Integrand size = 17, antiderivative size = 529 \[ \int \frac {2+3 x}{\sqrt [3]{4+27 x^2}} \, dx=\frac {1}{12} \left (4+27 x^2\right )^{2/3}-\frac {6 x}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}+\frac {\sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{4+27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4+27 x^2}+\left (4+27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}} E\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}\right )|-7+4 \sqrt {3}\right )}{3\ 3^{3/4} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4+27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}}}-\frac {2\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{4+27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4+27 x^2}+\left (4+27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}\right ),-7+4 \sqrt {3}\right )}{9 \sqrt [4]{3} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4+27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}}} \]
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Time = 0.20 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {655, 241, 310, 225, 1893} \[ \int \frac {2+3 x}{\sqrt [3]{4+27 x^2}} \, dx=-\frac {2\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt {\frac {\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}\right ),-7+4 \sqrt {3}\right )}{9 \sqrt [4]{3} \sqrt {-\frac {2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}+\frac {\sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt {\frac {\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} E\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt {3}\right )}{3\ 3^{3/4} \sqrt {-\frac {2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}-\frac {6 x}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}+\frac {1}{12} \left (27 x^2+4\right )^{2/3} \]
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Rule 225
Rule 241
Rule 310
Rule 655
Rule 1893
Rubi steps \begin{align*} \text {integral}& = \frac {1}{12} \left (4+27 x^2\right )^{2/3}+2 \int \frac {1}{\sqrt [3]{4+27 x^2}} \, dx \\ & = \frac {1}{12} \left (4+27 x^2\right )^{2/3}+\frac {\sqrt {x^2} \text {Subst}\left (\int \frac {x}{\sqrt {-4+x^3}} \, dx,x,\sqrt [3]{4+27 x^2}\right )}{\sqrt {3} x} \\ & = \frac {1}{12} \left (4+27 x^2\right )^{2/3}-\frac {\sqrt {x^2} \text {Subst}\left (\int \frac {2^{2/3} \left (1+\sqrt {3}\right )-x}{\sqrt {-4+x^3}} \, dx,x,\sqrt [3]{4+27 x^2}\right )}{\sqrt {3} x}+\frac {\left (2^{2/3} \left (1+\sqrt {3}\right ) \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-4+x^3}} \, dx,x,\sqrt [3]{4+27 x^2}\right )}{\sqrt {3} x} \\ & = \frac {1}{12} \left (4+27 x^2\right )^{2/3}-\frac {6 x}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}+\frac {\sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{4+27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4+27 x^2}+\left (4+27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}\right )|-7+4 \sqrt {3}\right )}{3\ 3^{3/4} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4+27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}}}-\frac {2\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{4+27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4+27 x^2}+\left (4+27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}\right )|-7+4 \sqrt {3}\right )}{9 \sqrt [4]{3} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4+27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 9.92 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.08 \[ \int \frac {2+3 x}{\sqrt [3]{4+27 x^2}} \, dx=\frac {1}{12} \left (4+27 x^2\right )^{2/3}+\sqrt [3]{2} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},-\frac {27 x^2}{4}\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 2.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.05
method | result | size |
risch | \(\frac {\left (27 x^{2}+4\right )^{\frac {2}{3}}}{12}+2^{\frac {1}{3}} x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};-\frac {27 x^{2}}{4}\right )\) | \(29\) |
meijerg | \(2^{\frac {1}{3}} x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};-\frac {27 x^{2}}{4}\right )+\frac {3 \,2^{\frac {1}{3}} x^{2} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},1;2;-\frac {27 x^{2}}{4}\right )}{4}\) | \(37\) |
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\[ \int \frac {2+3 x}{\sqrt [3]{4+27 x^2}} \, dx=\int { \frac {3 \, x + 2}{{\left (27 \, x^{2} + 4\right )}^{\frac {1}{3}}} \,d x } \]
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Time = 1.18 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.07 \[ \int \frac {2+3 x}{\sqrt [3]{4+27 x^2}} \, dx=\sqrt [3]{2} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {27 x^{2} e^{i \pi }}{4}} \right )} + \frac {\left (27 x^{2} + 4\right )^{\frac {2}{3}}}{12} \]
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\[ \int \frac {2+3 x}{\sqrt [3]{4+27 x^2}} \, dx=\int { \frac {3 \, x + 2}{{\left (27 \, x^{2} + 4\right )}^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {2+3 x}{\sqrt [3]{4+27 x^2}} \, dx=\int { \frac {3 \, x + 2}{{\left (27 \, x^{2} + 4\right )}^{\frac {1}{3}}} \,d x } \]
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Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.05 \[ \int \frac {2+3 x}{\sqrt [3]{4+27 x^2}} \, dx=\frac {{\left (27\,x^2+4\right )}^{2/3}}{12}+2^{1/3}\,x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {1}{2};\ \frac {3}{2};\ -\frac {27\,x^2}{4}\right ) \]
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