Optimal. Leaf size=50 \[ \frac{2 \tanh ^{-1}\left (\frac{\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt{a+b x^3}}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}} \]
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Rubi [A] time = 0.132317, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.047, Rules used = {2138, 206} \[ \frac{2 \tanh ^{-1}\left (\frac{\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt{a+b x^3}}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}} \]
Antiderivative was successfully verified.
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Rule 2138
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt [3]{a}+\sqrt [3]{b} x}{\left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{a+b x^3}} \, dx &=\frac{\left (2 \sqrt [3]{a}\right ) \operatorname{Subst}\left (\int \frac{1}{9-a x^2} \, dx,x,\frac{\left (1+\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2}{\sqrt{a+b x^3}}\right )}{\sqrt [3]{b}}\\ &=\frac{2 \tanh ^{-1}\left (\frac{\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt{a+b x^3}}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}}\\ \end{align*}
Mathematica [A] time = 0.0247096, size = 51, normalized size = 1.02 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} \left (\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )^2}{3 \sqrt{a+b x^3}}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.085, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \sqrt [3]{a}+\sqrt [3]{b}x \right ) \left ( 2\,\sqrt [3]{a}-\sqrt [3]{b}x \right ) ^{-1}{\frac{1}{\sqrt{b{x}^{3}+a}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{b^{\frac{1}{3}} x + a^{\frac{1}{3}}}{\sqrt{b x^{3} + a}{\left (b^{\frac{1}{3}} x - 2 \, a^{\frac{1}{3}}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{\sqrt [3]{a}}{- 2 \sqrt [3]{a} \sqrt{a + b x^{3}} + \sqrt [3]{b} x \sqrt{a + b x^{3}}}\, dx - \int \frac{\sqrt [3]{b} x}{- 2 \sqrt [3]{a} \sqrt{a + b x^{3}} + \sqrt [3]{b} x \sqrt{a + b x^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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