3.2058 \(\int \frac{\sqrt{a+\frac{b}{x^4}}}{x} \, dx\)

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 = 0.0814702, 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 \[ \frac{1}{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )-\frac{1}{2} \sqrt{a+\frac{b}{x^4}} \]

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

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Rubi in Sympy [A]  time = 6.69369, size = 34, normalized size = 0.79 \[ \frac{\sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{4}}}}{\sqrt{a}} \right )}}{2} - \frac{\sqrt{a + \frac{b}{x^{4}}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**4)**(1/2)/x,x)

[Out]

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

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Mathematica [A]  time = 0.0563084, size = 71, normalized size = 1.65 \[ \frac{\sqrt{a} x^2 \sqrt{a+\frac{b}{x^4}} \tanh ^{-1}\left (\frac{\sqrt{a} x^2}{\sqrt{a x^4+b}}\right )}{2 \sqrt{a x^4+b}}-\frac{1}{2} \sqrt{a+\frac{b}{x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x^4]/x,x]

[Out]

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

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Maple [B]  time = 0.017, size = 80, normalized size = 1.9 \[{\frac{1}{2\,b}\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}} \left ( a{x}^{4}\sqrt{a{x}^{4}+b}+\sqrt{a}\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ){x}^{2}b- \left ( a{x}^{4}+b \right ) ^{{\frac{3}{2}}} \right ){\frac{1}{\sqrt{a{x}^{4}+b}}}} \]

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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x^4)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.255495, size = 1, normalized size = 0.02 \[ \left [\frac{1}{4} \, \sqrt{a} \log \left (-2 \, a x^{4} - 2 \, \sqrt{a} x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} - b\right ) - \frac{1}{2} \, \sqrt{\frac{a x^{4} + b}{x^{4}}}, \frac{1}{2} \, \sqrt{-a} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x^{4} + b}{x^{4}}}}\right ) - \frac{1}{2} \, \sqrt{\frac{a x^{4} + b}{x^{4}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x^4)/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(a/(sqrt(-a)*sqrt((a*x^4 + b)/x^4))) - 1/2*
sqrt((a*x^4 + b)/x^4)]

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Sympy [A]  time = 5.84477, size = 66, normalized size = 1.53 \[ \frac{\sqrt{a} \operatorname{asinh}{\left (\frac{\sqrt{a} x^{2}}{\sqrt{b}} \right )}}{2} - \frac{a x^{2}}{2 \sqrt{b} \sqrt{\frac{a x^{4}}{b} + 1}} - \frac{\sqrt{b}}{2 x^{2} \sqrt{\frac{a x^{4}}{b} + 1}} \]

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/XCAS [A]  time = 0.231832, size = 49, normalized size = 1.14 \[ -\frac{a \arctan \left (\frac{\sqrt{a + \frac{b}{x^{4}}}}{\sqrt{-a}}\right )}{2 \, \sqrt{-a}} - \frac{1}{2} \, \sqrt{a + \frac{b}{x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x^4)/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)