Optimal. Leaf size=758 \[ \frac{2 \sqrt{2} \sqrt [3]{a} \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}} \text{EllipticF}\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 )}{\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{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} \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}} E\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 )}{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 \sqrt{a-b x^3}}{b^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac{\sqrt [4]{3} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (1+\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a-b x^3}}\right )}{2 \sqrt{2} b^{2/3}}-\frac{\sqrt [4]{3} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [4]{3} \sqrt [6]{a} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+2 \sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a-b x^3}}\right )}{\sqrt{2} b^{2/3}}+\frac{3^{3/4} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt [4]{3} \left (1-\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a-b x^3}}\right )}{2 \sqrt{2} b^{2/3}}+\frac{\sqrt [6]{a} \tanh ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt{a-b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right )}{\sqrt{2} \sqrt [4]{3} b^{2/3}} \]
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Rubi [A] time = 0.236822, antiderivative size = 758, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {489, 303, 218, 1877, 487} \[ \frac{2 \sqrt{2} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{a-b x^3}}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{a-b x^3}}+\frac{2 \sqrt{a-b x^3}}{b^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac{\sqrt [4]{3} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (1+\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a-b x^3}}\right )}{2 \sqrt{2} b^{2/3}}-\frac{\sqrt [4]{3} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [4]{3} \sqrt [6]{a} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+2 \sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a-b x^3}}\right )}{\sqrt{2} b^{2/3}}+\frac{3^{3/4} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt [4]{3} \left (1-\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a-b x^3}}\right )}{2 \sqrt{2} b^{2/3}}+\frac{\sqrt [6]{a} \tanh ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt{a-b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right )}{\sqrt{2} \sqrt [4]{3} b^{2/3}} \]
Antiderivative was successfully verified.
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Rule 489
Rule 303
Rule 218
Rule 1877
Rule 487
Rubi steps
\begin{align*} \int \frac{x \sqrt{a-b x^3}}{2 \left (5-3 \sqrt{3}\right ) a-b x^3} \, dx &=-\left (\left (3 \left (3-2 \sqrt{3}\right ) a\right ) \int \frac{x}{\sqrt{a-b x^3} \left (2 \left (5-3 \sqrt{3}\right ) a-b x^3\right )} \, dx\right )+\int \frac{x}{\sqrt{a-b x^3}} \, dx\\ &=-\frac{\sqrt [4]{3} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (1+\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a-b x^3}}\right )}{2 \sqrt{2} b^{2/3}}-\frac{\sqrt [4]{3} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [4]{3} \sqrt [6]{a} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+2 \sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a-b x^3}}\right )}{\sqrt{2} b^{2/3}}+\frac{3^{3/4} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt [4]{3} \left (1-\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a-b x^3}}\right )}{2 \sqrt{2} b^{2/3}}+\frac{\sqrt [6]{a} \tanh ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt{a-b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right )}{\sqrt{2} \sqrt [4]{3} b^{2/3}}-\frac{\int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt{a-b x^3}} \, dx}{\sqrt [3]{b}}-\frac{\left (\sqrt{2 \left (2-\sqrt{3}\right )} \sqrt [3]{a}\right ) \int \frac{1}{\sqrt{a-b x^3}} \, dx}{\sqrt [3]{b}}\\ &=\frac{2 \sqrt{a-b x^3}}{b^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac{\sqrt [4]{3} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (1+\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a-b x^3}}\right )}{2 \sqrt{2} b^{2/3}}-\frac{\sqrt [4]{3} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [4]{3} \sqrt [6]{a} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+2 \sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a-b x^3}}\right )}{\sqrt{2} b^{2/3}}+\frac{3^{3/4} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt [4]{3} \left (1-\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a-b x^3}}\right )}{2 \sqrt{2} b^{2/3}}+\frac{\sqrt [6]{a} \tanh ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt{a-b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right )}{\sqrt{2} \sqrt [4]{3} b^{2/3}}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} \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}} E\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 )}{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 \sqrt{2} \sqrt [3]{a} \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 )}{\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 = 0.0779448, size = 80, normalized size = 0.11 \[ \frac{x^2 \sqrt{1-\frac{b x^3}{a}} F_1\left (\frac{2}{3};-\frac{1}{2},1;\frac{5}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right )}{\left (20-12 \sqrt{3}\right ) \sqrt{a-b x^3}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.134, size = 924, normalized size = 1.2 \begin{align*} \text{result too large to display} \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{\sqrt{-b x^{3} + a} x}{b x^{3} + 2 \, a{\left (3 \, \sqrt{3} - 5\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}}}{- 10 a + 6 \sqrt{3} a + b x^{3}}\, 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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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