Optimal. Leaf size=585 \[ -\frac{2 \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right ),4 \sqrt{3}-7\right )}{21\ 3^{3/4} \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}+\frac{1}{21} (3 x+2) \left (27 x^2+54 x+28\right )^{2/3}-\frac{5}{42} \left (27 x^2+54 x+28\right )^{2/3}+\frac{\sqrt{2+\sqrt{3}} \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} E\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt{3}\right )}{21 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}-\frac{108 (x+1)}{7 \left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )} \]
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Rubi [A] time = 0.485929, antiderivative size = 585, normalized size of antiderivative = 1., 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 742
Rule 640
Rule 619
Rule 235
Rule 304
Rule 219
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*}
Mathematica [C] time = 0.0184156, 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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Maple [F] time = 1.506, size = 0, normalized size = 0. \begin{align*} \int{ \left ( 2+3\,x \right ) ^{2}{\frac{1}{\sqrt [3]{27\,{x}^{2}+54\,x+28}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (3 \, x + 2\right )}^{2}}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{9 \, x^{2} + 12 \, x + 4}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac{1}{3}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (3 x + 2\right )^{2}}{\sqrt [3]{27 x^{2} + 54 x + 28}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (3 \, x + 2\right )}^{2}}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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