3.287 \(\int \frac{1}{\sqrt{a x^2+b x^5}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0266031, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^5}}\right )}{3 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[a*x^2 + b*x^5],x]

[Out]

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

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Rubi in Sympy [A]  time = 2.32906, size = 31, normalized size = 0.97 \[ - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a} x}{\sqrt{a x^{2} + b x^{5}}} \right )}}{3 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0613647, size = 54, normalized size = 1.69 \[ -\frac{2 \sqrt{x^2 \left (a+b x^3\right )} \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{3 a x \sqrt{\frac{b x^3}{a}+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[a*x^2 + b*x^5],x]

[Out]

(-2*Sqrt[x^2*(a + b*x^3)]*ArcTanh[Sqrt[1 + (b*x^3)/a]])/(3*a*x*Sqrt[1 + (b*x^3)/
a])

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Maple [A]  time = 0.008, size = 43, normalized size = 1.3 \[ -{\frac{2\,x}{3}\sqrt{b{x}^{3}+a}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{b{x}^{5}+a{x}^{2}}}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^5+a*x^2)^(1/2),x)

[Out]

-2/3/(b*x^5+a*x^2)^(1/2)*x*(b*x^3+a)^(1/2)*arctanh((b*x^3+a)^(1/2)/a^(1/2))/a^(1
/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(1/sqrt(b*x^5 + a*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223321, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (\frac{{\left (b x^{4} + 2 \, a x\right )} \sqrt{a} - 2 \, \sqrt{b x^{5} + a x^{2}} a}{x^{4}}\right )}{3 \, \sqrt{a}}, -\frac{2 \, \sqrt{-a} \arctan \left (\frac{a x}{\sqrt{b x^{5} + a x^{2}} \sqrt{-a}}\right )}{3 \, a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(b*x^5 + a*x^2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a x^{2} + b x^{5}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/sqrt(a*x**2 + b*x**5), x)

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GIAC/XCAS [A]  time = 0.222081, size = 63, normalized size = 1.97 \[ -\frac{2 \, \arctan \left (\frac{\sqrt{a}}{\sqrt{-a}}\right ){\rm sign}\left (x\right )}{3 \, \sqrt{-a}} + \frac{2 \, \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{3 \, \sqrt{-a}{\rm sign}\left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(b*x^5 + a*x^2),x, algorithm="giac")

[Out]

-2/3*arctan(sqrt(a)/sqrt(-a))*sign(x)/sqrt(-a) + 2/3*arctan(sqrt(b*x^3 + a)/sqrt
(-a))/(sqrt(-a)*sign(x))