Optimal. Leaf size=146 \[ -\frac{3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c}{3 b^5 \left (a+b x^3\right )}+\frac{a \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}+\frac{\log \left (a+b x^3\right ) \left (6 a^2 f-3 a b e+b^2 d\right )}{3 b^5}+\frac{x^3 (b e-3 a f)}{3 b^4}+\frac{f x^6}{6 b^3} \]
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Rubi [A] time = 0.200995, antiderivative size = 146, 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} \[ -\frac{3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c}{3 b^5 \left (a+b x^3\right )}+\frac{a \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}+\frac{\log \left (a+b x^3\right ) \left (6 a^2 f-3 a b e+b^2 d\right )}{3 b^5}+\frac{x^3 (b e-3 a f)}{3 b^4}+\frac{f x^6}{6 b^3} \]
Antiderivative was successfully verified.
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Rule 1821
Rule 1620
Rubi steps
\begin{align*} \int \frac{x^5 \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 \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 e-3 a f}{b^4}+\frac{f x}{b^3}+\frac{a \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{b^4 (a+b x)^3}+\frac{b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f}{b^4 (a+b x)^2}+\frac{b^2 d-3 a b e+6 a^2 f}{b^4 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac{(b e-3 a f) x^3}{3 b^4}+\frac{f x^6}{6 b^3}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{6 b^5 \left (a+b x^3\right )^2}-\frac{b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f}{3 b^5 \left (a+b x^3\right )}+\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) \log \left (a+b x^3\right )}{3 b^5}\\ \end{align*}
Mathematica [A] time = 0.0737187, size = 145, normalized size = 0.99 \[ \frac{a^2 b^2 \left (3 d-4 e x^3-11 f x^6\right )+2 \left (a+b x^3\right )^2 \log \left (a+b x^3\right ) \left (6 a^2 f-3 a b e+b^2 d\right )+a^3 b \left (2 f x^3-5 e\right )+7 a^4 f-a b^3 \left (c-4 x^3 \left (d+e x^3-f x^6\right )\right )+b^4 x^3 \left (-2 c+2 e x^6+f x^9\right )}{6 b^5 \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 213, normalized size = 1.5 \begin{align*}{\frac{f{x}^{6}}{6\,{b}^{3}}}-{\frac{a{x}^{3}f}{{b}^{4}}}+{\frac{{x}^{3}e}{3\,{b}^{3}}}-{\frac{{a}^{4}f}{6\,{b}^{5} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{{a}^{3}e}{6\,{b}^{4} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{{a}^{2}d}{6\,{b}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{ac}{6\,{b}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+2\,{\frac{\ln \left ( b{x}^{3}+a \right ){a}^{2}f}{{b}^{5}}}-{\frac{\ln \left ( b{x}^{3}+a \right ) ae}{{b}^{4}}}+{\frac{\ln \left ( b{x}^{3}+a \right ) d}{3\,{b}^{3}}}+{\frac{4\,{a}^{3}f}{3\,{b}^{5} \left ( b{x}^{3}+a \right ) }}-{\frac{{a}^{2}e}{{b}^{4} \left ( b{x}^{3}+a \right ) }}+{\frac{2\,ad}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }}-{\frac{c}{3\,{b}^{2} \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 = 0.952873, size = 198, normalized size = 1.36 \begin{align*} -\frac{a b^{3} c - 3 \, a^{2} b^{2} d + 5 \, a^{3} b e - 7 \, a^{4} f + 2 \,{\left (b^{4} c - 2 \, a b^{3} d + 3 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{3}}{6 \,{\left (b^{7} x^{6} + 2 \, a b^{6} x^{3} + a^{2} b^{5}\right )}} + \frac{b f x^{6} + 2 \,{\left (b e - 3 \, a f\right )} x^{3}}{6 \, b^{4}} + \frac{{\left (b^{2} d - 3 \, a b e + 6 \, a^{2} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.25994, size = 470, normalized size = 3.22 \begin{align*} \frac{b^{4} f x^{12} + 2 \,{\left (b^{4} e - 2 \, a b^{3} f\right )} x^{9} +{\left (4 \, a b^{3} e - 11 \, a^{2} b^{2} f\right )} x^{6} - a b^{3} c + 3 \, a^{2} b^{2} d - 5 \, a^{3} b e + 7 \, a^{4} f - 2 \,{\left (b^{4} c - 2 \, a b^{3} d + 2 \, a^{2} b^{2} e - a^{3} b f\right )} x^{3} + 2 \,{\left ({\left (b^{4} d - 3 \, a b^{3} e + 6 \, a^{2} b^{2} f\right )} x^{6} + a^{2} b^{2} d - 3 \, a^{3} b e + 6 \, a^{4} f + 2 \,{\left (a b^{3} d - 3 \, a^{2} b^{2} e + 6 \, a^{3} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{6 \,{\left (b^{7} x^{6} + 2 \, a b^{6} x^{3} + a^{2} b^{5}\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 = 1.08806, size = 197, normalized size = 1.35 \begin{align*} \frac{{\left (b^{2} d + 6 \, a^{2} f - 3 \, a b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{5}} + \frac{b^{3} f x^{6} - 6 \, a b^{2} f x^{3} + 2 \, b^{3} x^{3} e}{6 \, b^{6}} - \frac{a b^{3} c - 3 \, a^{2} b^{2} d - 7 \, a^{4} f + 5 \, a^{3} b e + 2 \,{\left (b^{4} c - 2 \, a b^{3} d - 4 \, a^{3} b f + 3 \, a^{2} b^{2} e\right )} x^{3}}{6 \,{\left (b x^{3} + a\right )}^{2} b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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