Optimal. Leaf size=101 \[ -\frac{2 A b-a B}{3 a^3 \left (a+b x^3\right )}-\frac{A b-a B}{6 a^2 \left (a+b x^3\right )^2}+\frac{(3 A b-a B) \log \left (a+b x^3\right )}{3 a^4}-\frac{\log (x) (3 A b-a B)}{a^4}-\frac{A}{3 a^3 x^3} \]
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Rubi [A] time = 0.0998586, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {446, 77} \[ -\frac{2 A b-a B}{3 a^3 \left (a+b x^3\right )}-\frac{A b-a B}{6 a^2 \left (a+b x^3\right )^2}+\frac{(3 A b-a B) \log \left (a+b x^3\right )}{3 a^4}-\frac{\log (x) (3 A b-a B)}{a^4}-\frac{A}{3 a^3 x^3} \]
Antiderivative was successfully verified.
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Rule 446
Rule 77
Rubi steps
\begin{align*} \int \frac{A+B x^3}{x^4 \left (a+b x^3\right )^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{A+B x}{x^2 (a+b x)^3} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{A}{a^3 x^2}+\frac{-3 A b+a B}{a^4 x}-\frac{b (-A b+a B)}{a^2 (a+b x)^3}-\frac{b (-2 A b+a B)}{a^3 (a+b x)^2}-\frac{b (-3 A b+a B)}{a^4 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{A}{3 a^3 x^3}-\frac{A b-a B}{6 a^2 \left (a+b x^3\right )^2}-\frac{2 A b-a B}{3 a^3 \left (a+b x^3\right )}-\frac{(3 A b-a B) \log (x)}{a^4}+\frac{(3 A b-a B) \log \left (a+b x^3\right )}{3 a^4}\\ \end{align*}
Mathematica [A] time = 0.054895, size = 87, normalized size = 0.86 \[ \frac{\frac{a^2 (a B-A b)}{\left (a+b x^3\right )^2}+\frac{2 a (a B-2 A b)}{a+b x^3}+2 (3 A b-a B) \log \left (a+b x^3\right )+6 \log (x) (a B-3 A b)-\frac{2 a A}{x^3}}{6 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 117, normalized size = 1.2 \begin{align*} -{\frac{2\,Ab}{3\,{a}^{3} \left ( b{x}^{3}+a \right ) }}+{\frac{B}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}+{\frac{b\ln \left ( b{x}^{3}+a \right ) A}{{a}^{4}}}-{\frac{\ln \left ( b{x}^{3}+a \right ) B}{3\,{a}^{3}}}-{\frac{Ab}{6\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{B}{6\,a \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{A}{3\,{a}^{3}{x}^{3}}}-3\,{\frac{A\ln \left ( x \right ) b}{{a}^{4}}}+{\frac{\ln \left ( x \right ) B}{{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00189, size = 147, normalized size = 1.46 \begin{align*} \frac{2 \,{\left (B a b - 3 \, A b^{2}\right )} x^{6} + 3 \,{\left (B a^{2} - 3 \, A a b\right )} x^{3} - 2 \, A a^{2}}{6 \,{\left (a^{3} b^{2} x^{9} + 2 \, a^{4} b x^{6} + a^{5} x^{3}\right )}} - \frac{{\left (B a - 3 \, A b\right )} \log \left (b x^{3} + a\right )}{3 \, a^{4}} + \frac{{\left (B a - 3 \, A b\right )} \log \left (x^{3}\right )}{3 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.46795, size = 412, normalized size = 4.08 \begin{align*} \frac{2 \,{\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{6} - 2 \, A a^{3} + 3 \,{\left (B a^{3} - 3 \, A a^{2} b\right )} x^{3} - 2 \,{\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{9} + 2 \,{\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{6} +{\left (B a^{3} - 3 \, A a^{2} b\right )} x^{3}\right )} \log \left (b x^{3} + a\right ) + 6 \,{\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{9} + 2 \,{\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{6} +{\left (B a^{3} - 3 \, A a^{2} b\right )} x^{3}\right )} \log \left (x\right )}{6 \,{\left (a^{4} b^{2} x^{9} + 2 \, a^{5} b x^{6} + a^{6} x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.18212, size = 107, normalized size = 1.06 \begin{align*} \frac{- 2 A a^{2} + x^{6} \left (- 6 A b^{2} + 2 B a b\right ) + x^{3} \left (- 9 A a b + 3 B a^{2}\right )}{6 a^{5} x^{3} + 12 a^{4} b x^{6} + 6 a^{3} b^{2} x^{9}} + \frac{\left (- 3 A b + B a\right ) \log{\left (x \right )}}{a^{4}} - \frac{\left (- 3 A b + B a\right ) \log{\left (\frac{a}{b} + x^{3} \right )}}{3 a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13708, size = 184, normalized size = 1.82 \begin{align*} \frac{{\left (B a - 3 \, A b\right )} \log \left ({\left | x \right |}\right )}{a^{4}} - \frac{{\left (B a b - 3 \, A b^{2}\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{4} b} + \frac{3 \, B a b^{2} x^{6} - 9 \, A b^{3} x^{6} + 8 \, B a^{2} b x^{3} - 22 \, A a b^{2} x^{3} + 6 \, B a^{3} - 14 \, A a^{2} b}{6 \,{\left (b x^{3} + a\right )}^{2} a^{4}} - \frac{B a x^{3} - 3 \, A b x^{3} + A a}{3 \, a^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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