Optimal. Leaf size=564 \[ \frac{32\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right ),4 \sqrt{3}-7\right )}{63 \sqrt [4]{3} x \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}}-\frac{1}{30} i \left (4-27 x^2\right )^{2/3} (2+3 i x)^2-\frac{4}{35} (-4 x+7 i) \left (4-27 x^2\right )^{2/3}-\frac{96 x}{7 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )}-\frac{16 \sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{21\ 3^{3/4} x \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}} \]
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Rubi [A] time = 0.362809, antiderivative size = 564, 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, 780, 235, 304, 219, 1879} \[ -\frac{1}{30} i \left (4-27 x^2\right )^{2/3} (2+3 i x)^2-\frac{4}{35} (-4 x+7 i) \left (4-27 x^2\right )^{2/3}-\frac{96 x}{7 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )}+\frac{32\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{63 \sqrt [4]{3} x \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}}-\frac{16 \sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{21\ 3^{3/4} x \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}} \]
Antiderivative was successfully verified.
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Rule 743
Rule 780
Rule 235
Rule 304
Rule 219
Rule 1879
Rubi steps
\begin{align*} \int \frac{(2+3 i x)^3}{\sqrt [3]{4-27 x^2}} \, dx &=-\frac{1}{30} i (2+3 i x)^2 \left (4-27 x^2\right )^{2/3}-\frac{1}{90} \int \frac{(2+3 i x) (-288-864 i x)}{\sqrt [3]{4-27 x^2}} \, dx\\ &=-\frac{4}{35} (7 i-4 x) \left (4-27 x^2\right )^{2/3}-\frac{1}{30} i (2+3 i x)^2 \left (4-27 x^2\right )^{2/3}+\frac{32}{7} \int \frac{1}{\sqrt [3]{4-27 x^2}} \, dx\\ &=-\frac{4}{35} (7 i-4 x) \left (4-27 x^2\right )^{2/3}-\frac{1}{30} i (2+3 i x)^2 \left (4-27 x^2\right )^{2/3}-\frac{\left (16 \sqrt{-x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{-4+x^3}} \, dx,x,\sqrt [3]{4-27 x^2}\right )}{7 \sqrt{3} x}\\ &=-\frac{4}{35} (7 i-4 x) \left (4-27 x^2\right )^{2/3}-\frac{1}{30} i (2+3 i x)^2 \left (4-27 x^2\right )^{2/3}+\frac{\left (16 \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 \sqrt{3} x}-\frac{\left (32 \sqrt [6]{2} \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{3 \left (2-\sqrt{3}\right )} x}\\ &=-\frac{4}{35} (7 i-4 x) \left (4-27 x^2\right )^{2/3}-\frac{1}{30} i (2+3 i x)^2 \left (4-27 x^2\right )^{2/3}-\frac{96 x}{7 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )}-\frac{16 \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 )}{21\ 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{32\ 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 )}{63 \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 = 0.0351779, size = 60, normalized size = 0.11 \[ \frac{16}{7} \sqrt [3]{2} x \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{27 x^2}{4}\right )+\left (4-27 x^2\right )^{2/3} \left (\frac{3 i x^2}{10}+\frac{6 x}{7}-\frac{14 i}{15}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.267, size = 49, normalized size = 0.1 \begin{align*}{-{\frac{i}{210}} \left ( -180\,ix+63\,{x}^{2}-196 \right ) \left ( 27\,{x}^{2}-4 \right ){\frac{1}{\sqrt [3]{-27\,{x}^{2}+4}}}}+{\frac{16\,\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 )}^{3}}{{\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{630 \, x{\rm integral}\left (\frac{128 \,{\left (-27 \, x^{2} + 4\right )}^{\frac{2}{3}}}{63 \,{\left (27 \, x^{4} - 4 \, x^{2}\right )}}, x\right ) +{\left (189 i \, x^{3} + 540 \, x^{2} - 588 i \, x - 320\right )}{\left (-27 \, x^{2} + 4\right )}^{\frac{2}{3}}}{630 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.99207, size = 150, normalized size = 0.27 \begin{align*} - 9 \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 )} + 4 \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 )} - i \left (4 - 27 x^{2}\right )^{\frac{2}{3}} - 27 i \left (\begin{cases} \frac{x^{2} \left (27 x^{2} - 4\right )^{\frac{2}{3}} e^{- \frac{i \pi }{3}}}{90} + \frac{\left (27 x^{2} - 4\right )^{\frac{2}{3}} e^{- \frac{i \pi }{3}}}{405} & \text{for}\: \frac{27 \left |{x^{2}}\right |}{4} > 1 \\- \frac{x^{2} \left (4 - 27 x^{2}\right )^{\frac{2}{3}}}{90} - \frac{\left (4 - 27 x^{2}\right )^{\frac{2}{3}}}{405} & \text{otherwise} \end{cases}\right ) \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 )}^{3}}{{\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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