3.2059 \(\int \frac{\sqrt{a+\frac{b}{x^4}}}{x^3} \, dx\)

Optimal. Leaf size=50 \[ -\frac{\sqrt{a+\frac{b}{x^4}}}{4 x^2}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b}}{x^2 \sqrt{a+\frac{b}{x^4}}}\right )}{4 \sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.0992981, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{\sqrt{a+\frac{b}{x^4}}}{4 x^2}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b}}{x^2 \sqrt{a+\frac{b}{x^4}}}\right )}{4 \sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x^4]/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.81476, size = 44, normalized size = 0.88 \[ - \frac{a \operatorname{atanh}{\left (\frac{\sqrt{b}}{x^{2} \sqrt{a + \frac{b}{x^{4}}}} \right )}}{4 \sqrt{b}} - \frac{\sqrt{a + \frac{b}{x^{4}}}}{4 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**4)**(1/2)/x**3,x)

[Out]

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

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Mathematica [A]  time = 0.0838733, size = 60, normalized size = 1.2 \[ \frac{\sqrt{a+\frac{b}{x^4}} \left (-\frac{a x^4 \tanh ^{-1}\left (\frac{\sqrt{a x^4+b}}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{a x^4+b}}-1\right )}{4 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x^4]/x^3,x]

[Out]

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

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Maple [B]  time = 0.018, size = 90, normalized size = 1.8 \[ -{\frac{1}{4\,{x}^{2}}\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}} \left ( a\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{4}+b}+b}{{x}^{2}}} \right ){x}^{4}b-a\sqrt{a{x}^{4}+b}{x}^{4}\sqrt{b}+ \left ( a{x}^{4}+b \right ) ^{{\frac{3}{2}}}\sqrt{b} \right ){\frac{1}{\sqrt{a{x}^{4}+b}}}{b}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^4)^(1/2)/x^3,x)

[Out]

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

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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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.257514, size = 1, normalized size = 0.02 \[ \left [\frac{a \sqrt{b} x^{2} \log \left (-\frac{2 \, b x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}} -{\left (a x^{4} + 2 \, b\right )} \sqrt{b}}{x^{4}}\right ) - 2 \, b \sqrt{\frac{a x^{4} + b}{x^{4}}}}{8 \, b x^{2}}, -\frac{a \sqrt{-b} x^{2} \arctan \left (\frac{b}{\sqrt{-b} x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}}}\right ) + b \sqrt{\frac{a x^{4} + b}{x^{4}}}}{4 \, b x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x^4)/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 8.01696, size = 46, normalized size = 0.92 \[ - \frac{\sqrt{a} \sqrt{1 + \frac{b}{a x^{4}}}}{4 x^{2}} - \frac{a \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x^{2}} \right )}}{4 \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**4)**(1/2)/x**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.231559, size = 58, normalized size = 1.16 \[ \frac{1}{4} \, a{\left (\frac{\arctan \left (\frac{\sqrt{a x^{4} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{\sqrt{a x^{4} + b}}{a x^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x^4)/x^3,x, algorithm="giac")

[Out]

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