3.2060 \(\int \frac{\sqrt{a+\frac{b}{x^4}}}{x} \, dx\)

Optimal. Leaf size=43 \[ \frac{1}{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )-\frac{1}{2} \sqrt{a+\frac{b}{x^4}} \]

[Out]

-Sqrt[a + b/x^4]/2 + (Sqrt[a]*ArcTanh[Sqrt[a + b/x^4]/Sqrt[a]])/2

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Rubi [A]  time = 0.0237749, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 50, 63, 208} \[ \frac{1}{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )-\frac{1}{2} \sqrt{a+\frac{b}{x^4}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x^4]/x,x]

[Out]

-Sqrt[a + b/x^4]/2 + (Sqrt[a]*ArcTanh[Sqrt[a + b/x^4]/Sqrt[a]])/2

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 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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{\sqrt{a+\frac{b}{x^4}}}{x} \, dx &=-\left (\frac{1}{4} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\frac{1}{x^4}\right )\right )\\ &=-\frac{1}{2} \sqrt{a+\frac{b}{x^4}}-\frac{1}{4} a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x^4}\right )\\ &=-\frac{1}{2} \sqrt{a+\frac{b}{x^4}}-\frac{a \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x^4}}\right )}{2 b}\\ &=-\frac{1}{2} \sqrt{a+\frac{b}{x^4}}+\frac{1}{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0899466, size = 60, normalized size = 1.4 \[ \frac{1}{2} \sqrt{a+\frac{b}{x^4}} \left (\frac{\sqrt{a} x^2 \sinh ^{-1}\left (\frac{\sqrt{a} x^2}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{\frac{a x^4}{b}+1}}-1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x^4]/x,x]

[Out]

(Sqrt[a + b/x^4]*(-1 + (Sqrt[a]*x^2*ArcSinh[(Sqrt[a]*x^2)/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (a*x^4)/b])))/2

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Maple [B]  time = 0.009, size = 80, normalized size = 1.9 \begin{align*}{\frac{1}{2\,b}\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}} \left ( a{x}^{4}\sqrt{a{x}^{4}+b}+\sqrt{a}\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ){x}^{2}b- \left ( a{x}^{4}+b \right ) ^{{\frac{3}{2}}} \right ){\frac{1}{\sqrt{a{x}^{4}+b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x^4)^(1/2)/x,x)

[Out]

1/2*((a*x^4+b)/x^4)^(1/2)*(a*x^4*(a*x^4+b)^(1/2)+a^(1/2)*ln(x^2*a^(1/2)+(a*x^4+b)^(1/2))*x^2*b-(a*x^4+b)^(3/2)
)/(a*x^4+b)^(1/2)/b

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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((a+b/x^4)^(1/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.53487, size = 269, normalized size = 6.26 \begin{align*} \left [\frac{1}{4} \, \sqrt{a} \log \left (-2 \, a x^{4} - 2 \, \sqrt{a} x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} - b\right ) - \frac{1}{2} \, \sqrt{\frac{a x^{4} + b}{x^{4}}}, -\frac{1}{2} \, \sqrt{-a} \arctan \left (\frac{\sqrt{-a} x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{a x^{4} + b}\right ) - \frac{1}{2} \, \sqrt{\frac{a x^{4} + b}{x^{4}}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^4)^(1/2)/x,x, algorithm="fricas")

[Out]

[1/4*sqrt(a)*log(-2*a*x^4 - 2*sqrt(a)*x^4*sqrt((a*x^4 + b)/x^4) - b) - 1/2*sqrt((a*x^4 + b)/x^4), -1/2*sqrt(-a
)*arctan(sqrt(-a)*x^4*sqrt((a*x^4 + b)/x^4)/(a*x^4 + b)) - 1/2*sqrt((a*x^4 + b)/x^4)]

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Sympy [A]  time = 2.34012, size = 66, normalized size = 1.53 \begin{align*} \frac{\sqrt{a} \operatorname{asinh}{\left (\frac{\sqrt{a} x^{2}}{\sqrt{b}} \right )}}{2} - \frac{a x^{2}}{2 \sqrt{b} \sqrt{\frac{a x^{4}}{b} + 1}} - \frac{\sqrt{b}}{2 x^{2} \sqrt{\frac{a x^{4}}{b} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**4)**(1/2)/x,x)

[Out]

sqrt(a)*asinh(sqrt(a)*x**2/sqrt(b))/2 - a*x**2/(2*sqrt(b)*sqrt(a*x**4/b + 1)) - sqrt(b)/(2*x**2*sqrt(a*x**4/b
+ 1))

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Giac [A]  time = 1.09501, size = 49, normalized size = 1.14 \begin{align*} -\frac{a \arctan \left (\frac{\sqrt{a + \frac{b}{x^{4}}}}{\sqrt{-a}}\right )}{2 \, \sqrt{-a}} - \frac{1}{2} \, \sqrt{a + \frac{b}{x^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^4)^(1/2)/x,x, algorithm="giac")

[Out]

-1/2*a*arctan(sqrt(a + b/x^4)/sqrt(-a))/sqrt(-a) - 1/2*sqrt(a + b/x^4)