Optimal. Leaf size=66 \[ -\frac{A b-2 a B}{3 b^3 \left (a+b x^3\right )}+\frac{a (A b-a B)}{6 b^3 \left (a+b x^3\right )^2}+\frac{B \log \left (a+b x^3\right )}{3 b^3} \]
[Out]
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Rubi [A] time = 0.17726, antiderivative size = 66, 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 \[ -\frac{A b-2 a B}{3 b^3 \left (a+b x^3\right )}+\frac{a (A b-a B)}{6 b^3 \left (a+b x^3\right )^2}+\frac{B \log \left (a+b x^3\right )}{3 b^3} \]
Antiderivative was successfully verified.
[In] Int[(x^5*(A + B*x^3))/(a + b*x^3)^3,x]
[Out]
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Rubi in Sympy [A] time = 18.4061, size = 56, normalized size = 0.85 \[ \frac{B \log{\left (a + b x^{3} \right )}}{3 b^{3}} + \frac{a \left (A b - B a\right )}{6 b^{3} \left (a + b x^{3}\right )^{2}} - \frac{A b - 2 B a}{3 b^{3} \left (a + b x^{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5*(B*x**3+A)/(b*x**3+a)**3,x)
[Out]
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Mathematica [A] time = 0.0454318, size = 64, normalized size = 0.97 \[ \frac{3 a^2 B-a b \left (A-4 B x^3\right )+2 B \left (a+b x^3\right )^2 \log \left (a+b x^3\right )-2 A b^2 x^3}{6 b^3 \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
[In] Integrate[(x^5*(A + B*x^3))/(a + b*x^3)^3,x]
[Out]
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Maple [A] time = 0.008, size = 81, normalized size = 1.2 \[{\frac{aA}{6\,{b}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{{a}^{2}B}{6\,{b}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{B\ln \left ( b{x}^{3}+a \right ) }{3\,{b}^{3}}}-{\frac{A}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }}+{\frac{2\,Ba}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5*(B*x^3+A)/(b*x^3+a)^3,x)
[Out]
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Maxima [A] time = 1.40649, size = 97, normalized size = 1.47 \[ \frac{2 \,{\left (2 \, B a b - A b^{2}\right )} x^{3} + 3 \, B a^{2} - A a b}{6 \,{\left (b^{5} x^{6} + 2 \, a b^{4} x^{3} + a^{2} b^{3}\right )}} + \frac{B \log \left (b x^{3} + a\right )}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^5/(b*x^3 + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.22667, size = 120, normalized size = 1.82 \[ \frac{2 \,{\left (2 \, B a b - A b^{2}\right )} x^{3} + 3 \, B a^{2} - A a b + 2 \,{\left (B b^{2} x^{6} + 2 \, B a b x^{3} + B a^{2}\right )} \log \left (b x^{3} + a\right )}{6 \,{\left (b^{5} x^{6} + 2 \, a b^{4} x^{3} + a^{2} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^5/(b*x^3 + a)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 7.11545, size = 70, normalized size = 1.06 \[ \frac{B \log{\left (a + b x^{3} \right )}}{3 b^{3}} + \frac{- A a b + 3 B a^{2} + x^{3} \left (- 2 A b^{2} + 4 B a b\right )}{6 a^{2} b^{3} + 12 a b^{4} x^{3} + 6 b^{5} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5*(B*x**3+A)/(b*x**3+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.219721, size = 82, normalized size = 1.24 \[ \frac{B{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} + \frac{2 \,{\left (2 \, B a - A b\right )} x^{3} + \frac{3 \, B a^{2} - A a b}{b}}{6 \,{\left (b x^{3} + a\right )}^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^5/(b*x^3 + a)^3,x, algorithm="giac")
[Out]