Optimal. Leaf size=164 \[ \frac{b c-a d}{9 a^2 x^9}-\frac{a^2 e-a b d+b^2 c}{6 a^3 x^6}-\frac{b \log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^5}+\frac{b \log (x) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^5}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{3 a^4 x^3}-\frac{c}{12 a x^{12}} \]
[Out]
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Rubi [A] time = 0.358205, antiderivative size = 164, 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 \[ \frac{b c-a d}{9 a^2 x^9}-\frac{a^2 e-a b d+b^2 c}{6 a^3 x^6}-\frac{b \log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^5}+\frac{b \log (x) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^5}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{3 a^4 x^3}-\frac{c}{12 a x^{12}} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^13*(a + b*x^3)),x]
[Out]
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Rubi in Sympy [A] time = 56.3961, size = 153, normalized size = 0.93 \[ - \frac{c}{12 a x^{12}} - \frac{a d - b c}{9 a^{2} x^{9}} - \frac{a^{2} e - a b d + b^{2} c}{6 a^{3} x^{6}} - \frac{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c}{3 a^{4} x^{3}} - \frac{b \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (x^{3} \right )}}{3 a^{5}} + \frac{b \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a + b x^{3} \right )}}{3 a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x**9+e*x**6+d*x**3+c)/x**13/(b*x**3+a),x)
[Out]
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Mathematica [A] time = 0.131631, size = 164, normalized size = 1. \[ \frac{-a^4 \left (3 c+4 d x^3+6 e x^6+12 f x^9\right )+2 a^3 b x^3 \left (2 c+3 d x^3+6 e x^6\right )-6 a^2 b^2 x^6 \left (c+2 d x^3\right )+36 b x^{12} \log (x) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )-12 b x^{12} \log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )+12 a b^3 c x^9}{36 a^5 x^{12}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^13*(a + b*x^3)),x]
[Out]
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Maple [A] time = 0.014, size = 210, normalized size = 1.3 \[ -{\frac{c}{12\,a{x}^{12}}}-{\frac{d}{9\,a{x}^{9}}}+{\frac{bc}{9\,{a}^{2}{x}^{9}}}-{\frac{e}{6\,a{x}^{6}}}+{\frac{bd}{6\,{a}^{2}{x}^{6}}}-{\frac{{b}^{2}c}{6\,{a}^{3}{x}^{6}}}-{\frac{f}{3\,a{x}^{3}}}+{\frac{be}{3\,{a}^{2}{x}^{3}}}-{\frac{{b}^{2}d}{3\,{a}^{3}{x}^{3}}}+{\frac{{b}^{3}c}{3\,{a}^{4}{x}^{3}}}-{\frac{b\ln \left ( x \right ) f}{{a}^{2}}}+{\frac{{b}^{2}\ln \left ( x \right ) e}{{a}^{3}}}-{\frac{{b}^{3}\ln \left ( x \right ) d}{{a}^{4}}}+{\frac{{b}^{4}\ln \left ( x \right ) c}{{a}^{5}}}+{\frac{b\ln \left ( b{x}^{3}+a \right ) f}{3\,{a}^{2}}}-{\frac{{b}^{2}\ln \left ( b{x}^{3}+a \right ) e}{3\,{a}^{3}}}+{\frac{{b}^{3}\ln \left ( b{x}^{3}+a \right ) d}{3\,{a}^{4}}}-{\frac{{b}^{4}\ln \left ( b{x}^{3}+a \right ) c}{3\,{a}^{5}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x^9+e*x^6+d*x^3+c)/x^13/(b*x^3+a),x)
[Out]
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Maxima [A] time = 1.48564, size = 224, normalized size = 1.37 \[ -\frac{{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{5}} + \frac{{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} \log \left (x^{3}\right )}{3 \, a^{5}} + \frac{12 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{9} - 6 \,{\left (a b^{2} c - a^{2} b d + a^{3} e\right )} x^{6} - 3 \, a^{3} c + 4 \,{\left (a^{2} b c - a^{3} d\right )} x^{3}}{36 \, a^{4} x^{12}} \]
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^13),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.260637, size = 227, normalized size = 1.38 \[ -\frac{12 \,{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{12} \log \left (b x^{3} + a\right ) - 36 \,{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{12} \log \left (x\right ) - 12 \,{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{9} + 6 \,{\left (a^{2} b^{2} c - a^{3} b d + a^{4} e\right )} x^{6} + 3 \, a^{4} c - 4 \,{\left (a^{3} b c - a^{4} d\right )} x^{3}}{36 \, a^{5} x^{12}} \]
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^13),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**13/(b*x**3+a),x)
[Out]
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GIAC/XCAS [A] time = 0.214204, size = 317, normalized size = 1.93 \[ \frac{{\left (b^{4} c - a b^{3} d - a^{3} b f + a^{2} b^{2} e\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{5}} - \frac{{\left (b^{5} c - a b^{4} d - a^{3} b^{2} f + a^{2} b^{3} e\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{5} b} - \frac{25 \, b^{4} c x^{12} - 25 \, a b^{3} d x^{12} - 25 \, a^{3} b f x^{12} + 25 \, a^{2} b^{2} x^{12} e - 12 \, a b^{3} c x^{9} + 12 \, a^{2} b^{2} d x^{9} + 12 \, a^{4} f x^{9} - 12 \, a^{3} b x^{9} e + 6 \, a^{2} b^{2} c x^{6} - 6 \, a^{3} b d x^{6} + 6 \, a^{4} x^{6} e - 4 \, a^{3} b c x^{3} + 4 \, a^{4} d x^{3} + 3 \, a^{4} c}{36 \, a^{5} x^{12}} \]
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^13),x, algorithm="giac")
[Out]