Integrand size = 13, antiderivative size = 172 \[ \int \frac {t^3}{\sqrt {4+t^3}} \, dt=\frac {2}{5} t \sqrt {4+t^3}-\frac {8\ 2^{2/3} \sqrt {2+\sqrt {3}} \left (2^{2/3}+t\right ) \sqrt {\frac {2 \sqrt [3]{2}-2^{2/3} t+t^2}{\left (2^{2/3} \left (1+\sqrt {3}\right )+t\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1-\sqrt {3}\right )+t}{2^{2/3} \left (1+\sqrt {3}\right )+t}\right ),-7-4 \sqrt {3}\right )}{5 \sqrt [4]{3} \sqrt {\frac {2^{2/3}+t}{\left (2^{2/3} \left (1+\sqrt {3}\right )+t\right )^2}} \sqrt {4+t^3}} \]
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Time = 0.05 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {327, 224} \[ \int \frac {t^3}{\sqrt {4+t^3}} \, dt=\frac {2}{5} t \sqrt {t^3+4}-\frac {8\ 2^{2/3} \sqrt {2+\sqrt {3}} \left (t+2^{2/3}\right ) \sqrt {\frac {t^2-2^{2/3} t+2 \sqrt [3]{2}}{\left (t+2^{2/3} \left (1+\sqrt {3}\right )\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {t+2^{2/3} \left (1-\sqrt {3}\right )}{t+2^{2/3} \left (1+\sqrt {3}\right )}\right ),-7-4 \sqrt {3}\right )}{5 \sqrt [4]{3} \sqrt {\frac {t+2^{2/3}}{\left (t+2^{2/3} \left (1+\sqrt {3}\right )\right )^2}} \sqrt {t^3+4}} \]
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Rule 224
Rule 327
Rubi steps \begin{align*} \text {integral}& = \frac {2}{5} t \sqrt {4+t^3}-\frac {8}{5} \int \frac {1}{\sqrt {4+t^3}} \, dt \\ & = \frac {2}{5} t \sqrt {4+t^3}-\frac {8\ 2^{2/3} \sqrt {2+\sqrt {3}} \left (2^{2/3}+t\right ) \sqrt {\frac {2 \sqrt [3]{2}-2^{2/3} t+t^2}{\left (2^{2/3} \left (1+\sqrt {3}\right )+t\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1-\sqrt {3}\right )+t}{2^{2/3} \left (1+\sqrt {3}\right )+t}\right ),-7-4 \sqrt {3}\right )}{5 \sqrt [4]{3} \sqrt {\frac {2^{2/3}+t}{\left (2^{2/3} \left (1+\sqrt {3}\right )+t\right )^2}} \sqrt {4+t^3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 9.81 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.20 \[ \int \frac {t^3}{\sqrt {4+t^3}} \, dt=\frac {2}{5} t \left (\sqrt {4+t^3}-2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\frac {t^3}{4}\right )\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.10
method | result | size |
meijerg | \(\frac {t^{4} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},\frac {4}{3};\frac {7}{3};-\frac {t^{3}}{4}\right )}{8}\) | \(17\) |
default | \(\frac {2 t \sqrt {t^{3}+4}}{5}+\frac {8 i \sqrt {3}\, 2^{\frac {2}{3}} \sqrt {i \left (t -\frac {2^{\frac {2}{3}}}{2}-\frac {i \sqrt {3}\, 2^{\frac {2}{3}}}{2}\right ) \sqrt {3}\, 2^{\frac {1}{3}}}\, \sqrt {\frac {2^{\frac {2}{3}}+t}{\frac {3 \,2^{\frac {2}{3}}}{2}+\frac {i \sqrt {3}\, 2^{\frac {2}{3}}}{2}}}\, \sqrt {-i \left (t -\frac {2^{\frac {2}{3}}}{2}+\frac {i \sqrt {3}\, 2^{\frac {2}{3}}}{2}\right ) \sqrt {3}\, 2^{\frac {1}{3}}}\, F\left (\frac {\sqrt {6}\, \sqrt {i \left (t -\frac {2^{\frac {2}{3}}}{2}-\frac {i \sqrt {3}\, 2^{\frac {2}{3}}}{2}\right ) \sqrt {3}\, 2^{\frac {1}{3}}}}{6}, \sqrt {\frac {i \sqrt {3}\, 2^{\frac {2}{3}}}{\frac {3 \,2^{\frac {2}{3}}}{2}+\frac {i \sqrt {3}\, 2^{\frac {2}{3}}}{2}}}\right )}{15 \sqrt {t^{3}+4}}\) | \(168\) |
