3.3031 \(\int \frac{1}{a+b (c x^n)^{2/n}} \, dx\)

Optimal. Leaf size=44 \[ \frac{x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]

[Out]

(x*ArcTan[(Sqrt[b]*(c*x^n)^n^(-1))/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*(c*x^n)^n^(-1))

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Rubi [A]  time = 0.0123477, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {254, 205} \[ \frac{x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*(c*x^n)^(2/n))^(-1),x]

[Out]

(x*ArcTan[(Sqrt[b]*(c*x^n)^n^(-1))/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*(c*x^n)^n^(-1))

Rule 254

Int[((a_) + (b_.)*((c_.)*(x_)^(q_.))^(n_))^(p_.), x_Symbol] :> Dist[x/(c*x^q)^(1/q), Subst[Int[(a + b*x^(n*q))
^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, n, p, q}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{1}{a+b \left (c x^n\right )^{2/n}} \, dx &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\frac{x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.008214, size = 44, normalized size = 1. \[ \frac{x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*ArcTan[(Sqrt[b]*(c*x^n)^n^(-1))/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*(c*x^n)^n^(-1))

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Maple [F]  time = 0.675, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ( c{x}^{n} \right ) ^{2\,{n}^{-1}} \right ) ^{-1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (c x^{n}\right )^{\frac{2}{n}} b + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((c*x^n)^(2/n)*b + a), x)

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Fricas [A]  time = 1.3884, size = 238, normalized size = 5.41 \begin{align*} \left [-\frac{\sqrt{-a b c^{\frac{2}{n}}} \log \left (\frac{b c^{\frac{2}{n}} x^{2} - 2 \, \sqrt{-a b c^{\frac{2}{n}}} x - a}{b c^{\frac{2}{n}} x^{2} + a}\right )}{2 \, a b c^{\frac{2}{n}}}, \frac{\sqrt{a b c^{\frac{2}{n}}} \arctan \left (\frac{\sqrt{a b c^{\frac{2}{n}}} x}{a}\right )}{a b c^{\frac{2}{n}}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*sqrt(-a*b*c^(2/n))*log((b*c^(2/n)*x^2 - 2*sqrt(-a*b*c^(2/n))*x - a)/(b*c^(2/n)*x^2 + a))/(a*b*c^(2/n)),
sqrt(a*b*c^(2/n))*arctan(sqrt(a*b*c^(2/n))*x/a)/(a*b*c^(2/n))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{a + b \left (c x^{n}\right )^{\frac{2}{n}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(a + b*(c*x**n)**(2/n)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (c x^{n}\right )^{\frac{2}{n}} b + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((c*x^n)^(2/n)*b + a), x)