Optimal. Leaf size=76 \[ \frac{(2 A b-a B) \log \left (a+b x^3\right )}{3 a^3}-\frac{\log (x) (2 A b-a B)}{a^3}-\frac{A b-a B}{3 a^2 \left (a+b x^3\right )}-\frac{A}{3 a^2 x^3} \]
[Out]
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Rubi [A] time = 0.203712, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{(2 A b-a B) \log \left (a+b x^3\right )}{3 a^3}-\frac{\log (x) (2 A b-a B)}{a^3}-\frac{A b-a B}{3 a^2 \left (a+b x^3\right )}-\frac{A}{3 a^2 x^3} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^3)/(x^4*(a + b*x^3)^2),x]
[Out]
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Rubi in Sympy [A] time = 18.8255, size = 68, normalized size = 0.89 \[ - \frac{A}{3 a^{2} x^{3}} - \frac{A b - B a}{3 a^{2} \left (a + b x^{3}\right )} - \frac{\left (2 A b - B a\right ) \log{\left (x^{3} \right )}}{3 a^{3}} + \frac{\left (2 A b - B a\right ) \log{\left (a + b x^{3} \right )}}{3 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**3+A)/x**4/(b*x**3+a)**2,x)
[Out]
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Mathematica [A] time = 0.0882648, size = 64, normalized size = 0.84 \[ \frac{\frac{a (a B-A b)}{a+b x^3}+(2 A b-a B) \log \left (a+b x^3\right )+3 \log (x) (a B-2 A b)-\frac{a A}{x^3}}{3 a^3} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^3)/(x^4*(a + b*x^3)^2),x]
[Out]
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Maple [A] time = 0.015, size = 87, normalized size = 1.1 \[ -{\frac{A}{3\,{a}^{2}{x}^{3}}}-2\,{\frac{A\ln \left ( x \right ) b}{{a}^{3}}}+{\frac{B\ln \left ( x \right ) }{{a}^{2}}}+{\frac{2\,b\ln \left ( b{x}^{3}+a \right ) A}{3\,{a}^{3}}}-{\frac{\ln \left ( b{x}^{3}+a \right ) B}{3\,{a}^{2}}}-{\frac{Ab}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}+{\frac{B}{3\,a \left ( b{x}^{3}+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^3+A)/x^4/(b*x^3+a)^2,x)
[Out]
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Maxima [A] time = 1.37388, size = 103, normalized size = 1.36 \[ \frac{{\left (B a - 2 \, A b\right )} x^{3} - A a}{3 \,{\left (a^{2} b x^{6} + a^{3} x^{3}\right )}} - \frac{{\left (B a - 2 \, A b\right )} \log \left (b x^{3} + a\right )}{3 \, a^{3}} + \frac{{\left (B a - 2 \, A b\right )} \log \left (x^{3}\right )}{3 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)/((b*x^3 + a)^2*x^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.22906, size = 159, normalized size = 2.09 \[ \frac{{\left (B a^{2} - 2 \, A a b\right )} x^{3} - A a^{2} -{\left ({\left (B a b - 2 \, A b^{2}\right )} x^{6} +{\left (B a^{2} - 2 \, A a b\right )} x^{3}\right )} \log \left (b x^{3} + a\right ) + 3 \,{\left ({\left (B a b - 2 \, A b^{2}\right )} x^{6} +{\left (B a^{2} - 2 \, A a b\right )} x^{3}\right )} \log \left (x\right )}{3 \,{\left (a^{3} b x^{6} + a^{4} x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)/((b*x^3 + a)^2*x^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.5089, size = 70, normalized size = 0.92 \[ \frac{- A a + x^{3} \left (- 2 A b + B a\right )}{3 a^{3} x^{3} + 3 a^{2} b x^{6}} + \frac{\left (- 2 A b + B a\right ) \log{\left (x \right )}}{a^{3}} - \frac{\left (- 2 A b + B a\right ) \log{\left (\frac{a}{b} + x^{3} \right )}}{3 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**3+A)/x**4/(b*x**3+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.218866, size = 108, normalized size = 1.42 \[ \frac{{\left (B a - 2 \, A b\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} + \frac{B a x^{3} - 2 \, A b x^{3} - A a}{3 \,{\left (b x^{6} + a x^{3}\right )} a^{2}} - \frac{{\left (B a b - 2 \, A b^{2}\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)/((b*x^3 + a)^2*x^4),x, algorithm="giac")
[Out]