Optimal. Leaf size=16 \[ \frac{1}{3} \log \left (b x^3+c x^6\right ) \]
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Rubi [A] time = 0.0295228, antiderivative size = 15, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {1584, 446, 72} \[ \frac{1}{3} \log \left (b+c x^3\right )+\log (x) \]
Antiderivative was successfully verified.
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Rule 1584
Rule 446
Rule 72
Rubi steps
\begin{align*} \int \frac{x^2 \left (b+2 c x^3\right )}{b x^3+c x^6} \, dx &=\int \frac{b+2 c x^3}{x \left (b+c x^3\right )} \, dx\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{b+2 c x}{x (b+c x)} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{1}{x}+\frac{c}{b+c x}\right ) \, dx,x,x^3\right )\\ &=\log (x)+\frac{1}{3} \log \left (b+c x^3\right )\\ \end{align*}
Mathematica [A] time = 0.0070442, size = 15, normalized size = 0.94 \[ \frac{1}{3} \log \left (b+c x^3\right )+\log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 14, normalized size = 0.9 \begin{align*} \ln \left ( x \right ) +{\frac{\ln \left ( c{x}^{3}+b \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.202, size = 23, normalized size = 1.44 \begin{align*} \frac{1}{3} \, \log \left (c x^{3} + b\right ) + \frac{1}{3} \, \log \left (x^{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.16842, size = 39, normalized size = 2.44 \begin{align*} \frac{1}{3} \, \log \left (c x^{3} + b\right ) + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.348128, size = 12, normalized size = 0.75 \begin{align*} \log{\left (x \right )} + \frac{\log{\left (\frac{b}{c} + x^{3} \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12081, size = 20, normalized size = 1.25 \begin{align*} \frac{1}{3} \, \log \left ({\left | c x^{3} + b \right |}\right ) + \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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