Integrand size = 21, antiderivative size = 650 \[ \int \frac {1}{(2+3 i x)^2 \sqrt [3]{4-27 x^2}} \, dx=\frac {i \left (4-27 x^2\right )^{2/3}}{48 (2+3 i x)}-\frac {3 x}{16 \left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )}+\frac {i \arctan \left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-3 i x)}{\sqrt {3} \sqrt [3]{4-27 x^2}}\right )}{24 \sqrt [3]{2} \sqrt {3}}-\frac {\sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4-27 x^2}+\left (4-27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt {3}\right )}{48\ 2^{2/3} 3^{3/4} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}}+\frac {\left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4-27 x^2}+\left (4-27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right ),-7+4 \sqrt {3}\right )}{72 \sqrt [6]{2} \sqrt [4]{3} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}}+\frac {i \log (2+3 i x)}{48 \sqrt [3]{2}}-\frac {i \log \left (-54+81 i x+27\ 2^{2/3} \sqrt [3]{4-27 x^2}\right )}{48 \sqrt [3]{2}} \]
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Time = 0.28 (sec) , antiderivative size = 650, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {759, 858, 241, 310, 225, 1893, 765} \[ \int \frac {1}{(2+3 i x)^2 \sqrt [3]{4-27 x^2}} \, dx=\frac {\left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right ),-7+4 \sqrt {3}\right )}{72 \sqrt [6]{2} \sqrt [4]{3} \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} x}-\frac {\sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt {3}\right )}{48\ 2^{2/3} 3^{3/4} \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} x}+\frac {i \arctan \left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-3 i x)}{\sqrt {3} \sqrt [3]{4-27 x^2}}\right )}{24 \sqrt [3]{2} \sqrt {3}}-\frac {3 x}{16 \left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )}+\frac {i \left (4-27 x^2\right )^{2/3}}{48 (2+3 i x)}-\frac {i \log \left (27\ 2^{2/3} \sqrt [3]{4-27 x^2}+81 i x-54\right )}{48 \sqrt [3]{2}}+\frac {i \log (2+3 i x)}{48 \sqrt [3]{2}} \]
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Rule 225
Rule 241
Rule 310
Rule 759
Rule 765
Rule 858
Rule 1893
Rubi steps \begin{align*} \text {integral}& = \frac {i \left (4-27 x^2\right )^{2/3}}{48 (2+3 i x)}-\frac {3}{16} \int \frac {-2-i x}{(2+3 i x) \sqrt [3]{4-27 x^2}} \, dx \\ & = \frac {i \left (4-27 x^2\right )^{2/3}}{48 (2+3 i x)}+\frac {1}{16} \int \frac {1}{\sqrt [3]{4-27 x^2}} \, dx+\frac {1}{4} \int \frac {1}{(2+3 i x) \sqrt [3]{4-27 x^2}} \, dx \\ & = \frac {i \left (4-27 x^2\right )^{2/3}}{48 (2+3 i x)}+\frac {i \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-3 i x)}{\sqrt {3} \sqrt [3]{4-27 x^2}}\right )}{24 \sqrt [3]{2} \sqrt {3}}+\frac {i \log (2+3 i x)}{48 \sqrt [3]{2}}-\frac {i \log \left (-54+81 i x+27\ 2^{2/3} \sqrt [3]{4-27 x^2}\right )}{48 \sqrt [3]{2}}-\frac {\sqrt {-x^2} \text {Subst}\left (\int \frac {x}{\sqrt {-4+x^3}} \, dx,x,\sqrt [3]{4-27 x^2}\right )}{32 \sqrt {3} x} \\ & = \frac {i \left (4-27 x^2\right )^{2/3}}{48 (2+3 i x)}+\frac {i \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-3 i x)}{\sqrt {3} \sqrt [3]{4-27 x^2}}\right )}{24 \sqrt [3]{2} \sqrt {3}}+\frac {i \log (2+3 i x)}{48 \sqrt [3]{2}}-\frac {i \log \left (-54+81 i x+27\ 2^{2/3} \sqrt [3]{4-27 x^2}\right )}{48 \sqrt [3]{2}}+\frac {\sqrt {-x^2} \text {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 )}{32 \sqrt {3} x}-\frac {\left (\left (1+\sqrt {3}\right ) \sqrt {-x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-4+x^3}} \, dx,x,\sqrt [3]{4-27 x^2}\right )}{16 \sqrt [3]{2} \sqrt {3} x} \\ & = \frac {i \left (4-27 x^2\right )^{2/3}}{48 (2+3 i x)}-\frac {3 x}{16 \left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )}+\frac {i \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-3 i x)}{\sqrt {3} \sqrt [3]{4-27 x^2}}\right )}{24 \sqrt [3]{2} \sqrt {3}}-\frac {\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 )}{48\ 2^{2/3} 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 {\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 )}{72 \sqrt [6]{2} \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 {i \log (2+3 i x)}{48 \sqrt [3]{2}}-\frac {i \log \left (-54+81 i x+27\ 2^{2/3} \sqrt [3]{4-27 x^2}\right )}{48 \sqrt [3]{2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 6.24 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.20 \[ \int \frac {1}{(2+3 i x)^2 \sqrt [3]{4-27 x^2}} \, dx=\frac {\sqrt [3]{\frac {2 \sqrt {3}-9 x}{2 i-3 x}} \sqrt [3]{\frac {2 \sqrt {3}+9 x}{-2 i+3 x}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{3},\frac {1}{3},\frac {8}{3},\frac {2 \left (3 i+\sqrt {3}\right )}{6 i-9 x},\frac {2 \left (-3 i+\sqrt {3}\right )}{-6 i+9 x}\right )}{5\ 3^{2/3} (-2 i+3 x) \sqrt [3]{4-27 x^2}} \]
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\[\int \frac {1}{\left (3 i x +2\right )^{2} \left (-27 x^{2}+4\right )^{\frac {1}{3}}}d x\]
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Timed out. \[ \int \frac {1}{(2+3 i x)^2 \sqrt [3]{4-27 x^2}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(2+3 i x)^2 \sqrt [3]{4-27 x^2}} \, dx=- \int \frac {1}{9 x^{2} \sqrt [3]{4 - 27 x^{2}} - 12 i x \sqrt [3]{4 - 27 x^{2}} - 4 \sqrt [3]{4 - 27 x^{2}}}\, dx \]
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\[ \int \frac {1}{(2+3 i x)^2 \sqrt [3]{4-27 x^2}} \, dx=\int { \frac {1}{{\left (-27 \, x^{2} + 4\right )}^{\frac {1}{3}} {\left (3 i \, x + 2\right )}^{2}} \,d x } \]
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\[ \int \frac {1}{(2+3 i x)^2 \sqrt [3]{4-27 x^2}} \, dx=\int { \frac {1}{{\left (-27 \, x^{2} + 4\right )}^{\frac {1}{3}} {\left (3 i \, x + 2\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{(2+3 i x)^2 \sqrt [3]{4-27 x^2}} \, dx=\int \frac {1}{{\left (2+x\,3{}\mathrm {i}\right )}^2\,{\left (4-27\,x^2\right )}^{1/3}} \,d x \]
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