Optimal. Leaf size=170 \[ \frac{x^9 \left (a^2 f-a b e+b^2 d\right )}{9 b^3}+\frac{a^2 \log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^6}-\frac{a x^3 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^5}+\frac{x^6 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4}+\frac{x^{12} (b e-a f)}{12 b^2}+\frac{f x^{15}}{15 b} \]
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Rubi [A] time = 0.472533, antiderivative size = 170, 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 \[ \frac{x^9 \left (a^2 f-a b e+b^2 d\right )}{9 b^3}+\frac{a^2 \log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^6}-\frac{a x^3 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^5}+\frac{x^6 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4}+\frac{x^{12} (b e-a f)}{12 b^2}+\frac{f x^{15}}{15 b} \]
Antiderivative was successfully verified.
[In] Int[(x^8*(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. \[ - \frac{a^{2} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a + b x^{3} \right )}}{3 b^{6}} + \frac{f x^{15}}{15 b} - \frac{x^{12} \left (a f - b e\right )}{12 b^{2}} + \frac{x^{9} \left (a^{2} f - a b e + b^{2} d\right )}{9 b^{3}} - \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \int ^{x^{3}} x\, dx}{3 b^{4}} + \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \int ^{x^{3}} a\, dx}{3 b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**8*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a),x)
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Mathematica [A] time = 0.137552, size = 154, normalized size = 0.91 \[ \frac{b x^3 \left (60 a^4 f-30 a^3 b \left (2 e+f x^3\right )+10 a^2 b^2 \left (6 d+3 e x^3+2 f x^6\right )-5 a b^3 \left (12 c+6 d x^3+4 e x^6+3 f x^9\right )+b^4 x^3 \left (30 c+20 d x^3+15 e x^6+12 f x^9\right )\right )-60 a^2 \log \left (a+b x^3\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{180 b^6} \]
Antiderivative was successfully verified.
[In] Integrate[(x^8*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]
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Maple [A] time = 0.006, size = 218, normalized size = 1.3 \[{\frac{f{x}^{15}}{15\,b}}-{\frac{{x}^{12}af}{12\,{b}^{2}}}+{\frac{{x}^{12}e}{12\,b}}+{\frac{{x}^{9}{a}^{2}f}{9\,{b}^{3}}}-{\frac{{x}^{9}ae}{9\,{b}^{2}}}+{\frac{{x}^{9}d}{9\,b}}-{\frac{{a}^{3}f{x}^{6}}{6\,{b}^{4}}}+{\frac{{a}^{2}e{x}^{6}}{6\,{b}^{3}}}-{\frac{ad{x}^{6}}{6\,{b}^{2}}}+{\frac{{x}^{6}c}{6\,b}}+{\frac{{a}^{4}f{x}^{3}}{3\,{b}^{5}}}-{\frac{{a}^{3}e{x}^{3}}{3\,{b}^{4}}}+{\frac{{a}^{2}d{x}^{3}}{3\,{b}^{3}}}-{\frac{ac{x}^{3}}{3\,{b}^{2}}}-{\frac{{a}^{5}\ln \left ( b{x}^{3}+a \right ) f}{3\,{b}^{6}}}+{\frac{{a}^{4}\ln \left ( b{x}^{3}+a \right ) e}{3\,{b}^{5}}}-{\frac{{a}^{3}\ln \left ( b{x}^{3}+a \right ) d}{3\,{b}^{4}}}+{\frac{{a}^{2}\ln \left ( b{x}^{3}+a \right ) c}{3\,{b}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^8*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x)
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Maxima [A] time = 1.45629, size = 228, normalized size = 1.34 \[ \frac{12 \, b^{4} f x^{15} + 15 \,{\left (b^{4} e - a b^{3} f\right )} x^{12} + 20 \,{\left (b^{4} d - a b^{3} e + a^{2} b^{2} f\right )} x^{9} + 30 \,{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{6} - 60 \,{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{3}}{180 \, b^{5}} + \frac{{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^8/(b*x^3 + a),x, algorithm="maxima")
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Fricas [A] time = 0.227734, size = 230, normalized size = 1.35 \[ \frac{12 \, b^{5} f x^{15} + 15 \,{\left (b^{5} e - a b^{4} f\right )} x^{12} + 20 \,{\left (b^{5} d - a b^{4} e + a^{2} b^{3} f\right )} x^{9} + 30 \,{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{6} - 60 \,{\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{3} + 60 \,{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} \log \left (b x^{3} + a\right )}{180 \, b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^8/(b*x^3 + a),x, algorithm="fricas")
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Sympy [A] time = 2.32156, size = 155, normalized size = 0.91 \[ - \frac{a^{2} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a + b x^{3} \right )}}{3 b^{6}} + \frac{f x^{15}}{15 b} - \frac{x^{12} \left (a f - b e\right )}{12 b^{2}} + \frac{x^{9} \left (a^{2} f - a b e + b^{2} d\right )}{9 b^{3}} - \frac{x^{6} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{6 b^{4}} + \frac{x^{3} \left (a^{4} f - a^{3} b e + a^{2} b^{2} d - a b^{3} c\right )}{3 b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**8*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a),x)
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GIAC/XCAS [A] time = 0.214389, size = 266, normalized size = 1.56 \[ \frac{12 \, b^{4} f x^{15} - 15 \, a b^{3} f x^{12} + 15 \, b^{4} x^{12} e + 20 \, b^{4} d x^{9} + 20 \, a^{2} b^{2} f x^{9} - 20 \, a b^{3} x^{9} e + 30 \, b^{4} c x^{6} - 30 \, a b^{3} d x^{6} - 30 \, a^{3} b f x^{6} + 30 \, a^{2} b^{2} x^{6} e - 60 \, a b^{3} c x^{3} + 60 \, a^{2} b^{2} d x^{3} + 60 \, a^{4} f x^{3} - 60 \, a^{3} b x^{3} e}{180 \, b^{5}} + \frac{{\left (a^{2} b^{3} c - a^{3} b^{2} d - a^{5} f + a^{4} b e\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^8/(b*x^3 + a),x, algorithm="giac")
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