Optimal. Leaf size=557 \[ \frac{8\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right ),4 \sqrt{3}-7\right )}{21 \sqrt [4]{3} x \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}}-\frac{1}{21} i \left (4-27 x^2\right )^{2/3} (2+3 i x)-\frac{5}{21} i \left (4-27 x^2\right )^{2/3}-\frac{72 x}{7 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )}-\frac{4 \sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{7\ 3^{3/4} x \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}} \]
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Rubi [A] time = 0.315223, antiderivative size = 557, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {743, 641, 235, 304, 219, 1879} \[ -\frac{1}{21} i \left (4-27 x^2\right )^{2/3} (2+3 i x)-\frac{5}{21} i \left (4-27 x^2\right )^{2/3}-\frac{72 x}{7 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )}+\frac{8\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{21 \sqrt [4]{3} x \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}}-\frac{4 \sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{7\ 3^{3/4} x \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}} \]
Antiderivative was successfully verified.
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Rule 743
Rule 641
Rule 235
Rule 304
Rule 219
Rule 1879
Rubi steps
\begin{align*} \int \frac{(2+3 i x)^2}{\sqrt [3]{4-27 x^2}} \, dx &=-\frac{1}{21} i (2+3 i x) \left (4-27 x^2\right )^{2/3}-\frac{1}{63} \int \frac{-216-540 i x}{\sqrt [3]{4-27 x^2}} \, dx\\ &=-\frac{5}{21} i \left (4-27 x^2\right )^{2/3}-\frac{1}{21} i (2+3 i x) \left (4-27 x^2\right )^{2/3}+\frac{24}{7} \int \frac{1}{\sqrt [3]{4-27 x^2}} \, dx\\ &=-\frac{5}{21} i \left (4-27 x^2\right )^{2/3}-\frac{1}{21} i (2+3 i x) \left (4-27 x^2\right )^{2/3}-\frac{\left (4 \sqrt{3} \sqrt{-x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{-4+x^3}} \, dx,x,\sqrt [3]{4-27 x^2}\right )}{7 x}\\ &=-\frac{5}{21} i \left (4-27 x^2\right )^{2/3}-\frac{1}{21} i (2+3 i x) \left (4-27 x^2\right )^{2/3}+\frac{\left (4 \sqrt{3} \sqrt{-x^2}\right ) \operatorname{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 )}{7 x}-\frac{\left (8 \sqrt [6]{2} \sqrt{3} \sqrt{-x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-4+x^3}} \, dx,x,\sqrt [3]{4-27 x^2}\right )}{7 \sqrt{2-\sqrt{3}} x}\\ &=-\frac{5}{21} i \left (4-27 x^2\right )^{2/3}-\frac{1}{21} i (2+3 i x) \left (4-27 x^2\right )^{2/3}-\frac{72 x}{7 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )}-\frac{4 \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 )}{7\ 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{8\ 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 )}{21 \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*}
Mathematica [C] time = 0.0313351, size = 51, normalized size = 0.09 \[ \frac{12}{7} \sqrt [3]{2} x \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{27 x^2}{4}\right )+\left (4-27 x^2\right )^{2/3} \left (\frac{x}{7}-\frac{i}{3}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.273, size = 43, normalized size = 0.1 \begin{align*} -{\frac{ \left ( -7\,i+3\,x \right ) \left ( 27\,{x}^{2}-4 \right ) }{21}{\frac{1}{\sqrt [3]{-27\,{x}^{2}+4}}}}+{\frac{12\,\sqrt [3]{2}x}{7}{\mbox{$_2$F$_1$}({\frac{1}{3}},{\frac{1}{2}};\,{\frac{3}{2}};\,{\frac{27\,{x}^{2}}{4}})}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (3 i \, x + 2\right )}^{2}}{{\left (-27 \, x^{2} + 4\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{21 \, x{\rm integral}\left (\frac{32 \,{\left (-27 \, x^{2} + 4\right )}^{\frac{2}{3}}}{21 \,{\left (27 \, x^{4} - 4 \, x^{2}\right )}}, x\right ) +{\left (3 \, x^{2} - 7 i \, x - 8\right )}{\left (-27 \, x^{2} + 4\right )}^{\frac{2}{3}}}{21 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.47145, size = 73, normalized size = 0.13 \begin{align*} - \frac{3 \sqrt [3]{2} x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{27 x^{2} e^{2 i \pi }}{4}} \right )}}{2} + 2 \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^{2 i \pi }}{4}} \right )} - \frac{i \left (4 - 27 x^{2}\right )^{\frac{2}{3}}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (3 i \, x + 2\right )}^{2}}{{\left (-27 \, x^{2} + 4\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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