Optimal. Leaf size=628 \[ \frac{1}{21} (3 x+2) \left (27 x^2-54 x+52\right )^{2/3}+\frac{25}{42} \left (27 x^2-54 x+52\right )^{2/3}+\frac{2700 \sqrt [3]{5} (1-x)}{7 \left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )}+\frac{5\ 5^{5/6} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} F\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{63\ 3^{3/4} \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)}-\frac{5\ 5^{5/6} \sqrt{2+\sqrt{3}} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} E\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{126 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)} \]
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Rubi [A] time = 1.11493, antiderivative size = 628, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \frac{1}{21} (3 x+2) \left (27 x^2-54 x+52\right )^{2/3}+\frac{25}{42} \left (27 x^2-54 x+52\right )^{2/3}+\frac{2700 \sqrt [3]{5} (1-x)}{7 \left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )}+\frac{5\ 5^{5/6} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} F\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{63\ 3^{3/4} \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)}-\frac{5\ 5^{5/6} \sqrt{2+\sqrt{3}} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} E\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{126 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)} \]
Antiderivative was successfully verified.
[In] Int[(2 + 3*x)^2/(52 - 54*x + 27*x^2)^(1/3),x]
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Rubi in Sympy [A] time = 29.7476, size = 447, normalized size = 0.71 \[ \frac{5 \sqrt [3]{5} \left (- 54 x + 54\right )}{21 \left (- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} - \sqrt{3} + 1\right )} + \frac{\left (9 x + 6\right ) \left (27 x^{2} - 54 x + 52\right )^{\frac{2}{3}}}{63} + \frac{25 \left (27 x^{2} - 54 x + 52\right )^{\frac{2}{3}}}{42} - \frac{2250 \sqrt [4]{3} \sqrt [3]{5} \sqrt{\frac{\left (\frac{\left (54 x - 54\right )^{2}}{2700} + 1\right )^{\frac{2}{3}} + \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} + 1}{\left (- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} + 1\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} + 1 + \sqrt{3}}{- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{7 \sqrt{\frac{\sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} - 1}{\left (- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} - \sqrt{3} + 1\right )^{2}}} \left (- 54 x + 54\right )} + \frac{1500 \sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt [3]{5} \sqrt{\frac{\left (\frac{\left (54 x - 54\right )^{2}}{2700} + 1\right )^{\frac{2}{3}} + \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} + 1}{\left (- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} + 1\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} + 1 + \sqrt{3}}{- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{7 \sqrt{\frac{\sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} - 1}{\left (- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} - \sqrt{3} + 1\right )^{2}}} \left (- 54 x + 54\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+3*x)**2/(27*x**2-54*x+52)**(1/3),x)
[Out]
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Mathematica [C] time = 0.183292, size = 119, normalized size = 0.19 \[ \frac{15 \sqrt [3]{3} 10^{2/3} \sqrt [3]{-9 i x+5 \sqrt{3}+9 i} \left (3 \sqrt{3} x-3 \sqrt{3}-5 i\right ) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{9 i x+5 \sqrt{3}-9 i}{10 \sqrt{3}}\right )+162 x^3+459 x^2-1254 x+1508}{42 \sqrt [3]{27 x^2-54 x+52}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(2 + 3*x)^2/(52 - 54*x + 27*x^2)^(1/3),x]
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Maple [F] time = 0.095, size = 0, normalized size = 0. \[ \int{ \left ( 2+3\,x \right ) ^{2}{\frac{1}{\sqrt [3]{27\,{x}^{2}-54\,x+52}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+3*x)^2/(27*x^2-54*x+52)^(1/3),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{2}}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac{1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^2/(27*x^2 - 54*x + 52)^(1/3),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{9 \, x^{2} + 12 \, x + 4}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac{1}{3}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^2/(27*x^2 - 54*x + 52)^(1/3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x + 2\right )^{2}}{\sqrt [3]{27 x^{2} - 54 x + 52}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2+3*x)**2/(27*x**2-54*x+52)**(1/3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{2}}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac{1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^2/(27*x^2 - 54*x + 52)^(1/3),x, algorithm="giac")
[Out]