3.2662 \(\int \frac{1}{x \sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=28 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{\sqrt{a} n} \]

[Out]

(-2*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/(Sqrt[a]*n)

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Rubi [A]  time = 0.0164945, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {266, 63, 208} \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{\sqrt{a} n} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[a + b*x^n]),x]

[Out]

(-2*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/(Sqrt[a]*n)

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

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

Mathematica [A]  time = 0.004407, size = 28, normalized size = 1. \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{\sqrt{a} n} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[a + b*x^n]),x]

[Out]

(-2*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/(Sqrt[a]*n)

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Maple [A]  time = 0., size = 23, normalized size = 0.8 \begin{align*} -2\,{\frac{1}{n\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{a+b{x}^{n}}}{\sqrt{a}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.05136, size = 163, normalized size = 5.82 \begin{align*} \left [\frac{\log \left (\frac{b x^{n} - 2 \, \sqrt{b x^{n} + a} \sqrt{a} + 2 \, a}{x^{n}}\right )}{\sqrt{a} n}, \frac{2 \, \sqrt{-a} \arctan \left (\frac{\sqrt{b x^{n} + a} \sqrt{-a}}{a}\right )}{a n}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.36282, size = 26, normalized size = 0.93 \begin{align*} - \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{a} x^{- \frac{n}{2}}}{\sqrt{b}} \right )}}{\sqrt{a} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*asinh(sqrt(a)*x**(-n/2)/sqrt(b))/(sqrt(a)*n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(b*x^n + a)*x), x)