3.288 \(\int \frac{1}{x^3 \sqrt{a x^2+b x^5}} \, dx\)

Optimal. Leaf size=59 \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^5}}\right )}{3 a^{3/2}}-\frac{\sqrt{a x^2+b x^5}}{3 a x^4} \]

[Out]

-Sqrt[a*x^2 + b*x^5]/(3*a*x^4) + (b*ArcTanh[(Sqrt[a]*x)/Sqrt[a*x^2 + b*x^5]])/(3
*a^(3/2))

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Rubi [A]  time = 0.0997995, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^5}}\right )}{3 a^{3/2}}-\frac{\sqrt{a x^2+b x^5}}{3 a x^4} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[a*x^2 + b*x^5]/(3*a*x^4) + (b*ArcTanh[(Sqrt[a]*x)/Sqrt[a*x^2 + b*x^5]])/(3
*a^(3/2))

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Rubi in Sympy [A]  time = 9.05481, size = 49, normalized size = 0.83 \[ - \frac{\sqrt{a x^{2} + b x^{5}}}{3 a x^{4}} + \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a} x}{\sqrt{a x^{2} + b x^{5}}} \right )}}{3 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a*x**2 + b*x**5)/(3*a*x**4) + b*atanh(sqrt(a)*x/sqrt(a*x**2 + b*x**5))/(3*
a**(3/2))

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Mathematica [A]  time = 0.204733, size = 63, normalized size = 1.07 \[ \frac{\sqrt{x^2 \left (a+b x^3\right )} \left (\frac{b x^3 \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{\sqrt{\frac{b x^3}{a}+1}}-a\right )}{3 a^2 x^4} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x^2*(a + b*x^3)]*(-a + (b*x^3*ArcTanh[Sqrt[1 + (b*x^3)/a]])/Sqrt[1 + (b*x^
3)/a]))/(3*a^2*x^4)

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Maple [A]  time = 0.008, size = 66, normalized size = 1.1 \[ -{\frac{1}{3\,{x}^{2}}\sqrt{b{x}^{3}+a} \left ( -b{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){x}^{3}a+\sqrt{b{x}^{3}+a}{a}^{{\frac{3}{2}}} \right ){\frac{1}{\sqrt{b{x}^{5}+a{x}^{2}}}}{a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3/x^2*(b*x^3+a)^(1/2)*(-b*arctanh((b*x^3+a)^(1/2)/a^(1/2))*x^3*a+(b*x^3+a)^(1
/2)*a^(3/2))/(b*x^5+a*x^2)^(1/2)/a^(5/2)

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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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231216, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{a} b x^{4} \log \left (\frac{{\left (b x^{4} + 2 \, a x\right )} \sqrt{a} + 2 \, \sqrt{b x^{5} + a x^{2}} a}{x^{4}}\right ) - 2 \, \sqrt{b x^{5} + a x^{2}} a}{6 \, a^{2} x^{4}}, \frac{\sqrt{-a} b x^{4} \arctan \left (\frac{a x}{\sqrt{b x^{5} + a x^{2}} \sqrt{-a}}\right ) - \sqrt{b x^{5} + a x^{2}} a}{3 \, a^{2} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^5 + a*x^2)*x^3),x, algorithm="fricas")

[Out]

[1/6*(sqrt(a)*b*x^4*log(((b*x^4 + 2*a*x)*sqrt(a) + 2*sqrt(b*x^5 + a*x^2)*a)/x^4)
- 2*sqrt(b*x^5 + a*x^2)*a)/(a^2*x^4), 1/3*(sqrt(-a)*b*x^4*arctan(a*x/(sqrt(b*x^
5 + a*x^2)*sqrt(-a))) - sqrt(b*x^5 + a*x^2)*a)/(a^2*x^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \sqrt{x^{2} \left (a + b x^{3}\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**3*sqrt(x**2*(a + b*x**3))), x)

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GIAC/XCAS [A]  time = 0.234218, size = 77, normalized size = 1.31 \[ -\frac{b \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{3 \, \sqrt{-a} a{\rm sign}\left (x\right )} - \frac{\sqrt{\frac{b}{x} + \frac{a}{x^{4}}}}{3 \, a x{\rm sign}\left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^5 + a*x^2)*x^3),x, algorithm="giac")

[Out]

-1/3*b*arctan(sqrt(b*x^3 + a)/sqrt(-a))/(sqrt(-a)*a*sign(x)) - 1/3*sqrt(b/x + a/
x^4)/(a*x*sign(x))