Optimal. Leaf size=60 \[ \frac{a (A b-a B)}{3 b^3 \left (a+b x^3\right )}+\frac{(A b-2 a B) \log \left (a+b x^3\right )}{3 b^3}+\frac{B x^3}{3 b^2} \]
[Out]
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Rubi [A] time = 0.178759, antiderivative size = 60, 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 \[ \frac{a (A b-a B)}{3 b^3 \left (a+b x^3\right )}+\frac{(A b-2 a B) \log \left (a+b x^3\right )}{3 b^3}+\frac{B x^3}{3 b^2} \]
Antiderivative was successfully verified.
[In] Int[(x^5*(A + B*x^3))/(a + b*x^3)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{a \left (A b - B a\right )}{3 b^{3} \left (a + b x^{3}\right )} + \frac{\int ^{x^{3}} B\, dx}{3 b^{2}} + \frac{\left (A b - 2 B a\right ) \log{\left (a + b x^{3} \right )}}{3 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5*(B*x**3+A)/(b*x**3+a)**2,x)
[Out]
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Mathematica [A] time = 0.0661222, size = 50, normalized size = 0.83 \[ \frac{\frac{a (A b-a B)}{a+b x^3}+(A b-2 a B) \log \left (a+b x^3\right )+b B x^3}{3 b^3} \]
Antiderivative was successfully verified.
[In] Integrate[(x^5*(A + B*x^3))/(a + b*x^3)^2,x]
[Out]
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Maple [A] time = 0.009, size = 74, normalized size = 1.2 \[{\frac{B{x}^{3}}{3\,{b}^{2}}}+{\frac{\ln \left ( b{x}^{3}+a \right ) A}{3\,{b}^{2}}}-{\frac{2\,\ln \left ( b{x}^{3}+a \right ) Ba}{3\,{b}^{3}}}+{\frac{aA}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{{a}^{2}B}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5*(B*x^3+A)/(b*x^3+a)^2,x)
[Out]
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Maxima [A] time = 1.37334, size = 81, normalized size = 1.35 \[ \frac{B x^{3}}{3 \, b^{2}} - \frac{B a^{2} - A a b}{3 \,{\left (b^{4} x^{3} + a b^{3}\right )}} - \frac{{\left (2 \, B a - A b\right )} \log \left (b x^{3} + a\right )}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^5/(b*x^3 + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.223549, size = 109, normalized size = 1.82 \[ \frac{B b^{2} x^{6} + B a b x^{3} - B a^{2} + A a b -{\left ({\left (2 \, B a b - A b^{2}\right )} x^{3} + 2 \, B a^{2} - A a b\right )} \log \left (b x^{3} + a\right )}{3 \,{\left (b^{4} x^{3} + a b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^5/(b*x^3 + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.83167, size = 56, normalized size = 0.93 \[ \frac{B x^{3}}{3 b^{2}} - \frac{- A a b + B a^{2}}{3 a b^{3} + 3 b^{4} x^{3}} - \frac{\left (- A b + 2 B a\right ) \log{\left (a + b x^{3} \right )}}{3 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5*(B*x**3+A)/(b*x**3+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.217873, size = 123, normalized size = 2.05 \[ \frac{\frac{{\left (b x^{3} + a\right )} B}{b^{2}} + \frac{{\left (2 \, B a - A b\right )}{\rm ln}\left (\frac{{\left | b x^{3} + a \right |}}{{\left (b x^{3} + a\right )}^{2}{\left | b \right |}}\right )}{b^{2}} - \frac{\frac{B a^{2} b}{b x^{3} + a} - \frac{A a b^{2}}{b x^{3} + a}}{b^{3}}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^5/(b*x^3 + a)^2,x, algorithm="giac")
[Out]