Integrand size = 33, antiderivative size = 256 \[ \int \frac {1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\sqrt {-a+b x^3}} \, dx=\frac {2 \left (\frac {b}{a}\right )^{2/3} \sqrt {-a+b x^3}}{b \left (1-\sqrt {3}-\sqrt [3]{\frac {b}{a}} x\right )}-\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{\frac {b}{a}} x\right ) \sqrt {\frac {1+\sqrt [3]{\frac {b}{a}} x+\left (\frac {b}{a}\right )^{2/3} x^2}{\left (1-\sqrt {3}-\sqrt [3]{\frac {b}{a}} x\right )^2}} E\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{1-\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}\right )|-7+4 \sqrt {3}\right )}{\sqrt [3]{\frac {b}{a}} \sqrt {-\frac {1-\sqrt [3]{\frac {b}{a}} x}{\left (1-\sqrt {3}-\sqrt [3]{\frac {b}{a}} x\right )^2}} \sqrt {-a+b x^3}} \]
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Time = 0.07 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {1893} \[ \int \frac {1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\sqrt {-a+b x^3}} \, dx=\frac {2 \left (\frac {b}{a}\right )^{2/3} \sqrt {b x^3-a}}{b \left (x \left (-\sqrt [3]{\frac {b}{a}}\right )-\sqrt {3}+1\right )}-\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-x \sqrt [3]{\frac {b}{a}}\right ) \sqrt {\frac {x^2 \left (\frac {b}{a}\right )^{2/3}+x \sqrt [3]{\frac {b}{a}}+1}{\left (x \left (-\sqrt [3]{\frac {b}{a}}\right )-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{\frac {b}{a}} x+\sqrt {3}+1}{-\sqrt [3]{\frac {b}{a}} x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt [3]{\frac {b}{a}} \sqrt {-\frac {1-x \sqrt [3]{\frac {b}{a}}}{\left (x \left (-\sqrt [3]{\frac {b}{a}}\right )-\sqrt {3}+1\right )^2}} \sqrt {b x^3-a}} \]
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Rule 1893
Rubi steps \begin{align*} \text {integral}& = \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*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.35 \[ \int \frac {1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\sqrt {-a+b x^3}} \, dx=-\frac {x \sqrt {1-\frac {b x^3}{a}} \left (-2 \left (1+\sqrt {3}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\frac {b x^3}{a}\right )+\sqrt [3]{\frac {b}{a}} x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\frac {b x^3}{a}\right )\right )}{2 \sqrt {-a+b x^3}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 952 vs. \(2 (211 ) = 422\).
Time = 1.67 (sec) , antiderivative size = 953, normalized size of antiderivative = 3.72
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.20 \[ \int \frac {1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\sqrt {-a+b x^3}} \, dx=\frac {2 \, {\left (\sqrt {b} {\left (\sqrt {3} + 1\right )} {\rm weierstrassPInverse}\left (0, \frac {4 \, a}{b}, x\right ) + \sqrt {b} \left (\frac {b}{a}\right )^{\frac {1}{3}} {\rm weierstrassZeta}\left (0, \frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, \frac {4 \, a}{b}, x\right )\right )\right )}}{b} \]
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Time = 1.40 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.45 \[ \int \frac {1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\sqrt {-a+b x^3}} \, dx=\frac {i x^{2} \sqrt [3]{\frac {b}{a}} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3}}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {5}{3}\right )} - \frac {\sqrt {3} i x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3}}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {4}{3}\right )} - \frac {i x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3}}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {4}{3}\right )} \]
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\[ \int \frac {1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\sqrt {-a+b x^3}} \, dx=\int { -\frac {x \left (\frac {b}{a}\right )^{\frac {1}{3}} - \sqrt {3} - 1}{\sqrt {b x^{3} - a}} \,d x } \]
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Exception generated. \[ \int \frac {1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\sqrt {-a+b x^3}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\sqrt {-a+b x^3}} \, dx=\int \frac {\sqrt {3}-x\,{\left (\frac {b}{a}\right )}^{1/3}+1}{\sqrt {b\,x^3-a}} \,d x \]
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