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]

2/5*arctanh((a+b/x^5)^(1/2)/a^(1/2))/a^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 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]

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]]

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*}

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Mathematica [B]  time = 0.02, 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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fricas [B]  time = 2.14, size = 102, normalized size = 3.78 \[ \left [\frac {\log \left (-8 \, a^{2} x^{10} - 8 \, a b x^{5} - b^{2} - 4 \, {\left (2 \, a x^{10} + b x^{5}\right )} \sqrt {a} \sqrt {\frac {a x^{5} + b}{x^{5}}}\right )}{10 \, \sqrt {a}}, -\frac {\sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} x^{5} \sqrt {\frac {a x^{5} + b}{x^{5}}}}{2 \, a x^{5} + b}\right )}{5 \, a}\right ] \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + \frac {b}{x^{5}}} x}\,{d x} \]

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)

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a +\frac {b}{x^{5}}}\, x}\, dx \]

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 [A]  time = 1.93, size = 37, normalized size = 1.37 \[ -\frac {\log \left (\frac {\sqrt {a + \frac {b}{x^{5}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{5}}} + \sqrt {a}}\right )}{5 \, \sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*log((sqrt(a + b/x^5) - sqrt(a))/(sqrt(a + b/x^5) + sqrt(a)))/sqrt(a)

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mupad [B]  time = 1.35, size = 19, normalized size = 0.70 \[ \frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^5}}}{\sqrt {a}}\right )}{5\,\sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b/x^5)^(1/2)),x)

[Out]

(2*atanh((a + b/x^5)^(1/2)/a^(1/2)))/(5*a^(1/2))

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sympy [A]  time = 1.44, size = 24, normalized size = 0.89 \[ \frac {2 \operatorname {asinh}{\left (\frac {\sqrt {a} x^{\frac {5}{2}}}{\sqrt {b}} \right )}}{5 \sqrt {a}} \]

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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