Optimal. Leaf size=107 \[ \frac{a^3 (A b-a B)}{6 b^5 \left (a+b x^3\right )^2}-\frac{a^2 (3 A b-4 a B)}{3 b^5 \left (a+b x^3\right )}-\frac{a (A b-2 a B) \log \left (a+b x^3\right )}{b^5}+\frac{x^3 (A b-3 a B)}{3 b^4}+\frac{B x^6}{6 b^3} \]
[Out]
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Rubi [A] time = 0.346, antiderivative size = 107, 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 \[ \frac{a^3 (A b-a B)}{6 b^5 \left (a+b x^3\right )^2}-\frac{a^2 (3 A b-4 a B)}{3 b^5 \left (a+b x^3\right )}-\frac{a (A b-2 a B) \log \left (a+b x^3\right )}{b^5}+\frac{x^3 (A b-3 a B)}{3 b^4}+\frac{B x^6}{6 b^3} \]
Antiderivative was successfully verified.
[In] Int[(x^11*(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. \[ \frac{B \int ^{x^{3}} x\, dx}{3 b^{3}} + \frac{a^{3} \left (A b - B a\right )}{6 b^{5} \left (a + b x^{3}\right )^{2}} - \frac{a^{2} \left (3 A b - 4 B a\right )}{3 b^{5} \left (a + b x^{3}\right )} - \frac{a \left (A b - 2 B a\right ) \log{\left (a + b x^{3} \right )}}{b^{5}} + \left (\frac{A b}{3} - B a\right ) \int ^{x^{3}} \frac{1}{b^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**11*(B*x**3+A)/(b*x**3+a)**3,x)
[Out]
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Mathematica [A] time = 0.131595, size = 94, normalized size = 0.88 \[ \frac{\frac{a^3 (A b-a B)}{\left (a+b x^3\right )^2}+\frac{2 a^2 (4 a B-3 A b)}{a+b x^3}+2 b x^3 (A b-3 a B)+6 a (2 a B-A b) \log \left (a+b x^3\right )+b^2 B x^6}{6 b^5} \]
Antiderivative was successfully verified.
[In] Integrate[(x^11*(A + B*x^3))/(a + b*x^3)^3,x]
[Out]
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Maple [A] time = 0.012, size = 134, normalized size = 1.3 \[{\frac{B{x}^{6}}{6\,{b}^{3}}}+{\frac{A{x}^{3}}{3\,{b}^{3}}}-{\frac{B{x}^{3}a}{{b}^{4}}}+{\frac{A{a}^{3}}{6\,{b}^{4} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{B{a}^{4}}{6\,{b}^{5} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{a\ln \left ( b{x}^{3}+a \right ) A}{{b}^{4}}}+2\,{\frac{{a}^{2}\ln \left ( b{x}^{3}+a \right ) B}{{b}^{5}}}-{\frac{A{a}^{2}}{{b}^{4} \left ( b{x}^{3}+a \right ) }}+{\frac{4\,B{a}^{3}}{3\,{b}^{5} \left ( b{x}^{3}+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^11*(B*x^3+A)/(b*x^3+a)^3,x)
[Out]
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Maxima [A] time = 1.41061, size = 155, normalized size = 1.45 \[ \frac{7 \, B a^{4} - 5 \, A a^{3} b + 2 \,{\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{3}}{6 \,{\left (b^{7} x^{6} + 2 \, a b^{6} x^{3} + a^{2} b^{5}\right )}} + \frac{B b x^{6} - 2 \,{\left (3 \, B a - A b\right )} x^{3}}{6 \, b^{4}} + \frac{{\left (2 \, B a^{2} - A a b\right )} \log \left (b x^{3} + a\right )}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^11/(b*x^3 + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.220749, size = 242, normalized size = 2.26 \[ \frac{B b^{4} x^{12} - 2 \,{\left (2 \, B a b^{3} - A b^{4}\right )} x^{9} -{\left (11 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{6} + 7 \, B a^{4} - 5 \, A a^{3} b + 2 \,{\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{3} + 6 \,{\left ({\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} x^{6} + 2 \, B a^{4} - A a^{3} b + 2 \,{\left (2 \, B a^{3} b - A a^{2} b^{2}\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{6 \,{\left (b^{7} x^{6} + 2 \, a b^{6} x^{3} + a^{2} b^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^11/(b*x^3 + a)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 10.6876, size = 112, normalized size = 1.05 \[ \frac{B x^{6}}{6 b^{3}} + \frac{a \left (- A b + 2 B a\right ) \log{\left (a + b x^{3} \right )}}{b^{5}} + \frac{- 5 A a^{3} b + 7 B a^{4} + x^{3} \left (- 6 A a^{2} b^{2} + 8 B a^{3} b\right )}{6 a^{2} b^{5} + 12 a b^{6} x^{3} + 6 b^{7} x^{6}} - \frac{x^{3} \left (- A b + 3 B a\right )}{3 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**11*(B*x**3+A)/(b*x**3+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.221361, size = 177, normalized size = 1.65 \[ \frac{{\left (2 \, B a^{2} - A a b\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{b^{5}} + \frac{B b^{3} x^{6} - 6 \, B a b^{2} x^{3} + 2 \, A b^{3} x^{3}}{6 \, b^{6}} - \frac{18 \, B a^{2} b^{2} x^{6} - 9 \, A a b^{3} x^{6} + 28 \, B a^{3} b x^{3} - 12 \, A a^{2} b^{2} x^{3} + 11 \, B a^{4} - 4 \, A a^{3} b}{6 \,{\left (b x^{3} + a\right )}^{2} b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^11/(b*x^3 + a)^3,x, algorithm="giac")
[Out]