Optimal. Leaf size=330 \[ \frac{\left (2-\sqrt{3}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{3} \sqrt [6]{a} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{-a-b x^3}}\right )}{3 \sqrt{2} \sqrt [4]{3} a^{5/6} b^{2/3}}+\frac{\left (2-\sqrt{3}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (1-\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{-a-b x^3}}\right )}{6 \sqrt{2} \sqrt [4]{3} a^{5/6} b^{2/3}}+\frac{\left (2-\sqrt{3}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{3} \left (1+\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{-a-b x^3}}\right )}{2 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}-\frac{\left (2-\sqrt{3}\right ) \tanh ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt{-a-b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right )}{3 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}} \]
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Rubi [A] time = 0.0666485, antiderivative size = 330, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.027, Rules used = {488} \[ \frac{\left (2-\sqrt{3}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{3} \sqrt [6]{a} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{-a-b x^3}}\right )}{3 \sqrt{2} \sqrt [4]{3} a^{5/6} b^{2/3}}+\frac{\left (2-\sqrt{3}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (1-\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{-a-b x^3}}\right )}{6 \sqrt{2} \sqrt [4]{3} a^{5/6} b^{2/3}}+\frac{\left (2-\sqrt{3}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{3} \left (1+\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{-a-b x^3}}\right )}{2 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}-\frac{\left (2-\sqrt{3}\right ) \tanh ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt{-a-b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right )}{3 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}} \]
Antiderivative was successfully verified.
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Rule 488
Rubi steps
\begin{align*} \int \frac{x}{\sqrt{-a-b x^3} \left (-2 \left (5+3 \sqrt{3}\right ) a-b x^3\right )} \, dx &=\frac{\left (2-\sqrt{3}\right ) \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 )}{3 \sqrt{2} \sqrt [4]{3} a^{5/6} b^{2/3}}+\frac{\left (2-\sqrt{3}\right ) \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 )}{6 \sqrt{2} \sqrt [4]{3} a^{5/6} b^{2/3}}+\frac{\left (2-\sqrt{3}\right ) \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} 3^{3/4} a^{5/6} b^{2/3}}-\frac{\left (2-\sqrt{3}\right ) \tanh ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt{-a-b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right )}{3 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}\\ \end{align*}
Mathematica [C] time = 0.0671752, size = 87, normalized size = 0.26 \[ -\frac{x^2 \sqrt{\frac{b x^3}{a}+1} F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right )}{\left (12 \sqrt{3} a+20 a\right ) \sqrt{-a-b x^3}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.059, size = 541, normalized size = 1.6 \begin{align*}{\frac{{\frac{i}{27}}\sqrt{2}}{a{b}^{3}}\sum _{{\it \_alpha}={\it RootOf} \left ( b{{\it \_Z}}^{3}+6\,a\sqrt{3}+10\,a \right ) }{\frac{1}{{\it \_alpha}}\sqrt [3]{-a{b}^{2}}\sqrt{{{\frac{i}{2}}b \left ( 2\,x+{\frac{1}{b} \left ( -i\sqrt{3}\sqrt [3]{-a{b}^{2}}+\sqrt [3]{-a{b}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}}\sqrt{{b \left ( x-{\frac{1}{b}\sqrt [3]{-a{b}^{2}}} \right ) \left ( -3\,\sqrt [3]{-a{b}^{2}}+i\sqrt{3}\sqrt [3]{-a{b}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}b \left ( 2\,x+{\frac{1}{b} \left ( i\sqrt{3}\sqrt [3]{-a{b}^{2}}+\sqrt [3]{-a{b}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}} \left ( -3\,i\sqrt [3]{-a{b}^{2}}{\it \_alpha}\,\sqrt{3}b+4\,{b}^{2}{{\it \_alpha}}^{2}\sqrt{3}+3\,i \left ( -a{b}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}-2\,\sqrt{3}\sqrt [3]{-a{b}^{2}}{\it \_alpha}\,b+6\,i\sqrt [3]{-a{b}^{2}}{\it \_alpha}\,b-6\,{b}^{2}{{\it \_alpha}}^{2}-2\,\sqrt{3} \left ( -a{b}^{2} \right ) ^{2/3}-6\,i \left ( -a{b}^{2} \right ) ^{{\frac{2}{3}}}+3\,\sqrt [3]{-a{b}^{2}}{\it \_alpha}\,b+3\, \left ( -a{b}^{2} \right ) ^{2/3} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}b \left ( x+{\frac{1}{2\,b}\sqrt [3]{-a{b}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}}},-{\frac{1}{6\,ab} \left ( 2\,i\sqrt{3}\sqrt [3]{-a{b}^{2}}{{\it \_alpha}}^{2}b-i\sqrt{3} \left ( -a{b}^{2} \right ) ^{{\frac{2}{3}}}{\it \_alpha}-4\,i\sqrt [3]{-a{b}^{2}}{{\it \_alpha}}^{2}b+2\,\sqrt{3} \left ( -a{b}^{2} \right ) ^{2/3}{\it \_alpha}+i\sqrt{3}ab+2\,i \left ( -a{b}^{2} \right ) ^{{\frac{2}{3}}}{\it \_alpha}+2\,\sqrt{3}ab-3\, \left ( -a{b}^{2} \right ) ^{2/3}{\it \_alpha}-2\,iab-3\,ab \right ) },\sqrt{{\frac{i\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}} \left ( -{\frac{3}{2\,b}\sqrt [3]{-a{b}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{-b{x}^{3}-a}}}}} \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}{{\left (b x^{3} + 2 \, a{\left (3 \, \sqrt{3} + 5\right )}\right )} \sqrt{-b x^{3} - a}}\,{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}{10 a \sqrt{- a - b x^{3}} + 6 \sqrt{3} a \sqrt{- a - b x^{3}} + b x^{3} \sqrt{- 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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