3.409 \(\int \frac{1}{\sqrt{\frac{a+b x^5}{x^3}}} \, dx\)

Optimal. Leaf size=32 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{\frac{a}{x^3}+b x^2}}\right )}{5 \sqrt{b}} \]

[Out]

(2*ArcTanh[(Sqrt[b]*x)/Sqrt[a/x^3 + b*x^2]])/(5*Sqrt[b])

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Rubi [A]  time = 0.0341457, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{\frac{a}{x^3}+b x^2}}\right )}{5 \sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[(a + b*x^5)/x^3],x]

[Out]

(2*ArcTanh[(Sqrt[b]*x)/Sqrt[a/x^3 + b*x^2]])/(5*Sqrt[b])

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Rubi in Sympy [A]  time = 2.51158, size = 29, normalized size = 0.91 \[ \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{\frac{a}{x^{3}} + b x^{2}}} \right )}}{5 \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/((b*x**5+a)/x**3)**(1/2),x)

[Out]

2*atanh(sqrt(b)*x/sqrt(a/x**3 + b*x**2))/(5*sqrt(b))

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Mathematica [A]  time = 0.0378524, size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\frac{a+b x^5}{x^3}}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[1/Sqrt[(a + b*x^5)/x^3],x]

[Out]

Integrate[1/Sqrt[(a + b*x^5)/x^3], x]

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \[ \int{\frac{1}{\sqrt{{\frac{b{x}^{5}+a}{{x}^{3}}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/((b*x^5+a)/x^3)^(1/2),x)

[Out]

int(1/((b*x^5+a)/x^3)^(1/2),x)

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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(1/sqrt((b*x^5 + a)/x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.701436, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (-{\left (8 \, b^{2} x^{10} + 8 \, a b x^{5} + a^{2}\right )} \sqrt{b} - 4 \,{\left (2 \, b^{2} x^{9} + a b x^{4}\right )} \sqrt{\frac{b x^{5} + a}{x^{3}}}\right )}{10 \, \sqrt{b}}, -\frac{\sqrt{-b} \arctan \left (\frac{2 \, \sqrt{-b} x^{4} \sqrt{\frac{b x^{5} + a}{x^{3}}}}{2 \, b x^{5} + a}\right )}{5 \, b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt((b*x^5 + a)/x^3),x, algorithm="fricas")

[Out]

[1/10*log(-(8*b^2*x^10 + 8*a*b*x^5 + a^2)*sqrt(b) - 4*(2*b^2*x^9 + a*b*x^4)*sqrt
((b*x^5 + a)/x^3))/sqrt(b), -1/5*sqrt(-b)*arctan(2*sqrt(-b)*x^4*sqrt((b*x^5 + a)
/x^3)/(2*b*x^5 + a))/b]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x**5+a)/x**3)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt((b*x^5 + a)/x^3),x, algorithm="giac")

[Out]

Exception raised: TypeError