Integrand size = 22, antiderivative size = 585 \[ \int \frac {(2+3 x)^2}{\sqrt [3]{28+54 x+27 x^2}} \, dx=-\frac {5}{42} \left (28+54 x+27 x^2\right )^{2/3}+\frac {1}{21} (2+3 x) \left (28+54 x+27 x^2\right )^{2/3}-\frac {108 (1+x)}{7 \left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )}+\frac {\sqrt {2+\sqrt {3}} \left (6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right ) \sqrt {\frac {1+\sqrt [3]{28+54 x+27 x^2}+\left (28+54 x+27 x^2\right )^{2/3}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}} E\left (\arcsin \left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}\right )|-7+4 \sqrt {3}\right )}{21 \sqrt {2} \sqrt [4]{3} (1+x) \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}}}-\frac {2 \left (6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right ) \sqrt {\frac {1+\sqrt [3]{28+54 x+27 x^2}+\left (28+54 x+27 x^2\right )^{2/3}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}\right ),-7+4 \sqrt {3}\right )}{21\ 3^{3/4} (1+x) \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}}} \]
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Time = 0.32 (sec) , antiderivative size = 585, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {756, 654, 633, 241, 310, 225, 1893} \[ \int \frac {(2+3 x)^2}{\sqrt [3]{28+54 x+27 x^2}} \, dx=-\frac {2 \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt {\frac {\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right ),-7+4 \sqrt {3}\right )}{21\ 3^{3/4} \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}+\frac {\sqrt {2+\sqrt {3}} \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt {\frac {\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} E\left (\arcsin \left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt {3}\right )}{21 \sqrt {2} \sqrt [4]{3} \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}+\frac {1}{21} (3 x+2) \left (27 x^2+54 x+28\right )^{2/3}-\frac {5}{42} \left (27 x^2+54 x+28\right )^{2/3}-\frac {108 (x+1)}{7 \left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )} \]
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Rule 225
Rule 241
Rule 310
Rule 633
Rule 654
Rule 756
Rule 1893
Rubi steps \begin{align*} \text {integral}& = \frac {1}{21} (2+3 x) \left (28+54 x+27 x^2\right )^{2/3}+\frac {1}{63} \int \frac {-216-270 x}{\sqrt [3]{28+54 x+27 x^2}} \, dx \\ & = -\frac {5}{42} \left (28+54 x+27 x^2\right )^{2/3}+\frac {1}{21} (2+3 x) \left (28+54 x+27 x^2\right )^{2/3}+\frac {6}{7} \int \frac {1}{\sqrt [3]{28+54 x+27 x^2}} \, dx \\ & = -\frac {5}{42} \left (28+54 x+27 x^2\right )^{2/3}+\frac {1}{21} (2+3 x) \left (28+54 x+27 x^2\right )^{2/3}+\frac {1}{63} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+\frac {x^2}{108}}} \, dx,x,54+54 x\right ) \\ & = -\frac {5}{42} \left (28+54 x+27 x^2\right )^{2/3}+\frac {1}{21} (2+3 x) \left (28+54 x+27 x^2\right )^{2/3}+\frac {\left (\sqrt {3} \sqrt {(54+54 x)^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{28+54 x+27 x^2}\right )}{7 (54+54 x)} \\ & = -\frac {5}{42} \left (28+54 x+27 x^2\right )^{2/3}+\frac {1}{21} (2+3 x) \left (28+54 x+27 x^2\right )^{2/3}-\frac {\left (\sqrt {3} \sqrt {(54+54 x)^2}\right ) \text {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{28+54 x+27 x^2}\right )}{7 (54+54 x)}+\frac {\left (\sqrt {3} \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 )}{7 (54+54 x)} \\ & = -\frac {5}{42} \left (28+54 x+27 x^2\right )^{2/3}+\frac {1}{21} (2+3 x) \left (28+54 x+27 x^2\right )^{2/3}-\frac {18 (1+x)}{7 \left (1-\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}\right )}+\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 )}{7\ 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 {2 \sqrt {2} \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 )}{21 \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}}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.08 \[ \int \frac {(2+3 x)^2}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\frac {1}{42} \left ((-1+6 x) \left (28+54 x+27 x^2\right )^{2/3}+36 (1+x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},-27 (1+x)^2\right )\right ) \]
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\[\int \frac {\left (2+3 x \right )^{2}}{\left (27 x^{2}+54 x +28\right )^{\frac {1}{3}}}d x\]
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\[ \int \frac {(2+3 x)^2}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{2}}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {(2+3 x)^2}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\int \frac {\left (3 x + 2\right )^{2}}{\sqrt [3]{27 x^{2} + 54 x + 28}}\, dx \]
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\[ \int \frac {(2+3 x)^2}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{2}}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {(2+3 x)^2}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{2}}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {(2+3 x)^2}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\int \frac {{\left (3\,x+2\right )}^2}{{\left (27\,x^2+54\,x+28\right )}^{1/3}} \,d x \]
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