Optimal. Leaf size=75 \[ \frac{b x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{a \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]
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Rubi [A] time = 0.0195575, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1355, 14} \[ \frac{b x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{a \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]
Antiderivative was successfully verified.
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Rule 1355
Rule 14
Rubi steps
\begin{align*} \int \frac{\sqrt{a^2+2 a b x^3+b^2 x^6}}{x} \, dx &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \frac{a b+b^2 x^3}{x} \, dx}{a b+b^2 x^3}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \left (\frac{a b}{x}+b^2 x^2\right ) \, dx}{a b+b^2 x^3}\\ &=\frac{b x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6} \log (x)}{a+b x^3}\\ \end{align*}
Mathematica [A] time = 0.0113884, size = 37, normalized size = 0.49 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (3 a \log (x)+b x^3\right )}{3 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 34, normalized size = 0.5 \begin{align*}{\frac{b{x}^{3}+3\,a\ln \left ( x \right ) }{3\,b{x}^{3}+3\,a}\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 = 1.74956, size = 30, normalized size = 0.4 \begin{align*} \frac{1}{3} \, b x^{3} + a \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.116376, size = 10, normalized size = 0.13 \begin{align*} a \log{\left (x \right )} + \frac{b x^{3}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10445, size = 38, normalized size = 0.51 \begin{align*} \frac{1}{3} \, b x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + a \log \left ({\left | x \right |}\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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