3.2108 \(\int \frac{1}{\sqrt{-a+\frac{b}{x^5}} x} \, dx\)

Optimal. Leaf size=29 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{\frac{b}{x^5}-a}}{\sqrt{a}}\right )}{5 \sqrt{a}} \]

[Out]

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

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Rubi [A]  time = 0.0196157, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {266, 63, 205} \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{\frac{b}{x^5}-a}}{\sqrt{a}}\right )}{5 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[-a + b/x^5]*x),x]

[Out]

(-2*ArcTan[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 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}{\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 \tan ^{-1}\left (\frac{\sqrt{-a+\frac{b}{x^5}}}{\sqrt{a}}\right )}{5 \sqrt{a}}\\ \end{align*}

Mathematica [B]  time = 0.0343614, size = 65, normalized size = 2.24 \[ \frac{2 \sqrt{a x^5-b} \tanh ^{-1}\left (\frac{\sqrt{a} x^{5/2}}{\sqrt{a x^5-b}}\right )}{5 \sqrt{a} x^{5/2} \sqrt{\frac{b}{x^5}-a}} \]

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 = 0.031, 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 = 5.16746, size = 250, normalized size = 8.62 \begin{align*} \left [-\frac{\sqrt{-a} \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 \, a}, -\frac{\arctan \left (\frac{2 \, \sqrt{a} x^{5} \sqrt{-\frac{a x^{5} - b}{x^{5}}}}{2 \, a x^{5} - b}\right )}{5 \, \sqrt{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*sqrt(-a)*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))/a, -
1/5*arctan(2*sqrt(a)*x^5*sqrt(-(a*x^5 - b)/x^5)/(2*a*x^5 - b))/sqrt(a)]

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Sympy [A]  time = 1.56956, size = 61, normalized size = 2.1 \begin{align*} \begin{cases} - \frac{2 i \operatorname{acosh}{\left (\frac{\sqrt{a} x^{\frac{5}{2}}}{\sqrt{b}} \right )}}{5 \sqrt{a}} & \text{for}\: \frac{\left |{a x^{5}}\right |}{\left |{b}\right |} > 1 \\\frac{2 \operatorname{asin}{\left (\frac{\sqrt{a} x^{\frac{5}{2}}}{\sqrt{b}} \right )}}{5 \sqrt{a}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*I*acosh(sqrt(a)*x**(5/2)/sqrt(b))/(5*sqrt(a)), Abs(a*x**5)/Abs(b) > 1), (2*asin(sqrt(a)*x**(5/2)
/sqrt(b))/(5*sqrt(a)), True))

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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)