Integrand size = 22, antiderivative size = 696 \[ \int \frac {1}{(2+3 x)^3 \sqrt [3]{28+54 x+27 x^2}} \, dx=-\frac {\left (28+54 x+27 x^2\right )^{2/3}}{24 (2+3 x)^2}+\frac {\left (28+54 x+27 x^2\right )^{2/3}}{12 (2+3 x)}+\frac {9 (1+x)}{2 \left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )}-\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (4+3 x)}{\sqrt {3} \sqrt [3]{28+54 x+27 x^2}}\right )}{12\ 2^{2/3} \sqrt {3}}-\frac {\sqrt {2+\sqrt {3}} \left (6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right ) \sqrt {\frac {1+\sqrt [3]{28+54 x+27 x^2}+\left (28+54 x+27 x^2\right )^{2/3}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}} E\left (\arcsin \left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}\right )|-7+4 \sqrt {3}\right )}{72 \sqrt {2} \sqrt [4]{3} (1+x) \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}}}+\frac {\left (6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right ) \sqrt {\frac {1+\sqrt [3]{28+54 x+27 x^2}+\left (28+54 x+27 x^2\right )^{2/3}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}\right ),-7+4 \sqrt {3}\right )}{36\ 3^{3/4} (1+x) \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}}}-\frac {\log (2+3 x)}{24\ 2^{2/3}}+\frac {\log \left (-108-81 x+27 \sqrt [3]{2} \sqrt [3]{28+54 x+27 x^2}\right )}{24\ 2^{2/3}} \]
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Time = 0.39 (sec) , antiderivative size = 696, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {758, 848, 857, 633, 241, 310, 225, 1893, 766} \[ \int \frac {1}{(2+3 x)^3 \sqrt [3]{28+54 x+27 x^2}} \, dx=\frac {\left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt {\frac {\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right ),-7+4 \sqrt {3}\right )}{36\ 3^{3/4} \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}-\frac {\sqrt {2+\sqrt {3}} \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt {\frac {\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} E\left (\arcsin \left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt {3}\right )}{72 \sqrt {2} \sqrt [4]{3} \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}-\frac {\arctan \left (\frac {2^{2/3} (3 x+4)}{\sqrt {3} \sqrt [3]{27 x^2+54 x+28}}+\frac {1}{\sqrt {3}}\right )}{12\ 2^{2/3} \sqrt {3}}+\frac {\left (27 x^2+54 x+28\right )^{2/3}}{12 (3 x+2)}-\frac {\left (27 x^2+54 x+28\right )^{2/3}}{24 (3 x+2)^2}+\frac {\log \left (27 \sqrt [3]{2} \sqrt [3]{27 x^2+54 x+28}-81 x-108\right )}{24\ 2^{2/3}}+\frac {9 (x+1)}{2 \left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )}-\frac {\log (3 x+2)}{24\ 2^{2/3}} \]
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Rule 225
Rule 241
Rule 310
Rule 633
Rule 758
Rule 766
Rule 848
Rule 857
Rule 1893
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (28+54 x+27 x^2\right )^{2/3}}{24 (2+3 x)^2}-\frac {1}{72} \int \frac {108+54 x}{(2+3 x)^2 \sqrt [3]{28+54 x+27 x^2}} \, dx \\ & = -\frac {\left (28+54 x+27 x^2\right )^{2/3}}{24 (2+3 x)^2}+\frac {\left (28+54 x+27 x^2\right )^{2/3}}{12 (2+3 x)}+\frac {\int \frac {-648-1944 x}{(2+3 x) \sqrt [3]{28+54 x+27 x^2}} \, dx}{2592} \\ & = -\frac {\left (28+54 x+27 x^2\right )^{2/3}}{24 (2+3 x)^2}+\frac {\left (28+54 x+27 x^2\right )^{2/3}}{12 (2+3 x)}-\frac {1}{4} \int \frac {1}{\sqrt [3]{28+54 x+27 x^2}} \, dx+\frac {1}{4} \int \frac {1}{(2+3 x) \sqrt [3]{28+54 x+27 x^2}} \, dx \\ & = -\frac {\left (28+54 x+27 x^2\right )^{2/3}}{24 (2+3 x)^2}+\frac {\left (28+54 x+27 x^2\right )^{2/3}}{12 (2+3 x)}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (4+3 x)}{\sqrt {3} \sqrt [3]{28+54 x+27 x^2}}\right )}{12\ 2^{2/3} \sqrt {3}}-\frac {\log (2+3 x)}{24\ 2^{2/3}}+\frac {\log \left (-108-81 x+27 \sqrt [3]{2} \sqrt [3]{28+54 x+27 x^2}\right )}{24\ 2^{2/3}}-\frac {1}{216} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+\frac {x^2}{108}}} \, dx,x,54+54 x\right ) \\ & = -\frac {\left (28+54 x+27 x^2\right )^{2/3}}{24 (2+3 x)^2}+\frac {\left (28+54 x+27 x^2\right )^{2/3}}{12 (2+3 x)}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (4+3 x)}{\sqrt {3} \sqrt [3]{28+54 x+27 x^2}}\right )}{12\ 