Optimal. Leaf size=52 \[ \frac{2 b \log \left (a+b x^3\right )}{3 a^3}-\frac{2 b \log (x)}{a^3}-\frac{b}{3 a^2 \left (a+b x^3\right )}-\frac{1}{3 a^2 x^3} \]
[Out]
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Rubi [A] time = 0.0788349, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 b \log \left (a+b x^3\right )}{3 a^3}-\frac{2 b \log (x)}{a^3}-\frac{b}{3 a^2 \left (a+b x^3\right )}-\frac{1}{3 a^2 x^3} \]
Antiderivative was successfully verified.
[In] Int[1/(x^4*(a + b*x^3)^2),x]
[Out]
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Rubi in Sympy [A] time = 11.5762, size = 53, normalized size = 1.02 \[ - \frac{b}{3 a^{2} \left (a + b x^{3}\right )} - \frac{1}{3 a^{2} x^{3}} - \frac{2 b \log{\left (x^{3} \right )}}{3 a^{3}} + \frac{2 b \log{\left (a + b x^{3} \right )}}{3 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**4/(b*x**3+a)**2,x)
[Out]
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Mathematica [A] time = 0.0847641, size = 41, normalized size = 0.79 \[ -\frac{a \left (\frac{b}{a+b x^3}+\frac{1}{x^3}\right )-2 b \log \left (a+b x^3\right )+6 b \log (x)}{3 a^3} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^4*(a + b*x^3)^2),x]
[Out]
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Maple [A] time = 0.013, size = 47, normalized size = 0.9 \[ -{\frac{1}{3\,{x}^{3}{a}^{2}}}-{\frac{b}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}-2\,{\frac{b\ln \left ( x \right ) }{{a}^{3}}}+{\frac{2\,b\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^4/(b*x^3+a)^2,x)
[Out]
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Maxima [A] time = 1.45801, size = 72, normalized size = 1.38 \[ -\frac{2 \, b x^{3} + a}{3 \,{\left (a^{2} b x^{6} + a^{3} x^{3}\right )}} + \frac{2 \, b \log \left (b x^{3} + a\right )}{3 \, a^{3}} - \frac{2 \, b \log \left (x^{3}\right )}{3 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^3 + a)^2*x^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.238807, size = 99, normalized size = 1.9 \[ -\frac{2 \, a b x^{3} + a^{2} - 2 \,{\left (b^{2} x^{6} + a b x^{3}\right )} \log \left (b x^{3} + a\right ) + 6 \,{\left (b^{2} x^{6} + a b x^{3}\right )} \log \left (x\right )}{3 \,{\left (a^{3} b x^{6} + a^{4} x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^3 + a)^2*x^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.13246, size = 53, normalized size = 1.02 \[ - \frac{a + 2 b x^{3}}{3 a^{3} x^{3} + 3 a^{2} b x^{6}} - \frac{2 b \log{\left (x \right )}}{a^{3}} + \frac{2 b \log{\left (\frac{a}{b} + x^{3} \right )}}{3 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**4/(b*x**3+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.220423, size = 69, normalized size = 1.33 \[ \frac{2 \, b{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} - \frac{2 \, b{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} - \frac{2 \, b x^{3} + a}{3 \,{\left (b x^{6} + a x^{3}\right )} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^3 + a)^2*x^4),x, algorithm="giac")
[Out]