Optimal. Leaf size=82 \[ -\frac{a^2 (A b-a B)}{3 b^4 \left (a+b x^3\right )}-\frac{a (2 A b-3 a B) \log \left (a+b x^3\right )}{3 b^4}+\frac{x^3 (A b-2 a B)}{3 b^3}+\frac{B x^6}{6 b^2} \]
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Rubi [A] time = 0.252742, antiderivative size = 82, 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 \[ -\frac{a^2 (A b-a B)}{3 b^4 \left (a+b x^3\right )}-\frac{a (2 A b-3 a B) \log \left (a+b x^3\right )}{3 b^4}+\frac{x^3 (A b-2 a B)}{3 b^3}+\frac{B x^6}{6 b^2} \]
Antiderivative was successfully verified.
[In] Int[(x^8*(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. \[ \frac{B \int ^{x^{3}} x\, dx}{3 b^{2}} - \frac{a^{2} \left (A b - B a\right )}{3 b^{4} \left (a + b x^{3}\right )} - \frac{a \left (2 A b - 3 B a\right ) \log{\left (a + b x^{3} \right )}}{3 b^{4}} + \left (\frac{A b}{3} - \frac{2 B a}{3}\right ) \int ^{x^{3}} \frac{1}{b^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**8*(B*x**3+A)/(b*x**3+a)**2,x)
[Out]
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Mathematica [A] time = 0.130167, size = 72, normalized size = 0.88 \[ \frac{\frac{2 a^2 (a B-A b)}{a+b x^3}+2 b x^3 (A b-2 a B)+2 a (3 a B-2 A b) \log \left (a+b x^3\right )+b^2 B x^6}{6 b^4} \]
Antiderivative was successfully verified.
[In] Integrate[(x^8*(A + B*x^3))/(a + b*x^3)^2,x]
[Out]
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Maple [A] time = 0.008, size = 97, normalized size = 1.2 \[{\frac{B{x}^{6}}{6\,{b}^{2}}}+{\frac{A{x}^{3}}{3\,{b}^{2}}}-{\frac{2\,B{x}^{3}a}{3\,{b}^{3}}}-{\frac{2\,a\ln \left ( b{x}^{3}+a \right ) A}{3\,{b}^{3}}}+{\frac{{a}^{2}\ln \left ( b{x}^{3}+a \right ) B}{{b}^{4}}}-{\frac{A{a}^{2}}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }}+{\frac{B{a}^{3}}{3\,{b}^{4} \left ( b{x}^{3}+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^8*(B*x^3+A)/(b*x^3+a)^2,x)
[Out]
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Maxima [A] time = 1.45782, size = 111, normalized size = 1.35 \[ \frac{B a^{3} - A a^{2} b}{3 \,{\left (b^{5} x^{3} + a b^{4}\right )}} + \frac{B b x^{6} - 2 \,{\left (2 \, B a - A b\right )} x^{3}}{6 \, b^{3}} + \frac{{\left (3 \, B a^{2} - 2 \, A a b\right )} \log \left (b x^{3} + a\right )}{3 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^8/(b*x^3 + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.226716, size = 163, normalized size = 1.99 \[ \frac{B b^{3} x^{9} -{\left (3 \, B a b^{2} - 2 \, A b^{3}\right )} x^{6} + 2 \, B a^{3} - 2 \, A a^{2} b - 2 \,{\left (2 \, B a^{2} b - A a b^{2}\right )} x^{3} + 2 \,{\left (3 \, B a^{3} - 2 \, A a^{2} b +{\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{6 \,{\left (b^{5} x^{3} + a b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^8/(b*x^3 + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.22927, size = 78, normalized size = 0.95 \[ \frac{B x^{6}}{6 b^{2}} + \frac{a \left (- 2 A b + 3 B a\right ) \log{\left (a + b x^{3} \right )}}{3 b^{4}} + \frac{- A a^{2} b + B a^{3}}{3 a b^{4} + 3 b^{5} x^{3}} - \frac{x^{3} \left (- A b + 2 B a\right )}{3 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**8*(B*x**3+A)/(b*x**3+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.218305, size = 143, normalized size = 1.74 \[ \frac{{\left (3 \, B a^{2} - 2 \, A a b\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{4}} + \frac{B b^{2} x^{6} - 4 \, B a b x^{3} + 2 \, A b^{2} x^{3}}{6 \, b^{4}} - \frac{3 \, B a^{2} b x^{3} - 2 \, A a b^{2} x^{3} + 2 \, B a^{3} - A a^{2} b}{3 \,{\left (b x^{3} + a\right )} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^8/(b*x^3 + a)^2,x, algorithm="giac")
[Out]