2^{2/3} \sqrt {3}}-\frac {\log (2+3 x)}{24\ 2^{2/3}}+\frac {\log \left (-108-81 x+27 \sqrt [3]{2} \sqrt [3]{28+54 x+27 x^2}\right )}{24\ 2^{2/3}}-\frac {\sqrt {(54+54 x)^2} \text {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{28+54 x+27 x^2}\right )}{8 \sqrt {3} (54+54 x)} \\ & = -\frac {\left (28+54 x+27 x^2\right )^{2/3}}{24 (2+3 x)^2}+\frac {\left (28+54 x+27 x^2\right )^{2/3}}{12 (2+3 x)}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (4+3 x)}{\sqrt {3} \sqrt [3]{28+54 x+27 x^2}}\right )}{12\ 2^{2/3} \sqrt {3}}-\frac {\log (2+3 x)}{24\ 2^{2/3}}+\frac {\log \left (-108-81 x+27 \sqrt [3]{2} \sqrt [3]{28+54 x+27 x^2}\right )}{24\ 2^{2/3}}+\frac {\sqrt {(54+54 x)^2} \text {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{28+54 x+27 x^2}\right )}{8 \sqrt {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,\sqrt [3]{28+54 x+27 x^2}\right )}{8 \sqrt {3} (54+54 x)} \\ & = -\frac {\left (28+54 x+27 x^2\right )^{2/3}}{24 (2+3 x)^2}+\frac {\left (28+54 x+27 x^2\right )^{2/3}}{12 (2+3 x)}+\frac {3 (1+x)}{4 \left (1-\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}\right )}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (4+3 x)}{\sqrt {3} \sqrt [3]{28+54 x+27 x^2}}\right )}{12\ 2^{2/3} \sqrt {3}}-\frac {\sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{28+54 x+27 x^2}\right ) \sqrt {\frac {1+\sqrt [3]{28+54 x+27 x^2}+\left (28+54 x+27 x^2\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}}{1-\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}}\right )|-7+4 \sqrt {3}\right )}{24\ 3^{3/4} (1+x) \sqrt {-\frac {1-\sqrt [3]{28+54 x+27 x^2}}{\left (1-\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}\right )^2}}}+\frac {\left (1-\sqrt [3]{28+54 x+27 x^2}\right ) \sqrt {\frac {1+\sqrt [3]{28+54 x+27 x^2}+\left (28+54 x+27 x^2\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}}{1-\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}}\right )|-7+4 \sqrt {3}\right )}{18 \sqrt {2} \sqrt [4]{3} (1+x) \sqrt {-\frac {1-\sqrt [3]{28+54 x+27 x^2}}{\left (1-\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}\right )^2}}}-\frac {\log (2+3 x)}{24\ 2^{2/3}}+\frac {\log \left (-108-81 x+27 \sqrt [3]{2} \sqrt [3]{28+54 x+27 x^2}\right )}{24\ 2^{2/3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 16.12 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.33 \[ \int \frac {1}{(2+3 x)^3 \sqrt [3]{28+54 x+27 x^2}} \, dx=\frac {\frac {162 (1+2 x) \left (28+54 x+27 x^2\right )}{(2+3 x)^2}-54 \sqrt [3]{3} \sqrt [3]{\frac {9-i \sqrt {3}+9 x}{2+3 x}} \sqrt [3]{\frac {9+i \sqrt {3}+9 x}{2+3 x}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},\frac {1}{3},\frac {5}{3},-\frac {3+i \sqrt {3}}{6+9 x},\frac {-3+i \sqrt {3}}{6+9 x}\right )+9 i 2^{2/3} 3^{5/6} \sqrt [3]{-9 i+\sqrt {3}-9 i x} \left (9 i+\sqrt {3}+9 i x\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},\frac {9 i+\sqrt {3}+9 i x}{2 \sqrt {3}}\right )}{1296 \sqrt [3]{28+54 x+27 x^2}} \]
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\[\int \frac {1}{\left (2+3 x \right )^{3} \left (27 x^{2}+54 x +28\right )^{\frac {1}{3}}}d x\]
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\[ \int \frac {1}{(2+3 x)^3 \sqrt [3]{28+54 x+27 x^2}} \, dx=\int { \frac {1}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}^{3}} \,d x } \]
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\[ \int \frac {1}{(2+3 x)^3 \sqrt [3]{28+54 x+27 x^2}} \, dx=\int \frac {1}{\left (3 x + 2\right )^{3} \sqrt [3]{27 x^{2} + 54 x + 28}}\, dx \]
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\[ \int \frac {1}{(2+3 x)^3 \sqrt [3]{28+54 x+27 x^2}} \, dx=\int { \frac {1}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}^{3}} \,d x } \]
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\[ \int \frac {1}{(2+3 x)^3 \sqrt [3]{28+54 x+27 x^2}} \, dx=\int { \frac {1}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {1}{(2+3 x)^3 \sqrt [3]{28+54 x+27 x^2}} \, dx=\int \frac {1}{{\left (3\,x+2\right )}^3\,{\left (27\,x^2+54\,x+28\right )}^{1/3}} \,d x \]
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