Optimal. Leaf size=186 \[ \frac{a \left (4 a^2 b e-5 a^3 f-3 a b^2 d+2 b^3 c\right )}{3 b^6 \left (a+b x^3\right )}-\frac{a^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}+\frac{\log \left (a+b x^3\right ) \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )}{3 b^6}+\frac{x^3 \left (6 a^2 f-3 a b e+b^2 d\right )}{3 b^5}+\frac{x^6 (b e-3 a f)}{6 b^4}+\frac{f x^9}{9 b^3} \]
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Rubi [A] time = 0.268936, antiderivative size = 186, 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, Rules used = {1821, 1620} \[ \frac{a \left (4 a^2 b e-5 a^3 f-3 a b^2 d+2 b^3 c\right )}{3 b^6 \left (a+b x^3\right )}-\frac{a^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}+\frac{\log \left (a+b x^3\right ) \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )}{3 b^6}+\frac{x^3 \left (6 a^2 f-3 a b e+b^2 d\right )}{3 b^5}+\frac{x^6 (b e-3 a f)}{6 b^4}+\frac{f x^9}{9 b^3} \]
Antiderivative was successfully verified.
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Rule 1821
Rule 1620
Rubi steps
\begin{align*} \int \frac{x^8 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2 \left (c+d x+e x^2+f x^3\right )}{(a+b x)^3} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{b^2 d-3 a b e+6 a^2 f}{b^5}+\frac{(b e-3 a f) x}{b^4}+\frac{f x^2}{b^3}-\frac{a^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{b^5 (a+b x)^3}+\frac{a \left (-2 b^3 c+3 a b^2 d-4 a^2 b e+5 a^3 f\right )}{b^5 (a+b x)^2}+\frac{b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f}{b^5 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^3}{3 b^5}+\frac{(b e-3 a f) x^6}{6 b^4}+\frac{f x^9}{9 b^3}-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{6 b^6 \left (a+b x^3\right )^2}+\frac{a \left (2 b^3 c-3 a b^2 d+4 a^2 b e-5 a^3 f\right )}{3 b^6 \left (a+b x^3\right )}+\frac{\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) \log \left (a+b x^3\right )}{3 b^6}\\ \end{align*}
Mathematica [A] time = 0.0987472, size = 170, normalized size = 0.91 \[ \frac{-\frac{6 a \left (-4 a^2 b e+5 a^3 f+3 a b^2 d-2 b^3 c\right )}{a+b x^3}+\frac{3 a^2 \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{\left (a+b x^3\right )^2}+6 \log \left (a+b x^3\right ) \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )+6 b x^3 \left (6 a^2 f-3 a b e+b^2 d\right )+3 b^2 x^6 (b e-3 a f)+2 b^3 f x^9}{18 b^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 266, normalized size = 1.4 \begin{align*}{\frac{f{x}^{9}}{9\,{b}^{3}}}-{\frac{{x}^{6}af}{2\,{b}^{4}}}+{\frac{e{x}^{6}}{6\,{b}^{3}}}+2\,{\frac{{a}^{2}f{x}^{3}}{{b}^{5}}}-{\frac{ae{x}^{3}}{{b}^{4}}}+{\frac{d{x}^{3}}{3\,{b}^{3}}}+{\frac{{a}^{5}f}{6\,{b}^{6} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{{a}^{4}e}{6\,{b}^{5} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{{a}^{3}d}{6\,{b}^{4} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{{a}^{2}c}{6\,{b}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{10\,\ln \left ( b{x}^{3}+a \right ){a}^{3}f}{3\,{b}^{6}}}+2\,{\frac{\ln \left ( b{x}^{3}+a \right ){a}^{2}e}{{b}^{5}}}-{\frac{\ln \left ( b{x}^{3}+a \right ) ad}{{b}^{4}}}+{\frac{\ln \left ( b{x}^{3}+a \right ) c}{3\,{b}^{3}}}-{\frac{5\,{a}^{4}f}{3\,{b}^{6} \left ( b{x}^{3}+a \right ) }}+{\frac{4\,{a}^{3}e}{3\,{b}^{5} \left ( b{x}^{3}+a \right ) }}-{\frac{{a}^{2}d}{{b}^{4} \left ( b{x}^{3}+a \right ) }}+{\frac{2\,ac}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.95037, size = 258, normalized size = 1.39 \begin{align*} \frac{3 \, a^{2} b^{3} c - 5 \, a^{3} b^{2} d + 7 \, a^{4} b e - 9 \, a^{5} f + 2 \,{\left (2 \, a b^{4} c - 3 \, a^{2} b^{3} d + 4 \, a^{3} b^{2} e - 5 \, a^{4} b f\right )} x^{3}}{6 \,{\left (b^{8} x^{6} + 2 \, a b^{7} x^{3} + a^{2} b^{6}\right )}} + \frac{2 \, b^{2} f x^{9} + 3 \,{\left (b^{2} e - 3 \, a b f\right )} x^{6} + 6 \,{\left (b^{2} d - 3 \, a b e + 6 \, a^{2} f\right )} x^{3}}{18 \, b^{5}} + \frac{{\left (b^{3} c - 3 \, a b^{2} d + 6 \, a^{2} b e - 10 \, a^{3} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29363, size = 635, normalized size = 3.41 \begin{align*} \frac{2 \, b^{5} f x^{15} +{\left (3 \, b^{5} e - 5 \, a b^{4} f\right )} x^{12} + 2 \,{\left (3 \, b^{5} d - 6 \, a b^{4} e + 10 \, a^{2} b^{3} f\right )} x^{9} + 3 \,{\left (4 \, a b^{4} d - 11 \, a^{2} b^{3} e + 21 \, a^{3} b^{2} f\right )} x^{6} + 9 \, a^{2} b^{3} c - 15 \, a^{3} b^{2} d + 21 \, a^{4} b e - 27 \, a^{5} f + 6 \,{\left (2 \, a b^{4} c - 2 \, a^{2} b^{3} d + a^{3} b^{2} e + a^{4} b f\right )} x^{3} + 6 \,{\left ({\left (b^{5} c - 3 \, a b^{4} d + 6 \, a^{2} b^{3} e - 10 \, a^{3} b^{2} f\right )} x^{6} + a^{2} b^{3} c - 3 \, a^{3} b^{2} d + 6 \, a^{4} b e - 10 \, a^{5} f + 2 \,{\left (a b^{4} c - 3 \, a^{2} b^{3} d + 6 \, a^{3} b^{2} e - 10 \, a^{4} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{18 \,{\left (b^{8} x^{6} + 2 \, a b^{7} x^{3} + a^{2} b^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07923, size = 319, normalized size = 1.72 \begin{align*} \frac{{\left (b^{3} c - 3 \, a b^{2} d - 10 \, a^{3} f + 6 \, a^{2} b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{6}} - \frac{3 \, b^{5} c x^{6} - 9 \, a b^{4} d x^{6} - 30 \, a^{3} b^{2} f x^{6} + 18 \, a^{2} b^{3} x^{6} e + 2 \, a b^{4} c x^{3} - 12 \, a^{2} b^{3} d x^{3} - 50 \, a^{4} b f x^{3} + 28 \, a^{3} b^{2} x^{3} e - 4 \, a^{3} b^{2} d - 21 \, a^{5} f + 11 \, a^{4} b e}{6 \,{\left (b x^{3} + a\right )}^{2} b^{6}} + \frac{2 \, b^{6} f x^{9} - 9 \, a b^{5} f x^{6} + 3 \, b^{6} x^{6} e + 6 \, b^{6} d x^{3} + 36 \, a^{2} b^{4} f x^{3} - 18 \, a b^{5} x^{3} e}{18 \, b^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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