Optimal. Leaf size=656 \[ -\frac {3 x}{32 \left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )}-\frac {\left (27 x^2+4\right )^{2/3}}{96 (3 x+2)}-\frac {\left (27 x^2+4\right )^{2/3}}{96 (3 x+2)^2}+\frac {\log \left (-27\ 2^{2/3} \sqrt [3]{27 x^2+4}-81 x+54\right )}{192 \sqrt [3]{2}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{2} (2-3 x)}{\sqrt {3} \sqrt [3]{27 x^2+4}}+\frac {1}{\sqrt {3}}\right )}{96 \sqrt [3]{2} \sqrt {3}}-\frac {\left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt {\frac {\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} F\left (\sin ^{-1}\left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt {3}\right )}{144 \sqrt [6]{2} \sqrt [4]{3} \sqrt {-\frac {2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}+\frac {\sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt {\frac {\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} E\left (\sin ^{-1}\left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt {3}\right )}{96\ 2^{2/3} 3^{3/4} \sqrt {-\frac {2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}-\frac {\log (3 x+2)}{192 \sqrt [3]{2}} \]
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Rubi [A] time = 0.41, antiderivative size = 656, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {745, 835, 844, 235, 304, 219, 1879, 751} \[ -\frac {3 x}{32 \left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )}-\frac {\left (27 x^2+4\right )^{2/3}}{96 (3 x+2)}-\frac {\left (27 x^2+4\right )^{2/3}}{96 (3 x+2)^2}+\frac {\log \left (-27\ 2^{2/3} \sqrt [3]{27 x^2+4}-81 x+54\right )}{192 \sqrt [3]{2}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{2} (2-3 x)}{\sqrt {3} \sqrt [3]{27 x^2+4}}+\frac {1}{\sqrt {3}}\right )}{96 \sqrt [3]{2} \sqrt {3}}-\frac {\left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt {\frac {\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} F\left (\sin ^{-1}\left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt {3}\right )}{144 \sqrt [6]{2} \sqrt [4]{3} \sqrt {-\frac {2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}+\frac {\sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt {\frac {\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} E\left (\sin ^{-1}\left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt {3}\right )}{96\ 2^{2/3} 3^{3/4} \sqrt {-\frac {2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}-\frac {\log (3 x+2)}{192 \sqrt [3]{2}} \]
Antiderivative was successfully verified.
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Rule 219
Rule 235
Rule 304
Rule 745
Rule 751
Rule 835
Rule 844
Rule 1879
Rubi steps
\begin {align*} \int \frac {1}{(2+3 x)^3 \sqrt [3]{4+27 x^2}} \, dx &=-\frac {\left (4+27 x^2\right )^{2/3}}{96 (2+3 x)^2}-\frac {3}{32} \int \frac {-4+2 x}{(2+3 x)^2 \sqrt [3]{4+27 x^2}} \, dx\\ &=-\frac {\left (4+27 x^2\right )^{2/3}}{96 (2+3 x)^2}-\frac {\left (4+27 x^2\right )^{2/3}}{96 (2+3 x)}+\frac {\int \frac {192+144 x}{(2+3 x) \sqrt [3]{4+27 x^2}} \, dx}{1536}\\ &=-\frac {\left (4+27 x^2\right )^{2/3}}{96 (2+3 x)^2}-\frac {\left (4+27 x^2\right )^{2/3}}{96 (2+3 x)}+\frac {1}{32} \int \frac {1}{\sqrt [3]{4+27 x^2}} \, dx+\frac {1}{16} \int \frac {1}{(2+3 x) \sqrt [3]{4+27 x^2}} \, dx\\ &=-\frac {\left (4+27 x^2\right )^{2/3}}{96 (2+3 x)^2}-\frac {\left (4+27 x^2\right )^{2/3}}{96 (2+3 x)}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-3 x)}{\sqrt {3} \sqrt [3]{4+27 x^2}}\right )}{96 \sqrt [3]{2} \sqrt {3}}-\frac {\log (2+3 x)}{192 \sqrt [3]{2}}+\frac {\log \left (54-81 x-27\ 2^{2/3} \sqrt [3]{4+27 x^2}\right )}{192 \sqrt [3]{2}}+\frac {\sqrt {x^2} \operatorname {Subst}\left (\int \frac {x}{\sqrt {-4+x^3}} \, dx,x,\sqrt [3]{4+27 x^2}\right )}{64 \sqrt {3} x}\\ &=-\frac {\left (4+27 x^2\right )^{2/3}}{96 (2+3 x)^2}-\frac {\left (4+27 x^2\right )^{2/3}}{96 (2+3 x)}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-3 x)}{\sqrt {3} \sqrt [3]{4+27 x^2}}\right )}{96 \sqrt [3]{2} \sqrt {3}}-\frac {\log (2+3 x)}{192 \sqrt [3]{2}}+\frac {\log \left (54-81 x-27\ 2^{2/3} \sqrt [3]{4+27 x^2}\right )}{192 \sqrt [3]{2}}-\frac {\sqrt {x^2} \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 )}{64 \sqrt {3} x}+\frac {\sqrt {x^2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-4+x^3}} \, dx,x,\sqrt [3]{4+27 x^2}\right )}{16\ 2^{5/6} \sqrt {3 \left (2-\sqrt {3}\right )} x}\\ &=-\frac {\left (4+27 x^2\right )^{2/3}}{96 (2+3 x)^2}-\frac {\left (4+27 x^2\right )^{2/3}}{96 (2+3 x)}-\frac {3 x}{32 \left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-3 x)}{\sqrt {3} \sqrt [3]{4+27 x^2}}\right )}{96 \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 )}{96\ 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 )}{144 \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 {\log (2+3 x)}{192 \sqrt [3]{2}}+\frac {\log \left (54-81 x-27\ 2^{2/3} \sqrt [3]{4+27 x^2}\right )}{192 \sqrt [3]{2}}\\ \end {align*}
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Mathematica [C] time = 0.27, size = 223, normalized size = 0.34 \[ \frac {-4 \sqrt [3]{3} \sqrt [3]{\frac {9 x-2 i \sqrt {3}}{3 x+2}} \sqrt [3]{\frac {9 x+2 i \sqrt {3}}{3 x+2}} (3 x+2)^2 F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};\frac {6-2 i \sqrt {3}}{9 x+6},\frac {6+2 i \sqrt {3}}{9 x+6}\right )+\sqrt [3]{6} \sqrt [3]{2 \sqrt {3}-9 i x} \left (3 \sqrt {3} x-2 i\right ) (3 x+2)^2 \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};\frac {3}{4} i \sqrt {3} x+\frac {1}{2}\right )-12 \left (27 x^3+27 x^2+4 x+4\right )}{384 (3 x+2)^2 \sqrt [3]{27 x^2+4}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 5.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (27 \, x^{2} + 4\right )}^{\frac {2}{3}}}{729 \, x^{5} + 1458 \, x^{4} + 1080 \, x^{3} + 432 \, x^{2} + 144 \, x + 32}, x\right ) \]
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 \[ \int \frac {1}{{\left (27 \, x^{2} + 4\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.66, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (3 x +2\right )^{3} \left (27 x^{2}+4\right )^{\frac {1}{3}}}\, dx \]
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 \[ \int \frac {1}{{\left (27 \, x^{2} + 4\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}^{3}}\,{d x} \]
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 \[ \int \frac {1}{{\left (3\,x+2\right )}^3\,{\left (27\,x^2+4\right )}^{1/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (3 x + 2\right )^{3} \sqrt [3]{27 x^{2} + 4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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