Optimal. Leaf size=288 \[ \frac{2 \sqrt{2-\sqrt{3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{-a-b x^3}}-\frac{2\ 2^{2/3} \tanh ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{b} x\right )}{\sqrt{-a-b x^3}}\right )}{3 \sqrt{3} \sqrt [6]{a} b^{2/3}} \]
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Rubi [A] time = 0.370384, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108, Rules used = {2139, 219, 2137, 206} \[ \frac{2 \sqrt{2-\sqrt{3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{-a-b x^3}}-\frac{2\ 2^{2/3} \tanh ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{b} x\right )}{\sqrt{-a-b x^3}}\right )}{3 \sqrt{3} \sqrt [6]{a} b^{2/3}} \]
Antiderivative was successfully verified.
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Rule 2139
Rule 219
Rule 2137
Rule 206
Rubi steps
\begin{align*} \int \frac{x}{\left (2^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{-a-b x^3}} \, dx &=\frac{\int \frac{1}{\sqrt{-a-b x^3}} \, dx}{3 \sqrt [3]{b}}-\frac{\int \frac{2^{2/3} \sqrt [3]{a}-2 \sqrt [3]{b} x}{\left (2^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{-a-b x^3}} \, dx}{3 \sqrt [3]{b}}\\ &=\frac{2 \sqrt{2-\sqrt{3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{-a-b x^3}}-\frac{\left (2\ 2^{2/3} \sqrt [3]{a}\right ) \operatorname{Subst}\left (\int \frac{1}{1-3 a x^2} \, dx,x,\frac{1+\frac{\sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{-a-b x^3}}\right )}{3 b^{2/3}}\\ &=-\frac{2\ 2^{2/3} \tanh ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{b} x\right )}{\sqrt{-a-b x^3}}\right )}{3 \sqrt{3} \sqrt [6]{a} b^{2/3}}+\frac{2 \sqrt{2-\sqrt{3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{-a-b x^3}}\\ \end{align*}
Mathematica [C] time = 0.506538, size = 375, normalized size = 1.3 \[ \frac{2 \sqrt{\frac{\sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (\frac{\sqrt [3]{-1} 2^{2/3} \left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a} \sqrt{\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1} \Pi \left (\frac{i \sqrt{3}}{\sqrt [3]{-1}+2^{2/3}};\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{3}}-\frac{\left (\sqrt [3]{-1}+2^{2/3}\right ) \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\sqrt [6]{-1}-\frac{i \sqrt [3]{b} x}{\sqrt [3]{a}}} F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )}{\sqrt [4]{3}}\right )}{\left (\sqrt [3]{-1}+2^{2/3}\right ) b^{2/3} \sqrt{\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{-a-b x^3}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{x \left ({2}^{{\frac{2}{3}}}\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{x}{\sqrt{-b x^{3} - a}{\left (b^{\frac{1}{3}} x + 2^{\frac{2}{3}} 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{x}{\sqrt{- a - b x^{3}} \left (2^{\frac{2}{3}} \sqrt [3]{a} + \sqrt [3]{b} x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{-b x^{3} - a}{\left (b^{\frac{1}{3}} x + 2^{\frac{2}{3}} a^{\frac{1}{3}}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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