Optimal. Leaf size=103 \[ -\frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{3 b^4 \left (a+b x^3\right )}+\frac{\log \left (a+b x^3\right ) \left (3 a^2 f-2 a b e+b^2 d\right )}{3 b^4}+\frac{x^3 (b e-2 a f)}{3 b^3}+\frac{f x^6}{6 b^2} \]
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Rubi [A] time = 0.14512, antiderivative size = 103, 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 = {1819, 1850} \[ -\frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{3 b^4 \left (a+b x^3\right )}+\frac{\log \left (a+b x^3\right ) \left (3 a^2 f-2 a b e+b^2 d\right )}{3 b^4}+\frac{x^3 (b e-2 a f)}{3 b^3}+\frac{f x^6}{6 b^2} \]
Antiderivative was successfully verified.
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Rule 1819
Rule 1850
Rubi steps
\begin{align*} \int \frac{x^2 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{c+d x+e x^2+f x^3}{(a+b x)^2} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{b e-2 a f}{b^3}+\frac{f x}{b^2}+\frac{b^3 c-a b^2 d+a^2 b e-a^3 f}{b^3 (a+b x)^2}+\frac{b^2 d-2 a b e+3 a^2 f}{b^3 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac{(b e-2 a f) x^3}{3 b^3}+\frac{f x^6}{6 b^2}-\frac{b^3 c-a b^2 d+a^2 b e-a^3 f}{3 b^4 \left (a+b x^3\right )}+\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) \log \left (a+b x^3\right )}{3 b^4}\\ \end{align*}
Mathematica [A] time = 0.0584239, size = 93, normalized size = 0.9 \[ \frac{\frac{2 \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{a+b x^3}+2 \log \left (a+b x^3\right ) \left (3 a^2 f-2 a b e+b^2 d\right )+2 b x^3 (b e-2 a f)+b^2 f x^6}{6 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 142, normalized size = 1.4 \begin{align*}{\frac{f{x}^{6}}{6\,{b}^{2}}}-{\frac{2\,a{x}^{3}f}{3\,{b}^{3}}}+{\frac{{x}^{3}e}{3\,{b}^{2}}}+{\frac{\ln \left ( b{x}^{3}+a \right ){a}^{2}f}{{b}^{4}}}-{\frac{2\,\ln \left ( b{x}^{3}+a \right ) ae}{3\,{b}^{3}}}+{\frac{\ln \left ( b{x}^{3}+a \right ) d}{3\,{b}^{2}}}+{\frac{{a}^{3}f}{3\,{b}^{4} \left ( b{x}^{3}+a \right ) }}-{\frac{{a}^{2}e}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }}+{\frac{ad}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{c}{3\,b \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 = 0.959441, size = 132, normalized size = 1.28 \begin{align*} -\frac{b^{3} c - a b^{2} d + a^{2} b e - a^{3} f}{3 \,{\left (b^{5} x^{3} + a b^{4}\right )}} + \frac{b f x^{6} + 2 \,{\left (b e - 2 \, a f\right )} x^{3}}{6 \, b^{3}} + \frac{{\left (b^{2} d - 2 \, a b e + 3 \, a^{2} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.13548, size = 305, normalized size = 2.96 \begin{align*} \frac{b^{3} f x^{9} +{\left (2 \, b^{3} e - 3 \, a b^{2} f\right )} x^{6} - 2 \, b^{3} c + 2 \, a b^{2} d - 2 \, a^{2} b e + 2 \, a^{3} f + 2 \,{\left (a b^{2} e - 2 \, a^{2} b f\right )} x^{3} + 2 \,{\left (a b^{2} d - 2 \, a^{2} b e + 3 \, a^{3} f +{\left (b^{3} d - 2 \, a b^{2} e + 3 \, a^{2} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{6 \,{\left (b^{5} x^{3} + a b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.22015, size = 97, normalized size = 0.94 \begin{align*} \frac{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c}{3 a b^{4} + 3 b^{5} x^{3}} + \frac{f x^{6}}{6 b^{2}} - \frac{x^{3} \left (2 a f - b e\right )}{3 b^{3}} + \frac{\left (3 a^{2} f - 2 a b e + b^{2} d\right ) \log{\left (a + b x^{3} \right )}}{3 b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.05823, size = 278, normalized size = 2.7 \begin{align*} -\frac{1}{6} \, f{\left (\frac{{\left (b x^{3} + a\right )}^{2}{\left (\frac{6 \, a}{b x^{3} + a} - 1\right )}}{b^{4}} + \frac{6 \, a^{2} \log \left (\frac{{\left | b x^{3} + a \right |}}{{\left (b x^{3} + a\right )}^{2}{\left | b \right |}}\right )}{b^{4}} - \frac{2 \, a^{3}}{{\left (b x^{3} + a\right )} b^{4}}\right )} + \frac{1}{3} \,{\left (\frac{2 \, a \log \left (\frac{{\left | b x^{3} + a \right |}}{{\left (b x^{3} + a\right )}^{2}{\left | b \right |}}\right )}{b^{3}} + \frac{b x^{3} + a}{b^{3}} - \frac{a^{2}}{{\left (b x^{3} + a\right )} b^{3}}\right )} e - \frac{d{\left (\frac{\log \left (\frac{{\left | b x^{3} + a \right |}}{{\left (b x^{3} + a\right )}^{2}{\left | b \right |}}\right )}{b} - \frac{a}{{\left (b x^{3} + a\right )} b}\right )}}{3 \, b} - \frac{c}{3 \,{\left (b x^{3} + a\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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