3.2107 \(\int \frac{1}{\sqrt{a+\frac{b}{x^5}} x} \, dx\)

Optimal. Leaf size=27 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^5}}}{\sqrt{a}}\right )}{5 \sqrt{a}} \]

[Out]

(2*ArcTanh[Sqrt[a + b/x^5]/Sqrt[a]])/(5*Sqrt[a])

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Rubi [A]  time = 0.0194627, antiderivative size = 27, 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} \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^5}}}{\sqrt{a}}\right )}{5 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b/x^5]*x),x]

[Out]

(2*ArcTanh[Sqrt[a + b/x^5]/Sqrt[a]])/(5*Sqrt[a])

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}{\sqrt{a+\frac{b}{x^5}} x} \, dx &=-\left (\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x^5}\right )\right )\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x^5}}\right )}{5 b}\\ &=\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^5}}}{\sqrt{a}}\right )}{5 \sqrt{a}}\\ \end{align*}

Mathematica [B]  time = 0.0207192, size = 59, normalized size = 2.19 \[ \frac{2 \sqrt{a x^5+b} \tanh ^{-1}\left (\frac{\sqrt{a} x^{5/2}}{\sqrt{a x^5+b}}\right )}{5 \sqrt{a} x^{5/2} \sqrt{a+\frac{b}{x^5}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b/x^5]*x),x]

[Out]

(2*Sqrt[b + a*x^5]*ArcTanh[(Sqrt[a]*x^(5/2))/Sqrt[b + a*x^5]])/(5*Sqrt[a]*Sqrt[a + b/x^5]*x^(5/2))

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Maple [F]  time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}{\frac{1}{\sqrt{a+{\frac{b}{{x}^{5}}}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(a+b/x^5)^(1/2),x)

[Out]

int(1/x/(a+b/x^5)^(1/2),x)

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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^5)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 5.05926, size = 246, normalized size = 9.11 \begin{align*} \left [\frac{\log \left (-8 \, a^{2} x^{10} - 8 \, a b x^{5} - b^{2} - 4 \,{\left (2 \, a x^{10} + b x^{5}\right )} \sqrt{a} \sqrt{\frac{a x^{5} + b}{x^{5}}}\right )}{10 \, \sqrt{a}}, -\frac{\sqrt{-a} \arctan \left (\frac{2 \, \sqrt{-a} x^{5} \sqrt{\frac{a x^{5} + b}{x^{5}}}}{2 \, a x^{5} + b}\right )}{5 \, a}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+b/x^5)^(1/2),x, algorithm="fricas")

[Out]

[1/10*log(-8*a^2*x^10 - 8*a*b*x^5 - b^2 - 4*(2*a*x^10 + b*x^5)*sqrt(a)*sqrt((a*x^5 + b)/x^5))/sqrt(a), -1/5*sq
rt(-a)*arctan(2*sqrt(-a)*x^5*sqrt((a*x^5 + b)/x^5)/(2*a*x^5 + b))/a]

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Sympy [A]  time = 1.45086, size = 24, normalized size = 0.89 \begin{align*} \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{a} x^{\frac{5}{2}}}{\sqrt{b}} \right )}}{5 \sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+b/x**5)**(1/2),x)

[Out]

2*asinh(sqrt(a)*x**(5/2)/sqrt(b))/(5*sqrt(a))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + \frac{b}{x^{5}}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+b/x^5)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(a + b/x^5)*x), x)