Optimal. Leaf size=256 \[ \frac{2 \left (\frac{b}{a}\right )^{2/3} \sqrt{b x^3-a}}{b \left (x \left (-\sqrt [3]{\frac{b}{a}}\right )-\sqrt{3}+1\right )}-\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (1-x \sqrt [3]{\frac{b}{a}}\right ) \sqrt{\frac{x^2 \left (\frac{b}{a}\right )^{2/3}+x \sqrt [3]{\frac{b}{a}}+1}{\left (x \left (-\sqrt [3]{\frac{b}{a}}\right )-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{b}{a}} x+\sqrt{3}+1}{-\sqrt [3]{\frac{b}{a}} x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [3]{\frac{b}{a}} \sqrt{-\frac{1-x \sqrt [3]{\frac{b}{a}}}{\left (x \left (-\sqrt [3]{\frac{b}{a}}\right )-\sqrt{3}+1\right )^2}} \sqrt{b x^3-a}} \]
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Rubi [A] time = 0.0831561, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03, Rules used = {1879} \[ \frac{2 \left (\frac{b}{a}\right )^{2/3} \sqrt{b x^3-a}}{b \left (x \left (-\sqrt [3]{\frac{b}{a}}\right )-\sqrt{3}+1\right )}-\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (1-x \sqrt [3]{\frac{b}{a}}\right ) \sqrt{\frac{x^2 \left (\frac{b}{a}\right )^{2/3}+x \sqrt [3]{\frac{b}{a}}+1}{\left (x \left (-\sqrt [3]{\frac{b}{a}}\right )-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{b}{a}} x+\sqrt{3}+1}{-\sqrt [3]{\frac{b}{a}} x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [3]{\frac{b}{a}} \sqrt{-\frac{1-x \sqrt [3]{\frac{b}{a}}}{\left (x \left (-\sqrt [3]{\frac{b}{a}}\right )-\sqrt{3}+1\right )^2}} \sqrt{b x^3-a}} \]
Antiderivative was successfully verified.
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Rule 1879
Rubi steps
\begin{align*} \int \frac{1+\sqrt{3}-\sqrt [3]{\frac{b}{a}} x}{\sqrt{-a+b x^3}} \, dx &=\frac{2 \left (\frac{b}{a}\right )^{2/3} \sqrt{-a+b x^3}}{b \left (1-\sqrt{3}-\sqrt [3]{\frac{b}{a}} x\right )}-\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{\frac{b}{a}} x\right ) \sqrt{\frac{1+\sqrt [3]{\frac{b}{a}} x+\left (\frac{b}{a}\right )^{2/3} x^2}{\left (1-\sqrt{3}-\sqrt [3]{\frac{b}{a}} x\right )^2}} E\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-\sqrt [3]{\frac{b}{a}} x}{1-\sqrt{3}-\sqrt [3]{\frac{b}{a}} x}\right )|-7+4 \sqrt{3}\right )}{\sqrt [3]{\frac{b}{a}} \sqrt{-\frac{1-\sqrt [3]{\frac{b}{a}} x}{\left (1-\sqrt{3}-\sqrt [3]{\frac{b}{a}} x\right )^2}} \sqrt{-a+b x^3}}\\ \end{align*}
Mathematica [C] time = 0.032436, size = 90, normalized size = 0.35 \[ -\frac{x \sqrt{1-\frac{b x^3}{a}} \left (x \sqrt [3]{\frac{b}{a}} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\frac{b x^3}{a}\right )-2 \left (1+\sqrt{3}\right ) \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\frac{b x^3}{a}\right )\right )}{2 \sqrt{b x^3-a}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.019, size = 953, normalized size = 3.7 \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{x \left (\frac{b}{a}\right )^{\frac{1}{3}} - \sqrt{3} - 1}{\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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{x \left (\frac{b}{a}\right )^{\frac{1}{3}} - \sqrt{3} - 1}{\sqrt{b x^{3} - a}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.75303, size = 114, normalized size = 0.45 \begin{align*} \frac{i x^{2} \sqrt [3]{\frac{b}{a}} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\frac{b x^{3}}{a}} \right )}}{3 \sqrt{a} \Gamma \left (\frac{5}{3}\right )} - \frac{\sqrt{3} i x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3}}{a}} \right )}}{3 \sqrt{a} \Gamma \left (\frac{4}{3}\right )} - \frac{i x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3}}{a}} \right )}}{3 \sqrt{a} \Gamma \left (\frac{4}{3}\right )} \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 \left (\frac{b}{a}\right )^{\frac{1}{3}} - \sqrt{3} - 1}{\sqrt{b x^{3} - a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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