Optimal. Leaf size=603 \[ \frac{5^{5/6} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right ),4 \sqrt{3}-7\right )}{54\ 3^{3/4} \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)}+\frac{1}{12} \left (27 x^2-54 x+52\right )^{2/3}+\frac{90 \sqrt [3]{5} (1-x)}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}-\frac{5^{5/6} \sqrt{2+\sqrt{3}} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} E\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{108 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)} \]
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Rubi [A] time = 0.485279, antiderivative size = 603, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {640, 619, 235, 304, 219, 1879} \[ \frac{1}{12} \left (27 x^2-54 x+52\right )^{2/3}+\frac{90 \sqrt [3]{5} (1-x)}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}+\frac{5^{5/6} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} F\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{54\ 3^{3/4} \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)}-\frac{5^{5/6} \sqrt{2+\sqrt{3}} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} E\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{108 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)} \]
Antiderivative was successfully verified.
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Rule 640
Rule 619
Rule 235
Rule 304
Rule 219
Rule 1879
Rubi steps
\begin{align*} \int \frac{2+3 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx &=\frac{1}{12} \left (52-54 x+27 x^2\right )^{2/3}+5 \int \frac{1}{\sqrt [3]{52-54 x+27 x^2}} \, dx\\ &=\frac{1}{12} \left (52-54 x+27 x^2\right )^{2/3}+\frac{1}{54} \sqrt [3]{5} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1+\frac{x^2}{2700}}} \, dx,x,-54+54 x\right )\\ &=\frac{1}{12} \left (52-54 x+27 x^2\right )^{2/3}+\frac{\left (5 \sqrt [3]{5} \sqrt{(-54+54 x)^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+x^3}} \, dx,x,\frac{\sqrt [3]{2700+(-54+54 x)^2}}{3\ 10^{2/3}}\right )}{2 \sqrt{3} (-54+54 x)}\\ &=\frac{1}{12} \left (52-54 x+27 x^2\right )^{2/3}-\frac{\left (5 \sqrt [3]{5} \sqrt{(-54+54 x)^2}\right ) \operatorname{Subst}\left (\int \frac{1+\sqrt{3}-x}{\sqrt{-1+x^3}} \, dx,x,\frac{\sqrt [3]{2700+(-54+54 x)^2}}{3\ 10^{2/3}}\right )}{2 \sqrt{3} (-54+54 x)}+\frac{\left (5 \sqrt [3]{5} \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,\frac{\sqrt [3]{2700+(-54+54 x)^2}}{3\ 10^{2/3}}\right )}{-54+54 x}\\ &=\frac{1}{12} \left (52-54 x+27 x^2\right )^{2/3}+\frac{90 \sqrt [3]{5} (1-x)}{30-30 \sqrt{3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}-\frac{5^{5/6} \sqrt{2+\sqrt{3}} \left (30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right ) \sqrt{\frac{900+30 \sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}+10^{2/3} \left (2700+(-54+54 x)^2\right )^{2/3}}{\left (30-30 \sqrt{3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{30+30 \sqrt{3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{30-30 \sqrt{3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}\right )|-7+4 \sqrt{3}\right )}{108 \sqrt{2} \sqrt [4]{3} (1-x) \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{\left (30-30 \sqrt{3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}}}+\frac{5^{5/6} \left (30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right ) \sqrt{\frac{900+30 \sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}+10^{2/3} \left (2700+(-54+54 x)^2\right )^{2/3}}{\left (30-30 \sqrt{3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{30+30 \sqrt{3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{30-30 \sqrt{3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}\right )|-7+4 \sqrt{3}\right )}{54\ 3^{3/4} (1-x) \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{\left (30-30 \sqrt{3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}}}\\ \end{align*}
Mathematica [C] time = 0.0112111, size = 47, normalized size = 0.08 \[ \sqrt [3]{5} (x-1) \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};-\frac{27}{25} (x-1)^2\right )+\frac{1}{12} \left (27 x^2-54 x+52\right )^{2/3} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.403, size = 0, normalized size = 0. \begin{align*} \int{(2+3\,x){\frac{1}{\sqrt [3]{27\,{x}^{2}-54\,x+52}}}}\, 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{3 \, x + 2}{{\left (27 \, x^{2} - 54 \, x + 52\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{3 \, x + 2}{{\left (27 \, x^{2} - 54 \, x + 52\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{3 x + 2}{\sqrt [3]{27 x^{2} - 54 x + 52}}\, 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{3 \, x + 2}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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