Optimal. Leaf size=88 \[ -\frac{a^2 (A b-a B)}{6 b^4 \left (a+b x^3\right )^2}+\frac{a (2 A b-3 a B)}{3 b^4 \left (a+b x^3\right )}+\frac{(A b-3 a B) \log \left (a+b x^3\right )}{3 b^4}+\frac{B x^3}{3 b^3} \]
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Rubi [A] time = 0.25443, antiderivative size = 88, 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 \[ -\frac{a^2 (A b-a B)}{6 b^4 \left (a+b x^3\right )^2}+\frac{a (2 A b-3 a B)}{3 b^4 \left (a+b x^3\right )}+\frac{(A b-3 a B) \log \left (a+b x^3\right )}{3 b^4}+\frac{B x^3}{3 b^3} \]
Antiderivative was successfully verified.
[In] Int[(x^8*(A + B*x^3))/(a + b*x^3)^3,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2} \left (A b - B a\right )}{6 b^{4} \left (a + b x^{3}\right )^{2}} + \frac{a \left (2 A b - 3 B a\right )}{3 b^{4} \left (a + b x^{3}\right )} + \frac{\int ^{x^{3}} B\, dx}{3 b^{3}} + \frac{\left (A b - 3 B a\right ) \log{\left (a + b x^{3} \right )}}{3 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**8*(B*x**3+A)/(b*x**3+a)**3,x)
[Out]
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Mathematica [A] time = 0.0681161, size = 92, normalized size = 1.05 \[ \frac{2 a A b-3 a^2 B}{3 b^4 \left (a+b x^3\right )}+\frac{a^3 B-a^2 A b}{6 b^4 \left (a+b x^3\right )^2}+\frac{(A b-3 a B) \log \left (a+b x^3\right )}{3 b^4}+\frac{B x^3}{3 b^3} \]
Antiderivative was successfully verified.
[In] Integrate[(x^8*(A + B*x^3))/(a + b*x^3)^3,x]
[Out]
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Maple [A] time = 0.009, size = 110, normalized size = 1.3 \[{\frac{B{x}^{3}}{3\,{b}^{3}}}-{\frac{A{a}^{2}}{6\,{b}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{B{a}^{3}}{6\,{b}^{4} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{\ln \left ( b{x}^{3}+a \right ) A}{3\,{b}^{3}}}-{\frac{\ln \left ( b{x}^{3}+a \right ) Ba}{{b}^{4}}}+{\frac{2\,aA}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }}-{\frac{{a}^{2}B}{{b}^{4} \left ( b{x}^{3}+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^8*(B*x^3+A)/(b*x^3+a)^3,x)
[Out]
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Maxima [A] time = 1.37337, size = 127, normalized size = 1.44 \[ \frac{B x^{3}}{3 \, b^{3}} - \frac{5 \, B a^{3} - 3 \, A a^{2} b + 2 \,{\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{3}}{6 \,{\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} - \frac{{\left (3 \, B a - A b\right )} \log \left (b x^{3} + a\right )}{3 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^8/(b*x^3 + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.223666, size = 192, normalized size = 2.18 \[ \frac{2 \, B b^{3} x^{9} + 4 \, B a b^{2} x^{6} - 5 \, B a^{3} + 3 \, A a^{2} b - 4 \,{\left (B a^{2} b - A a b^{2}\right )} x^{3} - 2 \,{\left ({\left (3 \, B a b^{2} - A b^{3}\right )} x^{6} + 3 \, B a^{3} - A a^{2} b + 2 \,{\left (3 \, B a^{2} b - A a b^{2}\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{6 \,{\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^8/(b*x^3 + a)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 9.18242, size = 94, normalized size = 1.07 \[ \frac{B x^{3}}{3 b^{3}} - \frac{- 3 A a^{2} b + 5 B a^{3} + x^{3} \left (- 4 A a b^{2} + 6 B a^{2} b\right )}{6 a^{2} b^{4} + 12 a b^{5} x^{3} + 6 b^{6} x^{6}} - \frac{\left (- A b + 3 B a\right ) \log{\left (a + b x^{3} \right )}}{3 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**8*(B*x**3+A)/(b*x**3+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.218129, size = 126, normalized size = 1.43 \[ \frac{B x^{3}}{3 \, b^{3}} - \frac{{\left (3 \, B a - A b\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{4}} + \frac{9 \, B a b^{2} x^{6} - 3 \, A b^{3} x^{6} + 12 \, B a^{2} b x^{3} - 2 \, A a b^{2} x^{3} + 4 \, B a^{3}}{6 \,{\left (b x^{3} + a\right )}^{2} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^8/(b*x^3 + a)^3,x, algorithm="giac")
[Out]