Optimal. Leaf size=69 \[ \frac{A b-a B}{3 a^2 x^3}-\frac{b (A b-a B) \log \left (a+b x^3\right )}{3 a^3}+\frac{b \log (x) (A b-a B)}{a^3}-\frac{A}{6 a x^6} \]
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Rubi [A] time = 0.0640418, antiderivative size = 69, 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} \[ \frac{A b-a B}{3 a^2 x^3}-\frac{b (A b-a B) \log \left (a+b x^3\right )}{3 a^3}+\frac{b \log (x) (A b-a B)}{a^3}-\frac{A}{6 a x^6} \]
Antiderivative was successfully verified.
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Rule 446
Rule 77
Rubi steps
\begin{align*} \int \frac{A+B x^3}{x^7 \left (a+b x^3\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{A+B x}{x^3 (a+b x)} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{A}{a x^3}+\frac{-A b+a B}{a^2 x^2}-\frac{b (-A b+a B)}{a^3 x}+\frac{b^2 (-A b+a B)}{a^3 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{A}{6 a x^6}+\frac{A b-a B}{3 a^2 x^3}+\frac{b (A b-a B) \log (x)}{a^3}-\frac{b (A b-a B) \log \left (a+b x^3\right )}{3 a^3}\\ \end{align*}
Mathematica [A] time = 0.0284331, size = 70, normalized size = 1.01 \[ \frac{-a \left (a A+2 a B x^3-2 A b x^3\right )+6 b x^6 \log (x) (A b-a B)+2 b x^6 (a B-A b) \log \left (a+b x^3\right )}{6 a^3 x^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 81, normalized size = 1.2 \begin{align*} -{\frac{{b}^{2}\ln \left ( b{x}^{3}+a \right ) A}{3\,{a}^{3}}}+{\frac{b\ln \left ( b{x}^{3}+a \right ) B}{3\,{a}^{2}}}-{\frac{A}{6\,a{x}^{6}}}+{\frac{Ab}{3\,{a}^{2}{x}^{3}}}-{\frac{B}{3\,a{x}^{3}}}+{\frac{A\ln \left ( x \right ){b}^{2}}{{a}^{3}}}-{\frac{bB\ln \left ( x \right ) }{{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.36767, size = 95, normalized size = 1.38 \begin{align*} \frac{{\left (B a b - A b^{2}\right )} \log \left (b x^{3} + a\right )}{3 \, a^{3}} - \frac{{\left (B a b - A b^{2}\right )} \log \left (x^{3}\right )}{3 \, a^{3}} - \frac{2 \,{\left (B a - A b\right )} x^{3} + A a}{6 \, a^{2} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58467, size = 158, normalized size = 2.29 \begin{align*} \frac{2 \,{\left (B a b - A b^{2}\right )} x^{6} \log \left (b x^{3} + a\right ) - 6 \,{\left (B a b - A b^{2}\right )} x^{6} \log \left (x\right ) - 2 \,{\left (B a^{2} - A a b\right )} x^{3} - A a^{2}}{6 \, a^{3} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.77085, size = 61, normalized size = 0.88 \begin{align*} - \frac{A a + x^{3} \left (- 2 A b + 2 B a\right )}{6 a^{2} x^{6}} - \frac{b \left (- A b + B a\right ) \log{\left (x \right )}}{a^{3}} + \frac{b \left (- A b + B a\right ) \log{\left (\frac{a}{b} + x^{3} \right )}}{3 a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12729, size = 134, normalized size = 1.94 \begin{align*} -\frac{{\left (B a b - A b^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{3}} + \frac{{\left (B a b^{2} - A b^{3}\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3} b} + \frac{3 \, B a b x^{6} - 3 \, A b^{2} x^{6} - 2 \, B a^{2} x^{3} + 2 \, A a b x^{3} - A a^{2}}{6 \, a^{3} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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