Optimal. Leaf size=80 \[ \frac{\log (x) \left (a+b x^3\right )}{a \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a \sqrt{a^2+2 a b x^3+b^2 x^6}} \]
[Out]
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Rubi [A] time = 0.0961869, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{\log (x) \left (a+b x^3\right )}{a \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a \sqrt{a^2+2 a b x^3+b^2 x^6}} \]
Antiderivative was successfully verified.
[In] Int[1/(x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \sqrt{\left (a + b x^{3}\right )^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/((b*x**3+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0235559, size = 42, normalized size = 0.52 \[ \frac{\left (a+b x^3\right ) \left (3 \log (x)-\log \left (a+b x^3\right )\right )}{3 a \sqrt{\left (a+b x^3\right )^2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]),x]
[Out]
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Maple [A] time = 0.013, size = 37, normalized size = 0.5 \[ -{\frac{ \left ( b{x}^{3}+a \right ) \left ( \ln \left ( b{x}^{3}+a \right ) -3\,\ln \left ( x \right ) \right ) }{3\,a}{\frac{1}{\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/((b*x^3+a)^2)^(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)^2)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.274306, size = 24, normalized size = 0.3 \[ -\frac{\log \left (b x^{3} + a\right ) - 3 \, \log \left (x\right )}{3 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt((b*x^3 + a)^2)*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.721008, size = 15, normalized size = 0.19 \[ \frac{\log{\left (x \right )}}{a} - \frac{\log{\left (\frac{a}{b} + x^{3} \right )}}{3 a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/((b*x**3+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.277561, size = 43, normalized size = 0.54 \[ -\frac{1}{3} \,{\left (\frac{{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{a} - \frac{3 \,{\rm ln}\left ({\left | x \right |}\right )}{a}\right )}{\rm sign}\left (b x^{3} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt((b*x^3 + a)^2)*x),x, algorithm="giac")
[Out]