Optimal. Leaf size=122 \[ -\frac{a+b x^3}{3 a x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{b \log (x) \left (a+b x^3\right )}{a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{b \left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}} \]
[Out]
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Rubi [A] time = 0.124639, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{a+b x^3}{3 a x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{b \log (x) \left (a+b x^3\right )}{a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{b \left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^4*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. \[ \int \frac{1}{x^{4} \sqrt{\left (a + b x^{3}\right )^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**4/((b*x**3+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0280344, size = 54, normalized size = 0.44 \[ -\frac{\left (a+b x^3\right ) \left (-b x^3 \log \left (a+b x^3\right )+a+3 b x^3 \log (x)\right )}{3 a^2 x^3 \sqrt{\left (a+b x^3\right )^2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]),x]
[Out]
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Maple [A] time = 0.018, size = 52, normalized size = 0.4 \[{\frac{ \left ( b{x}^{3}+a \right ) \left ( b\ln \left ( b{x}^{3}+a \right ){x}^{3}-3\,b\ln \left ( x \right ){x}^{3}-a \right ) }{3\,{a}^{2}{x}^{3}}{\frac{1}{\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^4/((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^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.260216, size = 45, normalized size = 0.37 \[ \frac{b x^{3} \log \left (b x^{3} + a\right ) - 3 \, b x^{3} \log \left (x\right ) - a}{3 \, a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt((b*x^3 + a)^2)*x^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.94945, size = 31, normalized size = 0.25 \[ - \frac{1}{3 a x^{3}} - \frac{b \log{\left (x \right )}}{a^{2}} + \frac{b \log{\left (\frac{a}{b} + x^{3} \right )}}{3 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**4/((b*x**3+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.279935, size = 68, normalized size = 0.56 \[ \frac{1}{3} \,{\left (\frac{b{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{a^{2}} - \frac{3 \, b{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} + \frac{b x^{3} - a}{a^{2} x^{3}}\right )}{\rm sign}\left (b x^{3} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt((b*x^3 + a)^2)*x^4),x, algorithm="giac")
[Out]