Optimal. Leaf size=634 \[ -\frac{\left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt{\frac{\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}\right ),4 \sqrt{3}-7\right )}{72 \sqrt [6]{2} \sqrt [4]{3} \sqrt{-\frac{2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}-\frac{3 x}{16 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )}-\frac{\left (27 x^2+4\right )^{2/3}}{48 (3 x+2)}+\frac{\log \left (-27\ 2^{2/3} \sqrt [3]{27 x^2+4}-81 x+54\right )}{48 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2} (2-3 x)}{\sqrt{3} \sqrt [3]{27 x^2+4}}+\frac{1}{\sqrt{3}}\right )}{24 \sqrt [3]{2} \sqrt{3}}+\frac{\sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt{\frac{\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt{3}\right )}{48\ 2^{2/3} 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}-\frac{\log (3 x+2)}{48 \sqrt [3]{2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.364388, antiderivative size = 634, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {745, 844, 235, 304, 219, 1879, 751} \[ -\frac{3 x}{16 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )}-\frac{\left (27 x^2+4\right )^{2/3}}{48 (3 x+2)}+\frac{\log \left (-27\ 2^{2/3} \sqrt [3]{27 x^2+4}-81 x+54\right )}{48 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2} (2-3 x)}{\sqrt{3} \sqrt [3]{27 x^2+4}}+\frac{1}{\sqrt{3}}\right )}{24 \sqrt [3]{2} \sqrt{3}}-\frac{\left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt{\frac{\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt{3}\right )}{72 \sqrt [6]{2} \sqrt [4]{3} \sqrt{-\frac{2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}+\frac{\sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt{\frac{\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt{3}\right )}{48\ 2^{2/3} 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}-\frac{\log (3 x+2)}{48 \sqrt [3]{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 745
Rule 844
Rule 235
Rule 304
Rule 219
Rule 1879
Rule 751
Rubi steps
\begin{align*} \int \frac{1}{(2+3 x)^2 \sqrt [3]{4+27 x^2}} \, dx &=-\frac{\left (4+27 x^2\right )^{2/3}}{48 (2+3 x)}-\frac{3}{16} \int \frac{-2-x}{(2+3 x) \sqrt [3]{4+27 x^2}} \, dx\\ &=-\frac{\left (4+27 x^2\right )^{2/3}}{48 (2+3 x)}+\frac{1}{16} \int \frac{1}{\sqrt [3]{4+27 x^2}} \, dx+\frac{1}{4} \int \frac{1}{(2+3 x) \sqrt [3]{4+27 x^2}} \, dx\\ &=-\frac{\left (4+27 x^2\right )^{2/3}}{48 (2+3 x)}-\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{\sqrt [3]{2} (2-3 x)}{\sqrt{3} \sqrt [3]{4+27 x^2}}\right )}{24 \sqrt [3]{2} \sqrt{3}}-\frac{\log (2+3 x)}{48 \sqrt [3]{2}}+\frac{\log \left (54-81 x-27\ 2^{2/3} \sqrt [3]{4+27 x^2}\right )}{48 \sqrt [3]{2}}+\frac{\sqrt{x^2} \operatorname{Subst}\left (\int \frac{x}{\sqrt{-4+x^3}} \, dx,x,\sqrt [3]{4+27 x^2}\right )}{32 \sqrt{3} x}\\ &=-\frac{\left (4+27 x^2\right )^{2/3}}{48 (2+3 x)}-\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{\sqrt [3]{2} (2-3 x)}{\sqrt{3} \sqrt [3]{4+27 x^2}}\right )}{24 \sqrt [3]{2} \sqrt{3}}-\frac{\log (2+3 x)}{48 \sqrt [3]{2}}+\frac{\log \left (54-81 x-27\ 2^{2/3} \sqrt [3]{4+27 x^2}\right )}{48 \sqrt [3]{2}}-\frac{\sqrt{x^2} \operatorname{Subst}\left (\int \frac{2^{2/3} \left (1+\sqrt{3}\right )-x}{\sqrt{-4+x^3}} \, dx,x,\sqrt [3]{4+27 x^2}\right )}{32 \sqrt{3} x}+\frac{\sqrt{x^2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-4+x^3}} \, dx,x,\sqrt [3]{4+27 x^2}\right )}{8\ 2^{5/6} \sqrt{3 \left (2-\sqrt{3}\right )} x}\\ &=-\frac{\left (4+27 x^2\right )^{2/3}}{48 (2+3 x)}-\frac{3 x}{16 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4+27 x^2}\right )}-\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{\sqrt [3]{2} (2-3 x)}{\sqrt{3} \sqrt [3]{4+27 x^2}}\right )}{24 \sqrt [3]{2} \sqrt{3}}+\frac{\sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{4+27 x^2}\right ) \sqrt{\frac{2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4+27 x^2}+\left (4+27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4+27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4+27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4+27 x^2}}\right )|-7+4 \sqrt{3}\right )}{48\ 2^{2/3} 3^{3/4} x \sqrt{-\frac{2^{2/3}-\sqrt [3]{4+27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4+27 x^2}\right )^2}}}-\frac{\left (2^{2/3}-\sqrt [3]{4+27 x^2}\right ) \sqrt{\frac{2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4+27 x^2}+\left (4+27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4+27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4+27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4+27 x^2}}\right )|-7+4 \sqrt{3}\right )}{72 \sqrt [6]{2} \sqrt [4]{3} x \sqrt{-\frac{2^{2/3}-\sqrt [3]{4+27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4+27 x^2}\right )^2}}}-\frac{\log (2+3 x)}{48 \sqrt [3]{2}}+\frac{\log \left (54-81 x-27\ 2^{2/3} \sqrt [3]{4+27 x^2}\right )}{48 \sqrt [3]{2}}\\ \end{align*}
Mathematica [C] time = 0.242957, size = 211, normalized size = 0.33 \[ \frac{-8 \sqrt [3]{3} (3 x+2) \sqrt [3]{\frac{9 x-2 i \sqrt{3}}{3 x+2}} \sqrt [3]{\frac{9 x+2 i \sqrt{3}}{3 x+2}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{6-2 i \sqrt{3}}{9 x+6},\frac{6+2 i \sqrt{3}}{9 x+6}\right )+\sqrt [3]{6} \sqrt [3]{2 \sqrt{3}-9 i x} (3 x+2) \left (3 \sqrt{3} x-2 i\right ) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{3}{4} i \sqrt{3} x+\frac{1}{2}\right )-4 \left (27 x^2+4\right )}{192 (3 x+2) \sqrt [3]{27 x^2+4}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.378, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( 2+3\,x \right ) ^{2}}{\frac{1}{\sqrt [3]{27\,{x}^{2}+4}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (27 \, x^{2} + 4\right )}^{\frac{1}{3}}{\left (3 \, x + 2\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (27 \, x^{2} + 4\right )}^{\frac{2}{3}}}{243 \, x^{4} + 324 \, x^{3} + 144 \, x^{2} + 48 \, x + 16}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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} + 4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (27 \, x^{2} + 4\right )}^{\frac{1}{3}}{\left (3 \, x + 2\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]