3.409 \(\int \frac{1}{x^4 \sqrt{a+b x^3}} \, dx\)

Optimal. Leaf size=50 \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{\sqrt{a+b x^3}}{3 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.0751714, antiderivative size = 50, 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 \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{\sqrt{a+b x^3}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*Sqrt[a + b*x^3]),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.11963, size = 41, normalized size = 0.82 \[ - \frac{\sqrt{a + b x^{3}}}{3 a x^{3}} + \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{3 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.16101, size = 56, normalized size = 1.12 \[ \frac{\sqrt{a+b x^3} \left (\frac{b \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{\sqrt{\frac{b x^3}{a}+1}}-\frac{a}{x^3}\right )}{3 a^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*Sqrt[a + b*x^3]),x]

[Out]

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

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Maple [A]  time = 0.028, size = 39, normalized size = 0.8 \[{\frac{b}{3}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{3}{2}}}}-{\frac{1}{3\,a{x}^{3}}\sqrt{b{x}^{3}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x^3+a)^(1/2),x)

[Out]

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

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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^3 + a)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245458, size = 1, normalized size = 0.02 \[ \left [\frac{b x^{3} \log \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{3} + a} a}{x^{3}}\right ) - 2 \, \sqrt{b x^{3} + a} \sqrt{a}}{6 \, a^{\frac{3}{2}} x^{3}}, -\frac{b x^{3} \arctan \left (\frac{a}{\sqrt{b x^{3} + a} \sqrt{-a}}\right ) + \sqrt{b x^{3} + a} \sqrt{-a}}{3 \, \sqrt{-a} a x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^3 + a)*x^4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 8.07056, size = 49, normalized size = 0.98 \[ - \frac{\sqrt{b} \sqrt{\frac{a}{b x^{3}} + 1}}{3 a x^{\frac{3}{2}}} + \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{3 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(b)*sqrt(a/(b*x**3) + 1)/(3*a*x**(3/2)) + b*asinh(sqrt(a)/(sqrt(b)*x**(3/2)
))/(3*a**(3/2))

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GIAC/XCAS [A]  time = 0.229392, size = 65, normalized size = 1.3 \[ -\frac{1}{3} \, b{\left (\frac{\arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{\sqrt{b x^{3} + a}}{a b x^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^3 + a)*x^4),x, algorithm="giac")

[Out]

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