Optimal. Leaf size=80 \[ \frac{\log (x) \left (a+b x^3\right )}{a \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a \sqrt{a^2+2 a b x^3+b^2 x^6}} \]
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Rubi [A] time = 0.0339088, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {1355, 266, 36, 29, 31} \[ \frac{\log (x) \left (a+b x^3\right )}{a \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a \sqrt{a^2+2 a b x^3+b^2 x^6}} \]
Antiderivative was successfully verified.
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Rule 1355
Rule 266
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{a^2+2 a b x^3+b^2 x^6}} \, dx &=\frac{\left (a b+b^2 x^3\right ) \int \frac{1}{x \left (a b+b^2 x^3\right )} \, dx}{\sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{\left (a b+b^2 x^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (a b+b^2 x\right )} \, dx,x,x^3\right )}{3 \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{\left (a b+b^2 x^3\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^3\right )}{3 a b \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (b \left (a b+b^2 x^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a b+b^2 x} \, dx,x,x^3\right )}{3 a \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{\left (a+b x^3\right ) \log (x)}{a \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ \end{align*}
Mathematica [A] time = 0.011428, size = 42, normalized size = 0.52 \[ \frac{\left (a+b x^3\right ) \left (3 \log (x)-\log \left (a+b x^3\right )\right )}{3 a \sqrt{\left (a+b x^3\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 39, normalized size = 0.5 \begin{align*}{\frac{ \left ( b{x}^{3}+a \right ) \left ( 3\,\ln \left ( x \right ) -\ln \left ( b{x}^{3}+a \right ) \right ) }{3\,a}{\frac{1}{\sqrt{ \left ( b{x}^{3}+a \right ) ^{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 = 1.88243, size = 49, normalized size = 0.61 \begin{align*} -\frac{\log \left (b x^{3} + a\right ) - 3 \, \log \left (x\right )}{3 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.262047, size = 15, normalized size = 0.19 \begin{align*} \frac{\log{\left (x \right )}}{a} - \frac{\log{\left (\frac{a}{b} + x^{3} \right )}}{3 a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1237, size = 43, normalized size = 0.54 \begin{align*} -\frac{1}{3} \,{\left (\frac{\log \left ({\left | b x^{3} + a \right |}\right )}{a} - \frac{3 \, \log \left ({\left | x \right |}\right )}{a}\right )} \mathrm{sgn}\left (b x^{3} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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