risch | \(\frac {2 t \sqrt {t^{3}+4}}{5}+\frac {8 i \sqrt {3}\, 2^{\frac {2}{3}} \sqrt {i \left (t -\frac {2^{\frac {2}{3}}}{2}-\frac {i \sqrt {3}\, 2^{\frac {2}{3}}}{2}\right ) \sqrt {3}\, 2^{\frac {1}{3}}}\, \sqrt {\frac {2^{\frac {2}{3}}+t}{\frac {3 \,2^{\frac {2}{3}}}{2}+\frac {i \sqrt {3}\, 2^{\frac {2}{3}}}{2}}}\, \sqrt {-i \left (t -\frac {2^{\frac {2}{3}}}{2}+\frac {i \sqrt {3}\, 2^{\frac {2}{3}}}{2}\right ) \sqrt {3}\, 2^{\frac {1}{3}}}\, F\left (\frac {\sqrt {6}\, \sqrt {i \left (t -\frac {2^{\frac {2}{3}}}{2}-\frac {i \sqrt {3}\, 2^{\frac {2}{3}}}{2}\right ) \sqrt {3}\, 2^{\frac {1}{3}}}}{6}, \sqrt {\frac {i \sqrt {3}\, 2^{\frac {2}{3}}}{\frac {3 \,2^{\frac {2}{3}}}{2}+\frac {i \sqrt {3}\, 2^{\frac {2}{3}}}{2}}}\right )}{15 \sqrt {t^{3}+4}}\) | \(168\) |
elliptic | \(\frac {2 t \sqrt {t^{3}+4}}{5}+\frac {8 i \sqrt {3}\, 2^{\frac {2}{3}} \sqrt {i \left (t -\frac {2^{\frac {2}{3}}}{2}-\frac {i \sqrt {3}\, 2^{\frac {2}{3}}}{2}\right ) \sqrt {3}\, 2^{\frac {1}{3}}}\, \sqrt {\frac {2^{\frac {2}{3}}+t}{\frac {3 \,2^{\frac {2}{3}}}{2}+\frac {i \sqrt {3}\, 2^{\frac {2}{3}}}{2}}}\, \sqrt {-i \left (t -\frac {2^{\frac {2}{3}}}{2}+\frac {i \sqrt {3}\, 2^{\frac {2}{3}}}{2}\right ) \sqrt {3}\, 2^{\frac {1}{3}}}\, F\left (\frac {\sqrt {6}\, \sqrt {i \left (t -\frac {2^{\frac {2}{3}}}{2}-\frac {i \sqrt {3}\, 2^{\frac {2}{3}}}{2}\right ) \sqrt {3}\, 2^{\frac {1}{3}}}}{6}, \sqrt {\frac {i \sqrt {3}\, 2^{\frac {2}{3}}}{\frac {3 \,2^{\frac {2}{3}}}{2}+\frac {i \sqrt {3}\, 2^{\frac {2}{3}}}{2}}}\right )}{15 \sqrt {t^{3}+4}}\) | \(168\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.10 \[ \int \frac {t^3}{\sqrt {4+t^3}} \, dt=\frac {2}{5} \, \sqrt {t^{3} + 4} t - \frac {16}{5} \, {\rm weierstrassPInverse}\left (0, -16, t\right ) \]
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Time = 0.38 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.18 \[ \int \frac {t^3}{\sqrt {4+t^3}} \, dt=\frac {t^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {t^{3} e^{i \pi }}{4}} \right )}}{6 \Gamma \left (\frac {7}{3}\right )} \]
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\[ \int \frac {t^3}{\sqrt {4+t^3}} \, dt=\int { \frac {t^{3}}{\sqrt {t^{3} + 4}} \,d t } \]
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\[ \int \frac {t^3}{\sqrt {4+t^3}} \, dt=\int { \frac {t^{3}}{\sqrt {t^{3} + 4}} \,d t } \]
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Time = 0.08 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.75 \[ \int \frac {t^3}{\sqrt {4+t^3}} \, dt=\frac {2\,t\,\sqrt {t^3+4}}{5}-\frac {16\,\sqrt {-\frac {t-2^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2^{2/3}+2^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}}\,\sqrt {-\frac {t+2^{2/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2^{2/3}-2^{2/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}}\,\sqrt {\frac {t+2^{2/3}}{2^{2/3}+2^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}}\,\left (2^{2/3}+2^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {t+2^{2/3}}{2^{2/3}+2^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}}\right )\middle |\frac {2^{2/3}+2^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2^{2/3}-2^{2/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}\right )}{5\,\sqrt {t^3+\left (2^{2/3}+2^{2/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-2^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,t^2+\left (2\,2^{1/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-2\,2^{1/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-2\,2^{1/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,t-4\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
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