Optimal. Leaf size=163 \[ \frac{3 b c-a d}{3 a^4 x^3}-\frac{c}{6 a^3 x^6}-\frac{\log \left (a+b x^3\right ) \left (a^2 e-3 a b d+6 b^2 c\right )}{3 a^5}+\frac{\log (x) \left (a^2 e-3 a b d+6 b^2 c\right )}{a^5}+\frac{a^2 e-2 a b d+3 b^2 c}{3 a^4 \left (a+b x^3\right )}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{6 a^3 b \left (a+b x^3\right )^2} \]
[Out]
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Rubi [A] time = 0.402378, antiderivative size = 163, 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 \[ \frac{3 b c-a d}{3 a^4 x^3}-\frac{c}{6 a^3 x^6}-\frac{\log \left (a+b x^3\right ) \left (a^2 e-3 a b d+6 b^2 c\right )}{3 a^5}+\frac{\log (x) \left (a^2 e-3 a b d+6 b^2 c\right )}{a^5}+\frac{a^2 e-2 a b d+3 b^2 c}{3 a^4 \left (a+b x^3\right )}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{6 a^3 b \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^7*(a + b*x^3)^3),x]
[Out]
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Rubi in Sympy [A] time = 61.9752, size = 156, normalized size = 0.96 \[ - \frac{c}{6 a^{3} x^{6}} - \frac{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c}{6 a^{3} b \left (a + b x^{3}\right )^{2}} + \frac{a^{2} e - 2 a b d + 3 b^{2} c}{3 a^{4} \left (a + b x^{3}\right )} - \frac{a d - 3 b c}{3 a^{4} x^{3}} + \frac{\left (a^{2} e - 3 a b d + 6 b^{2} c\right ) \log{\left (x^{3} \right )}}{3 a^{5}} - \frac{\left (a^{2} e - 3 a b d + 6 b^{2} c\right ) \log{\left (a + b x^{3} \right )}}{3 a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x**9+e*x**6+d*x**3+c)/x**7/(b*x**3+a)**3,x)
[Out]
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Mathematica [A] time = 0.212299, size = 149, normalized size = 0.91 \[ \frac{\frac{2 a \left (a^2 e-2 a b d+3 b^2 c\right )}{a+b x^3}-2 \log \left (a+b x^3\right ) \left (a^2 e-3 a b d+6 b^2 c\right )+6 \log (x) \left (a^2 e-3 a b d+6 b^2 c\right )-\frac{a^2 c}{x^6}+\frac{a^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b \left (a+b x^3\right )^2}-\frac{2 a (a d-3 b c)}{x^3}}{6 a^5} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^7*(a + b*x^3)^3),x]
[Out]
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Maple [A] time = 0.025, size = 213, normalized size = 1.3 \[ -{\frac{c}{6\,{a}^{3}{x}^{6}}}-{\frac{d}{3\,{a}^{3}{x}^{3}}}+{\frac{bc}{{a}^{4}{x}^{3}}}+{\frac{e\ln \left ( x \right ) }{{a}^{3}}}-3\,{\frac{\ln \left ( x \right ) bd}{{a}^{4}}}+6\,{\frac{\ln \left ( x \right ){b}^{2}c}{{a}^{5}}}-{\frac{e\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{3}}}+{\frac{\ln \left ( b{x}^{3}+a \right ) bd}{{a}^{4}}}-2\,{\frac{\ln \left ( b{x}^{3}+a \right ){b}^{2}c}{{a}^{5}}}-{\frac{f}{6\,b \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{e}{6\,a \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{bd}{6\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{{b}^{2}c}{6\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{e}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{2\,bd}{3\,{a}^{3} \left ( b{x}^{3}+a \right ) }}+{\frac{{b}^{2}c}{{a}^{4} \left ( b{x}^{3}+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x^9+e*x^6+d*x^3+c)/x^7/(b*x^3+a)^3,x)
[Out]
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Maxima [A] time = 1.38375, size = 246, normalized size = 1.51 \[ \frac{2 \,{\left (6 \, b^{4} c - 3 \, a b^{3} d + a^{2} b^{2} e\right )} x^{9} +{\left (18 \, a b^{3} c - 9 \, a^{2} b^{2} d + 3 \, a^{3} b e - a^{4} f\right )} x^{6} - a^{3} b c + 2 \,{\left (2 \, a^{2} b^{2} c - a^{3} b d\right )} x^{3}}{6 \,{\left (a^{4} b^{3} x^{12} + 2 \, a^{5} b^{2} x^{9} + a^{6} b x^{6}\right )}} - \frac{{\left (6 \, b^{2} c - 3 \, a b d + a^{2} e\right )} \log \left (b x^{3} + a\right )}{3 \, a^{5}} + \frac{{\left (6 \, b^{2} c - 3 \, a b d + a^{2} e\right )} \log \left (x^{3}\right )}{3 \, a^{5}} \]
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)^3*x^7),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.279899, size = 427, normalized size = 2.62 \[ \frac{2 \,{\left (6 \, a b^{4} c - 3 \, a^{2} b^{3} d + a^{3} b^{2} e\right )} x^{9} +{\left (18 \, a^{2} b^{3} c - 9 \, a^{3} b^{2} d + 3 \, a^{4} b e - a^{5} f\right )} x^{6} - a^{4} b c + 2 \,{\left (2 \, a^{3} b^{2} c - a^{4} b d\right )} x^{3} - 2 \,{\left ({\left (6 \, b^{5} c - 3 \, a b^{4} d + a^{2} b^{3} e\right )} x^{12} + 2 \,{\left (6 \, a b^{4} c - 3 \, a^{2} b^{3} d + a^{3} b^{2} e\right )} x^{9} +{\left (6 \, a^{2} b^{3} c - 3 \, a^{3} b^{2} d + a^{4} b e\right )} x^{6}\right )} \log \left (b x^{3} + a\right ) + 6 \,{\left ({\left (6 \, b^{5} c - 3 \, a b^{4} d + a^{2} b^{3} e\right )} x^{12} + 2 \,{\left (6 \, a b^{4} c - 3 \, a^{2} b^{3} d + a^{3} b^{2} e\right )} x^{9} +{\left (6 \, a^{2} b^{3} c - 3 \, a^{3} b^{2} d + a^{4} b e\right )} x^{6}\right )} \log \left (x\right )}{6 \,{\left (a^{5} b^{3} x^{12} + 2 \, a^{6} b^{2} x^{9} + a^{7} b x^{6}\right )}} \]
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)^3*x^7),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**7/(b*x**3+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.216499, size = 255, normalized size = 1.56 \[ \frac{{\left (6 \, b^{2} c - 3 \, a b d + a^{2} e\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{5}} - \frac{{\left (6 \, b^{3} c - 3 \, a b^{2} d + a^{2} b e\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{5} b} + \frac{12 \, b^{4} c x^{9} - 6 \, a b^{3} d x^{9} + 2 \, a^{2} b^{2} x^{9} e + 18 \, a b^{3} c x^{6} - 9 \, a^{2} b^{2} d x^{6} - a^{4} f x^{6} + 3 \, a^{3} b x^{6} e + 4 \, a^{2} b^{2} c x^{3} - 2 \, a^{3} b d x^{3} - a^{3} b c}{6 \,{\left (b x^{6} + a x^{3}\right )}^{2} a^{4} b} \]
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)^3*x^7),x, algorithm="giac")
[Out]