Optimal. Leaf size=172 \[ \frac {2}{5} t \sqrt {t^3+4}-\frac {8\ 2^{2/3} \sqrt {2+\sqrt {3}} \left (t+2^{2/3}\right ) \sqrt {\frac {t^2-2^{2/3} t+2 \sqrt [3]{2}}{\left (t+2^{2/3} \left (1+\sqrt {3}\right )\right )^2}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {t+2^{2/3} \left (1-\sqrt {3}\right )}{t+2^{2/3} \left (1+\sqrt {3}\right )}\right ),-7-4 \sqrt {3}\right )}{5 \sqrt [4]{3} \sqrt {\frac {t+2^{2/3}}{\left (t+2^{2/3} \left (1+\sqrt {3}\right )\right )^2}} \sqrt {t^3+4}} \]
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Rubi [A] time = 0.03, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {321, 218} \[ \frac {2}{5} t \sqrt {t^3+4}-\frac {8\ 2^{2/3} \sqrt {2+\sqrt {3}} \left (t+2^{2/3}\right ) \sqrt {\frac {t^2-2^{2/3} t+2 \sqrt [3]{2}}{\left (t+2^{2/3} \left (1+\sqrt {3}\right )\right )^2}} \text {EllipticF}\left (\sin ^{-1}\left (\frac {t+2^{2/3} \left (1-\sqrt {3}\right )}{t+2^{2/3} \left (1+\sqrt {3}\right )}\right ),-7-4 \sqrt {3}\right )}{5 \sqrt [4]{3} \sqrt {\frac {t+2^{2/3}}{\left (t+2^{2/3} \left (1+\sqrt {3}\right )\right )^2}} \sqrt {t^3+4}} \]
Antiderivative was successfully verified.
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Rule 218
Rule 321
Rubi steps
\begin {align*} \int \frac {t^3}{\sqrt {4+t^3}} \, dt &=\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}} F\left (\sin ^{-1}\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*}
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Mathematica [C] time = 0.01, size = 34, normalized size = 0.20 \[ \frac {2}{5} t \left (\sqrt {t^3+4}-2 \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};-\frac {t^3}{4}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {t^{3}}{\sqrt {t^{3} + 4}}, t\right ) \]
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 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 168, normalized size = 0.98 \[ \frac {2 \sqrt {t^{3}+4}\, t}{5}+\frac {8 i \sqrt {3}\, 2^{\frac {2}{3}} \sqrt {i \left (t -\frac {2^{\frac {2}{3}}}{2}-\frac {i \sqrt {3}\, 2^{\frac {2}{3}}}{2}\right ) \sqrt {3}\, 2^{\frac {1}{3}}}\, \sqrt {\frac {t +2^{\frac {2}{3}}}{\frac {3 \,2^{\frac {2}{3}}}{2}+\frac {i \sqrt {3}\, 2^{\frac {2}{3}}}{2}}}\, \sqrt {-i \left (t -\frac {2^{\frac {2}{3}}}{2}+\frac {i \sqrt {3}\, 2^{\frac {2}{3}}}{2}\right ) \sqrt {3}\, 2^{\frac {1}{3}}}\, \EllipticF \left (\frac {\sqrt {6}\, \sqrt {i \left (t -\frac {2^{\frac {2}{3}}}{2}-\frac {i \sqrt {3}\, 2^{\frac {2}{3}}}{2}\right ) \sqrt {3}\, 2^{\frac {1}{3}}}}{6}, \sqrt {\frac {i \sqrt {3}\, 2^{\frac {2}{3}}}{\frac {3 \,2^{\frac {2}{3}}}{2}+\frac {i \sqrt {3}\, 2^{\frac {2}{3}}}{2}}}\right )}{15 \sqrt {t^{3}+4}} \]
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 \[ \int \frac {t^{3}}{\sqrt {t^{3} + 4}}\,{d t} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 301, normalized size = 1.75 \[ \frac {2\,t\,\sqrt {t^3+4}}{5}-\frac {16\,\sqrt {-\frac {t-2^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2^{2/3}+2^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}}\,\sqrt {-\frac {t+2^{2/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2^{2/3}-2^{2/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}}\,\sqrt {\frac {t+2^{2/3}}{2^{2/3}+2^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}}\,\left (2^{2/3}+2^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {t+2^{2/3}}{2^{2/3}+2^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}}\right )\middle |\frac {2^{2/3}+2^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2^{2/3}-2^{2/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}\right )}{5\,\sqrt {t^3+\left (2^{2/3}+2^{2/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-2^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,t^2+\left (2\,2^{1/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-2\,2^{1/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-2\,2^{1/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,t-4\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.72, size = 31, normalized size = 0.18 \[ \frac {t^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {t^{3} e^{i \pi }}{4}} \right )}}{6 \Gamma \left (\frac {7}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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