}}{a}+x \right ) {\mathrm e}^{\frac {\left (-n -1\right ) \ln \left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right )}{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((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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mupad [B]  time = 1.30, size = 75, normalized size = 4.17 \[ \frac {b\,x^{n+1}\,\left (\frac {a}{b\,x^n}-{\left (\frac {a}{b\,x^n}+1\right )}^{\frac {n+1}{n}}+1\right )}{a\,n\,\left (\frac {n+1}{n}-1\right )\,{\left (a+b\,x^n\right )}^{\frac {n+1}{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x^n)^((n + 1)/n),x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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