Optimal. Leaf size=557 \[ -\frac {1}{21} i \left (4-27 x^2\right )^{2/3} (2+3 i x)-\frac {5}{21} i \left (4-27 x^2\right )^{2/3}-\frac {72 x}{7 \left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )}+\frac {8\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt {3}\right )}{21 \sqrt [4]{3} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}}-\frac {4 \sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt {3}\right )}{7\ 3^{3/4} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}} \]
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Rubi [A] time = 0.32, antiderivative size = 557, normalized size of antiderivative = 1.00, 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 219
Rule 235
Rule 304
Rule 641
Rule 743
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*}
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Mathematica [C] time = 0.03, 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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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ \frac {21 \, x {\rm integral}\left (\frac {32 \, {\left (-27 \, x^{2} + 4\right )}^{\frac {2}{3}}}{21 \, {\left (27 \, x^{4} - 4 \, x^{2}\right )}}, x\right ) + {\left (3 \, x^{2} - 7 i \, x - 8\right )} {\left (-27 \, x^{2} + 4\right )}^{\frac {2}{3}}}{21 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (3 i \, x + 2\right )}^{2}}{{\left (-27 \, x^{2} + 4\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.54, size = 43, normalized size = 0.08 \[ \frac {12 \,2^{\frac {1}{3}} x \hypergeom \left (\left [\frac {1}{3}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], \frac {27 x^{2}}{4}\right )}{7}-\frac {\left (3 x -7 i\right ) \left (27 x^{2}-4\right )}{21 \left (-27 x^{2}+4\right )^{\frac {1}{3}}} \]
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 \[ \int \frac {{\left (3 i \, x + 2\right )}^{2}}{{\left (-27 \, x^{2} + 4\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (2+x\,3{}\mathrm {i}\right )}^2}{{\left (4-27\,x^2\right )}^{1/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.72, size = 73, normalized size = 0.13 \[ - \frac {3 \sqrt [3]{2} x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {27 x^{2} e^{2 i \pi }}{4}} \right )}}{2} + 2 \sqrt [3]{2} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {27 x^{2} e^{2 i \pi }}{4}} \right )} - \frac {i \left (4 - 27 x^{2}\right )^{\frac {2}{3}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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