Optimal. Leaf size=251 \[ \frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (x \sqrt [3]{\frac {b}{a}}+1\right ) \sqrt {\frac {x^2 \left (\frac {b}{a}\right )^{2/3}-x \sqrt [3]{\frac {b}{a}}+1}{\left (x \sqrt [3]{\frac {b}{a}}-\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {\sqrt [3]{\frac {b}{a}} x+\sqrt {3}+1}{\sqrt [3]{\frac {b}{a}} x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt [3]{\frac {b}{a}} \sqrt {-\frac {x \sqrt [3]{\frac {b}{a}}+1}{\left (x \sqrt [3]{\frac {b}{a}}-\sqrt {3}+1\right )^2}} \sqrt {-a-b x^3}}-\frac {2 \left (\frac {b}{a}\right )^{2/3} \sqrt {-a-b x^3}}{b \left (x \sqrt [3]{\frac {b}{a}}-\sqrt {3}+1\right )} \]
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Rubi [A] time = 0.06, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {1879} \[ \frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (x \sqrt [3]{\frac {b}{a}}+1\right ) \sqrt {\frac {x^2 \left (\frac {b}{a}\right )^{2/3}-x \sqrt [3]{\frac {b}{a}}+1}{\left (x \sqrt [3]{\frac {b}{a}}-\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {\sqrt [3]{\frac {b}{a}} x+\sqrt {3}+1}{\sqrt [3]{\frac {b}{a}} x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt [3]{\frac {b}{a}} \sqrt {-\frac {x \sqrt [3]{\frac {b}{a}}+1}{\left (x \sqrt [3]{\frac {b}{a}}-\sqrt {3}+1\right )^2}} \sqrt {-a-b x^3}}-\frac {2 \left (\frac {b}{a}\right )^{2/3} \sqrt {-a-b x^3}}{b \left (x \sqrt [3]{\frac {b}{a}}-\sqrt {3}+1\right )} \]
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*}
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Mathematica [C] time = 0.04, size = 92, normalized size = 0.37 \[ \frac {x \sqrt {\frac {b x^3}{a}+1} \left (2 \left (1+\sqrt {3}\right ) \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};-\frac {b x^3}{a}\right )+x \sqrt [3]{\frac {b}{a}} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )\right )}{2 \sqrt {-a-b x^3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-b x^{3} - a} x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \sqrt {-b x^{3} - a} {\left (\sqrt {3} + 1\right )}}{b x^{3} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 1013, normalized size = 4.04 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \sqrt {3} + 1}{\sqrt {-b x^{3} - a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {3}+x\,{\left (\frac {b}{a}\right )}^{1/3}+1}{\sqrt {-b\,x^3-a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.61, size = 131, normalized size = 0.52 \[ - \frac {i x^{2} \sqrt [3]{\frac {b}{a}} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {5}{3}\right )} - \frac {\sqrt {3} i x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {4}{3}\right )} - \frac {i x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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