Optimal. Leaf size=551 \[ -\frac{8\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt{\frac{\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}\right ),4 \sqrt{3}-7\right )}{21 \sqrt [4]{3} \sqrt{-\frac{2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}-\frac{72 x}{7 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )}+\frac{1}{21} (3 x+2) \left (27 x^2+4\right )^{2/3}+\frac{5}{21} \left (27 x^2+4\right )^{2/3}+\frac{4 \sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt{\frac{\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt{3}\right )}{7\ 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x} \]
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Rubi [A] time = 0.312908, antiderivative size = 551, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {743, 641, 235, 304, 219, 1879} \[ -\frac{72 x}{7 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )}+\frac{1}{21} (3 x+2) \left (27 x^2+4\right )^{2/3}+\frac{5}{21} \left (27 x^2+4\right )^{2/3}-\frac{8\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt{\frac{\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt{3}\right )}{21 \sqrt [4]{3} \sqrt{-\frac{2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}+\frac{4 \sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt{\frac{\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt{3}\right )}{7\ 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x} \]
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 x)^2}{\sqrt [3]{4+27 x^2}} \, dx &=\frac{1}{21} (2+3 x) \left (4+27 x^2\right )^{2/3}+\frac{1}{63} \int \frac{216+540 x}{\sqrt [3]{4+27 x^2}} \, dx\\ &=\frac{5}{21} \left (4+27 x^2\right )^{2/3}+\frac{1}{21} (2+3 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} \left (4+27 x^2\right )^{2/3}+\frac{1}{21} (2+3 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} \left (4+27 x^2\right )^{2/3}+\frac{1}{21} (2+3 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} \left (4+27 x^2\right )^{2/3}+\frac{1}{21} (2+3 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 = 0.0251864, size = 47, normalized size = 0.09 \[ \frac{1}{21} \left (36 \sqrt [3]{2} x \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};-\frac{27 x^2}{4}\right )+\left (27 x^2+4\right )^{2/3} (3 x+7)\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.218, size = 35, normalized size = 0.1 \begin{align*}{\frac{7+3\,x}{21} \left ( 27\,{x}^{2}+4 \right ) ^{{\frac{2}{3}}}}+{\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 \, 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*}{\rm integral}\left (\frac{9 \, x^{2} + 12 \, x + 4}{{\left (27 \, x^{2} + 4\right )}^{\frac{1}{3}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.433, size = 68, normalized size = 0.12 \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^{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^{i \pi }}{4}} \right )} + \frac{\left (27 x^{2} + 4\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 \, 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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