Optimal. Leaf size=205 \[ \frac{b c-a d}{12 a^2 x^{12}}-\frac{a^2 e-a b d+b^2 c}{9 a^3 x^9}+\frac{b^2 \log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^6}-\frac{b^2 \log (x) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^6}-\frac{b \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^5 x^3}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{6 a^4 x^6}-\frac{c}{15 a x^{15}} \]
[Out]
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Rubi [A] time = 0.44397, antiderivative size = 205, 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 \[ \frac{b c-a d}{12 a^2 x^{12}}-\frac{a^2 e-a b d+b^2 c}{9 a^3 x^9}+\frac{b^2 \log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^6}-\frac{b^2 \log (x) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^6}-\frac{b \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^5 x^3}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{6 a^4 x^6}-\frac{c}{15 a x^{15}} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^16*(a + b*x^3)),x]
[Out]
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Rubi in Sympy [A] time = 67.1214, size = 190, normalized size = 0.93 \[ - \frac{c}{15 a x^{15}} - \frac{a d - b c}{12 a^{2} x^{12}} - \frac{a^{2} e - a b d + b^{2} c}{9 a^{3} x^{9}} - \frac{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c}{6 a^{4} x^{6}} + \frac{b \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{3 a^{5} x^{3}} + \frac{b^{2} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (x^{3} \right )}}{3 a^{6}} - \frac{b^{2} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a + b x^{3} \right )}}{3 a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x**9+e*x**6+d*x**3+c)/x**16/(b*x**3+a),x)
[Out]
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Mathematica [A] time = 0.407158, size = 194, normalized size = 0.95 \[ -\frac{-60 b^2 \log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )+180 b^2 \log (x) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )+\frac{a \left (a^4 \left (12 c+15 d x^3+20 e x^6+30 f x^9\right )-5 a^3 b x^3 \left (3 c+4 d x^3+6 e x^6+12 f x^9\right )+10 a^2 b^2 x^6 \left (2 c+3 d x^3+6 e x^6\right )-30 a b^3 x^9 \left (c+2 d x^3\right )+60 b^4 c x^{12}\right )}{x^{15}}}{180 a^6} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^16*(a + b*x^3)),x]
[Out]
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Maple [A] time = 0.013, size = 260, normalized size = 1.3 \[ -{\frac{c}{15\,a{x}^{15}}}-{\frac{d}{12\,a{x}^{12}}}+{\frac{bc}{12\,{a}^{2}{x}^{12}}}-{\frac{e}{9\,a{x}^{9}}}+{\frac{bd}{9\,{a}^{2}{x}^{9}}}-{\frac{{b}^{2}c}{9\,{a}^{3}{x}^{9}}}-{\frac{f}{6\,a{x}^{6}}}+{\frac{be}{6\,{a}^{2}{x}^{6}}}-{\frac{{b}^{2}d}{6\,{a}^{3}{x}^{6}}}+{\frac{{b}^{3}c}{6\,{a}^{4}{x}^{6}}}+{\frac{{b}^{2}\ln \left ( x \right ) f}{{a}^{3}}}-{\frac{{b}^{3}\ln \left ( x \right ) e}{{a}^{4}}}+{\frac{{b}^{4}\ln \left ( x \right ) d}{{a}^{5}}}-{\frac{{b}^{5}\ln \left ( x \right ) c}{{a}^{6}}}+{\frac{bf}{3\,{a}^{2}{x}^{3}}}-{\frac{{b}^{2}e}{3\,{a}^{3}{x}^{3}}}+{\frac{{b}^{3}d}{3\,{a}^{4}{x}^{3}}}-{\frac{{b}^{4}c}{3\,{a}^{5}{x}^{3}}}-{\frac{{b}^{2}\ln \left ( b{x}^{3}+a \right ) f}{3\,{a}^{3}}}+{\frac{{b}^{3}\ln \left ( b{x}^{3}+a \right ) e}{3\,{a}^{4}}}-{\frac{{b}^{4}\ln \left ( b{x}^{3}+a \right ) d}{3\,{a}^{5}}}+{\frac{{b}^{5}\ln \left ( b{x}^{3}+a \right ) c}{3\,{a}^{6}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x^9+e*x^6+d*x^3+c)/x^16/(b*x^3+a),x)
[Out]
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Maxima [A] time = 1.4418, size = 281, normalized size = 1.37 \[ \frac{{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{6}} - \frac{{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} \log \left (x^{3}\right )}{3 \, a^{6}} - \frac{60 \,{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{12} - 30 \,{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{9} + 20 \,{\left (a^{2} b^{2} c - a^{3} b d + a^{4} e\right )} x^{6} + 12 \, a^{4} c - 15 \,{\left (a^{3} b c - a^{4} d\right )} x^{3}}{180 \, a^{5} x^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)/((b*x^3 + a)*x^16),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.283746, size = 284, normalized size = 1.39 \[ \frac{60 \,{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{15} \log \left (b x^{3} + a\right ) - 180 \,{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{15} \log \left (x\right ) - 60 \,{\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{12} + 30 \,{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x^{9} - 20 \,{\left (a^{3} b^{2} c - a^{4} b d + a^{5} e\right )} x^{6} - 12 \, a^{5} c + 15 \,{\left (a^{4} b c - a^{5} d\right )} x^{3}}{180 \, a^{6} x^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)/((b*x^3 + a)*x^16),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x**9+e*x**6+d*x**3+c)/x**16/(b*x**3+a),x)
[Out]
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GIAC/XCAS [A] time = 0.215204, size = 387, normalized size = 1.89 \[ -\frac{{\left (b^{5} c - a b^{4} d - a^{3} b^{2} f + a^{2} b^{3} e\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{6}} + \frac{{\left (b^{6} c - a b^{5} d - a^{3} b^{3} f + a^{2} b^{4} e\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{6} b} + \frac{137 \, b^{5} c x^{15} - 137 \, a b^{4} d x^{15} - 137 \, a^{3} b^{2} f x^{15} + 137 \, a^{2} b^{3} x^{15} e - 60 \, a b^{4} c x^{12} + 60 \, a^{2} b^{3} d x^{12} + 60 \, a^{4} b f x^{12} - 60 \, a^{3} b^{2} x^{12} e + 30 \, a^{2} b^{3} c x^{9} - 30 \, a^{3} b^{2} d x^{9} - 30 \, a^{5} f x^{9} + 30 \, a^{4} b x^{9} e - 20 \, a^{3} b^{2} c x^{6} + 20 \, a^{4} b d x^{6} - 20 \, a^{5} x^{6} e + 15 \, a^{4} b c x^{3} - 15 \, a^{5} d x^{3} - 12 \, a^{5} c}{180 \, a^{6} x^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)/((b*x^3 + a)*x^16),x, algorithm="giac")
[Out]