Optimal. Leaf size=69 \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right )}{3 a \sqrt{b^2-4 a c}}-\frac{\log \left (a+b x^3+c x^6\right )}{6 a}+\frac{\log (x)}{a} \]
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Rubi [A] time = 0.0695275, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {1357, 705, 29, 634, 618, 206, 628} \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right )}{3 a \sqrt{b^2-4 a c}}-\frac{\log \left (a+b x^3+c x^6\right )}{6 a}+\frac{\log (x)}{a} \]
Antiderivative was successfully verified.
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Rule 1357
Rule 705
Rule 29
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x \left (a+b x^3+c x^6\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x \left (a+b x+c x^2\right )} \, dx,x,x^3\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^3\right )}{3 a}+\frac{\operatorname{Subst}\left (\int \frac{-b-c x}{a+b x+c x^2} \, dx,x,x^3\right )}{3 a}\\ &=\frac{\log (x)}{a}-\frac{\operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^3\right )}{6 a}-\frac{b \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^3\right )}{6 a}\\ &=\frac{\log (x)}{a}-\frac{\log \left (a+b x^3+c x^6\right )}{6 a}+\frac{b \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^3\right )}{3 a}\\ &=\frac{b \tanh ^{-1}\left (\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right )}{3 a \sqrt{b^2-4 a c}}+\frac{\log (x)}{a}-\frac{\log \left (a+b x^3+c x^6\right )}{6 a}\\ \end{align*}
Mathematica [C] time = 0.0234794, size = 66, normalized size = 0.96 \[ \frac{\log (x)}{a}-\frac{\text{RootSum}\left [\text{$\#$1}^3 b+\text{$\#$1}^6 c+a\& ,\frac{\text{$\#$1}^3 c \log (x-\text{$\#$1})+b \log (x-\text{$\#$1})}{2 \text{$\#$1}^3 c+b}\& \right ]}{3 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 66, normalized size = 1. \begin{align*}{\frac{\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( c{x}^{6}+b{x}^{3}+a \right ) }{6\,a}}-{\frac{b}{3\,a}\arctan \left ({(2\,c{x}^{3}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61825, size = 510, normalized size = 7.39 \begin{align*} \left [\frac{\sqrt{b^{2} - 4 \, a c} b \log \left (\frac{2 \, c^{2} x^{6} + 2 \, b c x^{3} + b^{2} - 2 \, a c +{\left (2 \, c x^{3} + b\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{6} + b x^{3} + a}\right ) -{\left (b^{2} - 4 \, a c\right )} \log \left (c x^{6} + b x^{3} + a\right ) + 6 \,{\left (b^{2} - 4 \, a c\right )} \log \left (x\right )}{6 \,{\left (a b^{2} - 4 \, a^{2} c\right )}}, \frac{2 \, \sqrt{-b^{2} + 4 \, a c} b \arctan \left (-\frac{{\left (2 \, c x^{3} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) -{\left (b^{2} - 4 \, a c\right )} \log \left (c x^{6} + b x^{3} + a\right ) + 6 \,{\left (b^{2} - 4 \, a c\right )} \log \left (x\right )}{6 \,{\left (a b^{2} - 4 \, a^{2} c\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.6923, size = 253, normalized size = 3.67 \begin{align*} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{6 a \left (4 a c - b^{2}\right )} - \frac{1}{6 a}\right ) \log{\left (x^{3} + \frac{- 12 a^{2} c \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{6 a \left (4 a c - b^{2}\right )} - \frac{1}{6 a}\right ) + 3 a b^{2} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{6 a \left (4 a c - b^{2}\right )} - \frac{1}{6 a}\right ) - 2 a c + b^{2}}{b c} \right )} + \left (\frac{b \sqrt{- 4 a c + b^{2}}}{6 a \left (4 a c - b^{2}\right )} - \frac{1}{6 a}\right ) \log{\left (x^{3} + \frac{- 12 a^{2} c \left (\frac{b \sqrt{- 4 a c + b^{2}}}{6 a \left (4 a c - b^{2}\right )} - \frac{1}{6 a}\right ) + 3 a b^{2} \left (\frac{b \sqrt{- 4 a c + b^{2}}}{6 a \left (4 a c - b^{2}\right )} - \frac{1}{6 a}\right ) - 2 a c + b^{2}}{b c} \right )} + \frac{\log{\left (x \right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.43558, size = 89, normalized size = 1.29 \begin{align*} -\frac{b \arctan \left (\frac{2 \, c x^{3} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{3 \, \sqrt{-b^{2} + 4 \, a c} a} - \frac{\log \left (c x^{6} + b x^{3} + a\right )}{6 \, a} + \frac{\log \left ({\left | x \right |}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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