Integrand size = 22, antiderivative size = 719 \[ \int \frac {1}{(2+3 x)^2 \sqrt [3]{52-54 x+27 x^2}} \, dx=-\frac {\left (52-54 x+27 x^2\right )^{2/3}}{300 (2+3 x)}+\frac {9 (1-x)}{10\ 5^{2/3} \left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )}-\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (8-3 x)}{\sqrt {3} \sqrt [3]{5} \sqrt [3]{52-54 x+27 x^2}}\right )}{30 \sqrt {3} 10^{2/3}}-\frac {\sqrt {2+\sqrt {3}} \left (30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right ) \sqrt {\frac {900+30 \sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}+10^{2/3} \left (2700+(-54+54 x)^2\right )^{2/3}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}} E\left (\arcsin \left (\frac {30 \left (1+\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}\right )|-7+4 \sqrt {3}\right )}{10800 \sqrt {2} \sqrt [4]{3} \sqrt [6]{5} (1-x) \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}}}+\frac {\left (30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right ) \sqrt {\frac {900+30 \sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}+10^{2/3} \left (2700+(-54+54 x)^2\right )^{2/3}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {30 \left (1+\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}\right ),-7+4 \sqrt {3}\right )}{5400\ 3^{3/4} \sqrt [6]{5} (1-x) \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}}}-\frac {\log (2+3 x)}{60\ 10^{2/3}}+\frac {\log \left (216-81 x-27 \sqrt [3]{10} \sqrt [3]{52-54 x+27 x^2}\right )}{60\ 10^{2/3}} \]
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Time = 0.42 (sec) , antiderivative size = 719, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {758, 857, 633, 241, 310, 225, 1893, 764} \[ \int \frac {1}{(2+3 x)^2 \sqrt [3]{52-54 x+27 x^2}} \, dx=\frac {\left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt {\frac {10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {30 \left (1+\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right ),-7+4 \sqrt {3}\right )}{5400\ 3^{3/4} \sqrt [6]{5} \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)}-\frac {\sqrt {2+\sqrt {3}} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt {\frac {10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} E\left (\arcsin \left (\frac {30 \left (1+\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt {3}\right )}{10800 \sqrt {2} \sqrt [4]{3} \sqrt [6]{5} \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)}-\frac {\arctan \left (\frac {2^{2/3} (8-3 x)}{\sqrt {3} \sqrt [3]{5} \sqrt [3]{27 x^2-54 x+52}}+\frac {1}{\sqrt {3}}\right )}{30 \sqrt {3} 10^{2/3}}-\frac {\left (27 x^2-54 x+52\right )^{2/3}}{300 (3 x+2)}+\frac {\log \left (-27 \sqrt [3]{10} \sqrt [3]{27 x^2-54 x+52}-81 x+216\right )}{60\ 10^{2/3}}+\frac {9 (1-x)}{10\ 5^{2/3} \left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )}-\frac {\log (3 x+2)}{60\ 10^{2/3}} \]
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Rule 225
Rule 241
Rule 310
Rule 633
Rule 758
Rule 764
Rule 857
Rule 1893
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (52-54 x+27 x^2\right )^{2/3}}{300 (2+3 x)}-\frac {1}{900} \int \frac {-108-27 x}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx \\ & = -\frac {\left (52-54 x+27 x^2\right )^{2/3}}{300 (2+3 x)}+\frac {1}{100} \int \frac {1}{\sqrt [3]{52-54 x+27 x^2}} \, dx+\frac {1}{10} \int \frac {1}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx \\ & = -\frac {\left (52-54 x+27 x^2\right )^{2/3}}{300 (2+3 x)}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (8-3 x)}{\sqrt {3} \sqrt [3]{5} \sqrt [3]{52-54 x+27 x^2}}\right )}{30 \sqrt {3} 10^{2/3}}-\frac {\log (2+3 x)}{60\ 10^{2/3}}+\frac {\log \left (216-81 x-27 \sqrt [3]{10} \sqrt [3]{52-54 x+27 x^2}\right )}{60\ 10^{2/3}}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt [3]{1+\frac {x^2}{2700}}} \, dx,x,-54+54 x\right )}{5400\ 5^{2/3}} \\ & = -\frac {\left (52-54 x+27 x^2\right )^{2/3}}{300 (2+3 x)}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (8-3 x)}{\sqrt {3} \sqrt [3]{5} \sqrt [3]{52-54 x+27 x^2}}\right )}{30 \sqrt {3} 10^{2/3}}-\frac {\log (2+3 x)}{60\ 10^{2/3}}+\frac {\log \left (216-81 x-27 \sqrt [3]{10} \sqrt [3]{52-54 x+27 x^2}\right )}{60\ 10^{2/3}}+\frac {\sqrt {(-54+54 x)^2} \text {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,\frac {\sqrt [3]{2700+(-54+54 x)^2}}{3\ 10^{2/3}}\right )}{40 \sqrt {3} 5^{2/3} (-54+54 x)} \\ & = -\frac {\left (52-54 x+27 x^2\right )^{2/3}}{300 (2+3 x)}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (8-3 x)}{\sqrt {3} \sqrt [3]{5} \sqrt [3]{52-54 x+27 x^2}}\right )}{30 \sqrt {3} 10^{2/3}}-\frac {\log (2+3 x)}{60\ 10^{2/3}}+\frac {\log \left (216-81 x-27 \sqrt [3]{10} \sqrt [3]{52-54 x+27 x^2}\right )}{60\ 10^{2/3}}-\frac {\sqrt {(-54+54 x)^2} \text {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,\frac {\sqrt [3]{2700+(-54+54 x)^2}}{3\ 10^{2/3}}\right )}{40 \sqrt {3} 5^{2/3} (-54+54 x)}+\frac {\left (\left (1+\sqrt {3}\right ) \sqrt {(-54+54 x)^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,\frac {\sqrt [3]{2700+(-54+54 x)^2}}{3\ 10^{2/3}}\right )}{40 \sqrt {3} 5^{2/3} (-54+54 x)} \\ & = -\frac {\left (52-54 x+27 x^2\right )^{2/3}}{300 (2+3 x)}+\frac {9 (1-x)}{10\ 5^{2/3} \left (30-30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (8-3 x)}{\sqrt {3} \sqrt [3]{5} \sqrt [3]{52-54 x+27 x^2}}\right )}{30 \sqrt {3} 10^{2/3}}-\frac {\sqrt {2+\sqrt {3}} \left (30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right ) \sqrt {\frac {900+30 \sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}+10^{2/3} \left (2700+(-54+54 x)^2\right )^{2/3}}{\left (30-30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {30+30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{30-30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}\right )|-7+4 \sqrt {3}\right )}{10800 \sqrt {2} \sqrt [4]{3} \sqrt [6]{5} (1-x) \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{\left (30-30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}}}+\frac {\left (30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right ) \sqrt {\frac {900+30 \sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}+10^{2/3} \left (2700+(-54+54 x)^2\right )^{2/3}}{\left (30-30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {30+30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{30-30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}\right )|-7+4 \sqrt {3}\right )}{5400\ 3^{3/4} \sqrt [6]{5} (1-x) \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{\left (30-30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}}}-\frac {\log (2+3 x)}{60\ 10^{2/3}}+\frac {\log \left (216-81 x-27 \sqrt [3]{10} \sqrt [3]{52-54 x+27 x^2}\right )}{60\ 10^{2/3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.13 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.34 \[ \int \frac {1}{(2+3 x)^2 \sqrt [3]{52-54 x+27 x^2}} \, dx=\frac {-20 \left (52-54 x+27 x^2\right )-100 \sqrt [3]{3} (2+3 x) \sqrt [3]{\frac {-9-5 i \sqrt {3}+9 x}{2+3 x}} \sqrt [3]{\frac {-9+5 i \sqrt {3}+9 x}{2+3 x}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},\frac {1}{3},\frac {5}{3},\frac {15-5 i \sqrt {3}}{6+9 x},\frac {15+5 i \sqrt {3}}{6+9 x}\right )+\sqrt [3]{3} 10^{2/3} \sqrt [3]{9 i+5 \sqrt {3}-9 i x} (2+3 x) \left (-5 i-3 \sqrt {3}+3 \sqrt {3} x\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},\frac {-9 i+5 \sqrt {3}+9 i x}{10 \sqrt {3}}\right )}{6000 (2+3 x) \sqrt [3]{52-54 x+27 x^2}} \]
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\[\int \frac {1}{\left (2+3 x \right )^{2} \left (27 x^{2}-54 x +52\right )^{\frac {1}{3}}}d x\]
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\[ \int \frac {1}{(2+3 x)^2 \sqrt [3]{52-54 x+27 x^2}} \, dx=\int { \frac {1}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}^{2}} \,d x } \]
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\[ \int \frac {1}{(2+3 x)^2 \sqrt [3]{52-54 x+27 x^2}} \, dx=\int \frac {1}{\left (3 x + 2\right )^{2} \sqrt [3]{27 x^{2} - 54 x + 52}}\, dx \]
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\[ \int \frac {1}{(2+3 x)^2 \sqrt [3]{52-54 x+27 x^2}} \, dx=\int { \frac {1}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}^{2}} \,d x } \]
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\[ \int \frac {1}{(2+3 x)^2 \sqrt [3]{52-54 x+27 x^2}} \, dx=\int { \frac {1}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{(2+3 x)^2 \sqrt [3]{52-54 x+27 x^2}} \, dx=\int \frac {1}{{\left (3\,x+2\right )}^2\,{\left (27\,x^2-54\,x+52\right )}^{1/3}} \,d x \]
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