Optimal. Leaf size=96 \[ \frac{x^3 \left (a^2 f-a b e+b^2 d\right )}{3 b^3}+\frac{\log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^4}+\frac{x^6 (b e-a f)}{6 b^2}+\frac{f x^9}{9 b} \]
[Out]
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Rubi [A] time = 0.278955, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{x^3 \left (a^2 f-a b e+b^2 d\right )}{3 b^3}+\frac{\log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^4}+\frac{x^6 (b e-a f)}{6 b^2}+\frac{f x^9}{9 b} \]
Antiderivative was successfully verified.
[In] Int[(x^2*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \left (\frac{a^{2} f}{3} - \frac{a b e}{3} + \frac{b^{2} d}{3}\right ) \int ^{x^{3}} \frac{1}{b^{3}}\, dx + \frac{f x^{9}}{9 b} - \frac{\left (a f - b e\right ) \int ^{x^{3}} x\, dx}{3 b^{2}} - \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a + b x^{3} \right )}}{3 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a),x)
[Out]
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Mathematica [A] time = 0.0757291, size = 88, normalized size = 0.92 \[ \frac{b x^3 \left (6 a^2 f-3 a b \left (2 e+f x^3\right )+b^2 \left (6 d+3 e x^3+2 f x^6\right )\right )+6 \log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{18 b^4} \]
Antiderivative was successfully verified.
[In] Integrate[(x^2*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]
[Out]
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Maple [A] time = 0.004, size = 124, normalized size = 1.3 \[{\frac{f{x}^{9}}{9\,b}}-{\frac{{x}^{6}af}{6\,{b}^{2}}}+{\frac{e{x}^{6}}{6\,b}}+{\frac{{a}^{2}f{x}^{3}}{3\,{b}^{3}}}-{\frac{ae{x}^{3}}{3\,{b}^{2}}}+{\frac{d{x}^{3}}{3\,b}}-{\frac{\ln \left ( b{x}^{3}+a \right ){a}^{3}f}{3\,{b}^{4}}}+{\frac{\ln \left ( b{x}^{3}+a \right ){a}^{2}e}{3\,{b}^{3}}}-{\frac{\ln \left ( b{x}^{3}+a \right ) ad}{3\,{b}^{2}}}+{\frac{c\ln \left ( b{x}^{3}+a \right ) }{3\,b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x)
[Out]
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Maxima [A] time = 1.3992, size = 123, normalized size = 1.28 \[ \frac{2 \, b^{2} f x^{9} + 3 \,{\left (b^{2} e - a b f\right )} x^{6} + 6 \,{\left (b^{2} d - a b e + a^{2} f\right )} x^{3}}{18 \, b^{3}} + \frac{{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^2/(b*x^3 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.212525, size = 124, normalized size = 1.29 \[ \frac{2 \, b^{3} f x^{9} + 3 \,{\left (b^{3} e - a b^{2} f\right )} x^{6} + 6 \,{\left (b^{3} d - a b^{2} e + a^{2} b f\right )} x^{3} + 6 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (b x^{3} + a\right )}{18 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^2/(b*x^3 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.05452, size = 83, normalized size = 0.86 \[ \frac{f x^{9}}{9 b} - \frac{x^{6} \left (a f - b e\right )}{6 b^{2}} + \frac{x^{3} \left (a^{2} f - a b e + b^{2} d\right )}{3 b^{3}} - \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a + b x^{3} \right )}}{3 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a),x)
[Out]
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GIAC/XCAS [A] time = 0.213935, size = 136, normalized size = 1.42 \[ \frac{2 \, b^{2} f x^{9} - 3 \, a b f x^{6} + 3 \, b^{2} x^{6} e + 6 \, b^{2} d x^{3} + 6 \, a^{2} f x^{3} - 6 \, a b x^{3} e}{18 \, b^{3}} + \frac{{\left (b^{3} c - a b^{2} d - a^{3} f + a^{2} b e\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^2/(b*x^3 + a),x, algorithm="giac")
[Out]