Optimal. Leaf size=585 \[ \frac {1}{21} (3 x+2) \left (27 x^2+54 x+28\right )^{2/3}-\frac {5}{42} \left (27 x^2+54 x+28\right )^{2/3}-\frac {2 \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt {\frac {\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} F\left (\sin ^{-1}\left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt {3}\right )}{21\ 3^{3/4} \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}+\frac {\sqrt {2+\sqrt {3}} \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt {\frac {\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} E\left (\sin ^{-1}\left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt {3}\right )}{21 \sqrt {2} \sqrt [4]{3} \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}-\frac {108 (x+1)}{7 \left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )} \]
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Rubi [A] time = 0.49, antiderivative size = 585, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {742, 640, 619, 235, 304, 219, 1879} \[ \frac {1}{21} (3 x+2) \left (27 x^2+54 x+28\right )^{2/3}-\frac {5}{42} \left (27 x^2+54 x+28\right )^{2/3}-\frac {2 \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt {\frac {\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} F\left (\sin ^{-1}\left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt {3}\right )}{21\ 3^{3/4} \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}+\frac {\sqrt {2+\sqrt {3}} \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt {\frac {\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} E\left (\sin ^{-1}\left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt {3}\right )}{21 \sqrt {2} \sqrt [4]{3} \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}-\frac {108 (x+1)}{7 \left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )} \]
Antiderivative was successfully verified.
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Rule 219
Rule 235
Rule 304
Rule 619
Rule 640
Rule 742
Rule 1879
Rubi steps
\begin {align*} \int \frac {(2+3 x)^2}{\sqrt [3]{28+54 x+27 x^2}} \, dx &=\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} \operatorname {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 ) \operatorname {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 ) \operatorname {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 {6 \left (2+\sqrt {3}\right )} \sqrt {(54+54 x)^2}\right ) \operatorname {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*}
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Mathematica [C] time = 0.02, size = 47, normalized size = 0.08 \[ \frac {1}{42} \left (36 (x+1) \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};-27 (x+1)^2\right )+\left (27 x^2+54 x+28\right )^{2/3} (6 x-1)\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 1.19, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {9 \, x^{2} + 12 \, x + 4}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}}}, x\right ) \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{2}}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 3.21, size = 0, normalized size = 0.00 \[ \int \frac {\left (3 x +2\right )^{2}}{\left (27 x^{2}+54 x +28\right )^{\frac {1}{3}}}\, dx \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{2}}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}}}\,{d x} \]
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 \[ \int \frac {{\left (3\,x+2\right )}^2}{{\left (27\,x^2+54\,x+28\right )}^{1/3}} \,d x \]
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 \[ \int \frac {\left (3 x + 2\right )^{2}}{\sqrt [3]{27 x^{2} + 54 x + 28}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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