Optimal. Leaf size=134 \[ \frac{(3 b c-a d) \log \left (a+b x^3\right )}{3 a^4}-\frac{\log (x) (3 b c-a d)}{a^4}-\frac{a^3 f-a b^2 d+2 b^3 c}{3 a^3 b^2 \left (a+b x^3\right )}-\frac{c}{3 a^3 x^3}-\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{6 a^2 b^2 \left (a+b x^3\right )^2} \]
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Rubi [A] time = 0.34449, antiderivative size = 134, 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 \[ \frac{(3 b c-a d) \log \left (a+b x^3\right )}{3 a^4}-\frac{\log (x) (3 b c-a d)}{a^4}-\frac{a^3 f-a b^2 d+2 b^3 c}{3 a^3 b^2 \left (a+b x^3\right )}-\frac{c}{3 a^3 x^3}-\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{6 a^2 b^2 \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^4*(a + b*x^3)^3),x]
[Out]
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Rubi in Sympy [A] time = 57.1606, size = 124, normalized size = 0.93 \[ \frac{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c}{6 a^{2} b^{2} \left (a + b x^{3}\right )^{2}} - \frac{c}{3 a^{3} x^{3}} - \frac{a^{3} f - a b^{2} d + 2 b^{3} c}{3 a^{3} b^{2} \left (a + b x^{3}\right )} + \frac{\left (a d - 3 b c\right ) \log{\left (x^{3} \right )}}{3 a^{4}} - \frac{\left (a d - 3 b c\right ) \log{\left (a + b x^{3} \right )}}{3 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x**9+e*x**6+d*x**3+c)/x**4/(b*x**3+a)**3,x)
[Out]
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Mathematica [A] time = 0.172494, size = 121, normalized size = 0.9 \[ \frac{-\frac{2 a \left (a^3 f-a b^2 d+2 b^3 c\right )}{b^2 \left (a+b x^3\right )}+\frac{a^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{b^2 \left (a+b x^3\right )^2}+2 (3 b c-a d) \log \left (a+b x^3\right )+6 \log (x) (a d-3 b c)-\frac{2 a c}{x^3}}{6 a^4} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^4*(a + b*x^3)^3),x]
[Out]
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Maple [A] time = 0.025, size = 163, normalized size = 1.2 \[ -{\frac{c}{3\,{a}^{3}{x}^{3}}}+{\frac{d\ln \left ( x \right ) }{{a}^{3}}}-3\,{\frac{bc\ln \left ( x \right ) }{{a}^{4}}}-{\frac{d\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{3}}}+{\frac{bc\ln \left ( b{x}^{3}+a \right ) }{{a}^{4}}}+{\frac{af}{6\,{b}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{e}{6\,b \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{d}{6\,a \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{bc}{6\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{f}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }}+{\frac{d}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{2\,bc}{3\,{a}^{3} \left ( b{x}^{3}+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x^9+e*x^6+d*x^3+c)/x^4/(b*x^3+a)^3,x)
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Maxima [A] time = 1.38988, size = 194, normalized size = 1.45 \[ -\frac{2 \,{\left (3 \, b^{4} c - a b^{3} d + a^{3} b f\right )} x^{6} + 2 \, a^{2} b^{2} c +{\left (9 \, a b^{3} c - 3 \, a^{2} b^{2} d + a^{3} b e + a^{4} f\right )} x^{3}}{6 \,{\left (a^{3} b^{4} x^{9} + 2 \, a^{4} b^{3} x^{6} + a^{5} b^{2} x^{3}\right )}} + \frac{{\left (3 \, b c - a d\right )} \log \left (b x^{3} + a\right )}{3 \, a^{4}} - \frac{{\left (3 \, b c - a d\right )} \log \left (x^{3}\right )}{3 \, a^{4}} \]
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^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.273458, size = 338, normalized size = 2.52 \[ -\frac{2 \,{\left (3 \, a b^{4} c - a^{2} b^{3} d + a^{4} b f\right )} x^{6} + 2 \, a^{3} b^{2} c +{\left (9 \, a^{2} b^{3} c - 3 \, a^{3} b^{2} d + a^{4} b e + a^{5} f\right )} x^{3} - 2 \,{\left ({\left (3 \, b^{5} c - a b^{4} d\right )} x^{9} + 2 \,{\left (3 \, a b^{4} c - a^{2} b^{3} d\right )} x^{6} +{\left (3 \, a^{2} b^{3} c - a^{3} b^{2} d\right )} x^{3}\right )} \log \left (b x^{3} + a\right ) + 6 \,{\left ({\left (3 \, b^{5} c - a b^{4} d\right )} x^{9} + 2 \,{\left (3 \, a b^{4} c - a^{2} b^{3} d\right )} x^{6} +{\left (3 \, a^{2} b^{3} c - a^{3} b^{2} d\right )} x^{3}\right )} \log \left (x\right )}{6 \,{\left (a^{4} b^{4} x^{9} + 2 \, a^{5} b^{3} x^{6} + a^{6} b^{2} x^{3}\right )}} \]
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^4),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**4/(b*x**3+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.215929, size = 234, normalized size = 1.75 \[ -\frac{{\left (3 \, b c - a d\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} + \frac{{\left (3 \, b^{2} c - a b d\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{4} b} + \frac{3 \, b c x^{3} - a d x^{3} - a c}{3 \, a^{4} x^{3}} - \frac{9 \, b^{5} c x^{6} - 3 \, a b^{4} d x^{6} + 22 \, a b^{4} c x^{3} - 8 \, a^{2} b^{3} d x^{3} + 2 \, a^{4} b f x^{3} + 14 \, a^{2} b^{3} c - 6 \, a^{3} b^{2} d + a^{5} f + a^{4} b e}{6 \,{\left (b x^{3} + a\right )}^{2} a^{4} b^{2}} \]
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^4),x, algorithm="giac")
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