Optimal. Leaf size=671 \[ -\frac{\left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right ),4 \sqrt{3}-7\right )}{36\ 3^{3/4} \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}-\frac{\left (27 x^2+54 x+28\right )^{2/3}}{12 (3 x+2)}-\frac{\log \left (27 \sqrt [3]{2} \sqrt [3]{27 x^2+54 x+28}-81 x-108\right )}{12\ 2^{2/3}}+\frac{\tan ^{-1}\left (\frac{2^{2/3} (3 x+4)}{\sqrt{3} \sqrt [3]{27 x^2+54 x+28}}+\frac{1}{\sqrt{3}}\right )}{6\ 2^{2/3} \sqrt{3}}+\frac{\sqrt{2+\sqrt{3}} \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} E\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt{3}\right )}{72 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}-\frac{9 (x+1)}{2 \left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )}+\frac{\log (3 x+2)}{12\ 2^{2/3}} \]
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Rubi [A] time = 0.594944, antiderivative size = 671, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {744, 12, 843, 619, 235, 304, 219, 1879, 752} \[ -\frac{\left (27 x^2+54 x+28\right )^{2/3}}{12 (3 x+2)}-\frac{\log \left (27 \sqrt [3]{2} \sqrt [3]{27 x^2+54 x+28}-81 x-108\right )}{12\ 2^{2/3}}+\frac{\tan ^{-1}\left (\frac{2^{2/3} (3 x+4)}{\sqrt{3} \sqrt [3]{27 x^2+54 x+28}}+\frac{1}{\sqrt{3}}\right )}{6\ 2^{2/3} \sqrt{3}}-\frac{\left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} F\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt{3}\right )}{36\ 3^{3/4} \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}+\frac{\sqrt{2+\sqrt{3}} \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} E\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt{3}\right )}{72 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}-\frac{9 (x+1)}{2 \left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )}+\frac{\log (3 x+2)}{12\ 2^{2/3}} \]
Antiderivative was successfully verified.
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Rule 744
Rule 12
Rule 843
Rule 619
Rule 235
Rule 304
Rule 219
Rule 1879
Rule 752
Rubi steps
\begin{align*} \int \frac{1}{(2+3 x)^2 \sqrt [3]{28+54 x+27 x^2}} \, dx &=-\frac{\left (28+54 x+27 x^2\right )^{2/3}}{12 (2+3 x)}-\frac{1}{36} \int -\frac{27 x}{(2+3 x) \sqrt [3]{28+54 x+27 x^2}} \, dx\\ &=-\frac{\left (28+54 x+27 x^2\right )^{2/3}}{12 (2+3 x)}+\frac{3}{4} \int \frac{x}{(2+3 x) \sqrt [3]{28+54 x+27 x^2}} \, dx\\ &=-\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}{2} \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}}{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 )}{6\ 2^{2/3} \sqrt{3}}+\frac{\log (2+3 x)}{12\ 2^{2/3}}-\frac{\log \left (-108-81 x+27 \sqrt [3]{2} \sqrt [3]{28+54 x+27 x^2}\right )}{12\ 2^{2/3}}+\frac{1}{216} \operatorname{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}}{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 )}{6\ 2^{2/3} \sqrt{3}}+\frac{\log (2+3 x)}{12\ 2^{2/3}}-\frac{\log \left (-108-81 x+27 \sqrt [3]{2} \sqrt [3]{28+54 x+27 x^2}\right )}{12\ 2^{2/3}}+\frac{\sqrt{(54+54 x)^2} \operatorname{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}}{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 )}{6\ 2^{2/3} \sqrt{3}}+\frac{\log (2+3 x)}{12\ 2^{2/3}}-\frac{\log \left (-108-81 x+27 \sqrt [3]{2} \sqrt [3]{28+54 x+27 x^2}\right )}{12\ 2^{2/3}}-\frac{\sqrt{(54+54 x)^2} \operatorname{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 (\sqrt{\frac{1}{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 )}{4 (54+54 x)}\\ &=-\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 )}{6\ 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)}{12\ 2^{2/3}}-\frac{\log \left (-108-81 x+27 \sqrt [3]{2} \sqrt [3]{28+54 x+27 x^2}\right )}{12\ 2^{2/3}}\\ \end{align*}
Mathematica [C] time = 0.252011, size = 240, normalized size = 0.36 \[ \frac{4 \sqrt [3]{3} (3 x+2) \sqrt [3]{\frac{9 x-i \sqrt{3}+9}{3 x+2}} \sqrt [3]{\frac{9 x+i \sqrt{3}+9}{3 x+2}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};-\frac{3+i \sqrt{3}}{9 x+6},\frac{-3+i \sqrt{3}}{9 x+6}\right )+2^{2/3} \sqrt [3]{3} \sqrt [3]{-9 i x+\sqrt{3}-9 i} (3 x+2) \left (3 \sqrt{3} x+3 \sqrt{3}-i\right ) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{9 i x+\sqrt{3}+9 i}{2 \sqrt{3}}\right )-4 \left (27 x^2+54 x+28\right )}{48 (3 x+2) \sqrt [3]{27 x^2+54 x+28}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 1.715, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \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{1}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac{1}{3}}{\left (3 \, x + 2\right )}^{2}}\,{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{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac{2}{3}}}{243 \, x^{4} + 810 \, x^{3} + 1008 \, x^{2} + 552 \, x + 112}, 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{1}{\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{1}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac{1}{3}}{\left (3 \, x + 2\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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