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