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

Optimal. Leaf size=47 \[ -\frac{\sqrt{a+b x^3}}{3 x^3}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 \sqrt{a}} \]

[Out]

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

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Rubi [A]  time = 0.0718749, antiderivative size = 47, 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 \[ -\frac{\sqrt{a+b x^3}}{3 x^3}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x^3]/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 7.28328, size = 41, normalized size = 0.87 \[ - \frac{\sqrt{a + b x^{3}}}{3 x^{3}} - \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{3 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.105519, size = 59, normalized size = 1.26 \[ -\frac{b x^3 \sqrt{\frac{b x^3}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )+a+b x^3}{3 x^3 \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.026, size = 36, normalized size = 0.8 \[ -{\frac{b}{3}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a}}}}-{\frac{1}{3\,{x}^{3}}\sqrt{b{x}^{3}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^(1/2)/x^4,x)

[Out]

-1/3*b*arctanh((b*x^3+a)^(1/2)/a^(1/2))/a^(1/2)-1/3*(b*x^3+a)^(1/2)/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(sqrt(b*x^3 + a)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225095, size = 1, normalized size = 0.02 \[ \left [\frac{b x^{3} \log \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{3} + a} a}{x^{3}}\right ) - 2 \, \sqrt{b x^{3} + a} \sqrt{a}}{6 \, \sqrt{a} x^{3}}, \frac{b x^{3} \arctan \left (\frac{a}{\sqrt{b x^{3} + a} \sqrt{-a}}\right ) - \sqrt{b x^{3} + a} \sqrt{-a}}{3 \, \sqrt{-a} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(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))/(sqrt(a)*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)*x^3)]

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Sympy [A]  time = 6.70303, size = 49, normalized size = 1.04 \[ - \frac{\sqrt{b} \sqrt{\frac{a}{b x^{3}} + 1}}{3 x^{\frac{3}{2}}} - \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{3 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.260943, size = 58, normalized size = 1.23 \[ \frac{1}{3} \, b{\left (\frac{\arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{\sqrt{b x^{3} + a}}{b x^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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