Optimal. Leaf size=744 \[ -\frac{\left (27 x^2-54 x+52\right )^{2/3}}{1500 (3 x+2)}-\frac{\left (27 x^2-54 x+52\right )^{2/3}}{600 (3 x+2)^2}+\frac{\log \left (-27 \sqrt [3]{10} \sqrt [3]{27 x^2-54 x+52}-81 x+216\right )}{600\ 10^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (8-3 x)}{\sqrt{3} \sqrt [3]{5} \sqrt [3]{27 x^2-54 x+52}}+\frac{1}{\sqrt{3}}\right )}{300 \sqrt{3} 10^{2/3}}+\frac{9 (1-x)}{50\ 5^{2/3} \left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )}-\frac{\log (3 x+2)}{600\ 10^{2/3}}+\frac{\left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} F\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{27000\ 3^{3/4} \sqrt [6]{5} \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)}-\frac{\sqrt{2+\sqrt{3}} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} E\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{54000 \sqrt{2} \sqrt [4]{3} \sqrt [6]{5} \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)} \]
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Rubi [A] time = 1.3685, antiderivative size = 744, 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 \[ -\frac{\left (27 x^2-54 x+52\right )^{2/3}}{1500 (3 x+2)}-\frac{\left (27 x^2-54 x+52\right )^{2/3}}{600 (3 x+2)^2}+\frac{\log \left (-27 \sqrt [3]{10} \sqrt [3]{27 x^2-54 x+52}-81 x+216\right )}{600\ 10^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (8-3 x)}{\sqrt{3} \sqrt [3]{5} \sqrt [3]{27 x^2-54 x+52}}+\frac{1}{\sqrt{3}}\right )}{300 \sqrt{3} 10^{2/3}}+\frac{9 (1-x)}{50\ 5^{2/3} \left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )}-\frac{\log (3 x+2)}{600\ 10^{2/3}}+\frac{\left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} F\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{27000\ 3^{3/4} \sqrt [6]{5} \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)}-\frac{\sqrt{2+\sqrt{3}} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} E\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{54000 \sqrt{2} \sqrt [4]{3} \sqrt [6]{5} \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)} \]
Warning: Unable to verify antiderivative.
[In] Int[1/((2 + 3*x)^3*(52 - 54*x + 27*x^2)^(1/3)),x]
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Rubi in Sympy [A] time = 50.7373, size = 546, normalized size = 0.73 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(2+3*x)**3/(27*x**2-54*x+52)**(1/3),x)
[Out]
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Mathematica [C] time = 0.600578, size = 407, normalized size = 0.55 \[ \frac{-\frac{450 (3 x+2) \left (9 x-5 i \sqrt{3}-9\right ) \left (9 x+5 i \sqrt{3}-9\right ) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{15-5 i \sqrt{3}}{9 x+6},\frac{15+5 i \sqrt{3}}{9 x+6}\right )}{(9 x+6) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{15-5 i \sqrt{3}}{9 x+6},\frac{15+5 i \sqrt{3}}{9 x+6}\right )+\left (3+i \sqrt{3}\right ) F_1\left (\frac{5}{3};\frac{1}{3},\frac{4}{3};\frac{8}{3};\frac{15-5 i \sqrt{3}}{9 x+6},\frac{15+5 i \sqrt{3}}{9 x+6}\right )+\left (3-i \sqrt{3}\right ) F_1\left (\frac{5}{3};\frac{4}{3},\frac{1}{3};\frac{8}{3};\frac{15-5 i \sqrt{3}}{9 x+6},\frac{15+5 i \sqrt{3}}{9 x+6}\right )}+3^{5/6} 10^{2/3} \sqrt [3]{-9 i x+5 \sqrt{3}+9 i} \left (9 x-5 i \sqrt{3}-9\right ) \left (27 x^2-54 x+52\right ) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{9 i x+5 \sqrt{3}-9 i}{10 \sqrt{3}}\right )-\frac{90 (2 x+3) \left (27 x^2-54 x+52\right )^2}{(3 x+2)^2}}{90000 \left (27 x^2-54 x+52\right )^{4/3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/((2 + 3*x)^3*(52 - 54*x + 27*x^2)^(1/3)),x]
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Maple [F] time = 0.121, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( 2+3\,x \right ) ^{3}}{\frac{1}{\sqrt [3]{27\,{x}^{2}-54\,x+52}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(2+3*x)^3/(27*x^2-54*x+52)^(1/3),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac{1}{3}}{\left (3 \, x + 2\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((27*x^2 - 54*x + 52)^(1/3)*(3*x + 2)^3),x, algorithm="maxima")
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((27*x^2 - 54*x + 52)^(1/3)*(3*x + 2)^3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (3 x + 2\right )^{3} \sqrt [3]{27 x^{2} - 54 x + 52}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(2+3*x)**3/(27*x**2-54*x+52)**(1/3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac{1}{3}}{\left (3 \, x + 2\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((27*x^2 - 54*x + 52)^(1/3)*(3*x + 2)^3),x, algorithm="giac")
[Out]