3.2486 \(\int \frac{(2+3 x)^3}{\sqrt [3]{52-54 x+27 x^2}} \, dx\)

Optimal. Leaf size=635 \[ \frac{1}{30} \left (27 x^2-54 x+52\right )^{2/3} (3 x+2)^2+\frac{1}{7} (8 x+27) \left (27 x^2-54 x+52\right )^{2/3}+\frac{9000 \sqrt [3]{5} (1-x)}{7 \left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )}+\frac{50\ 5^{5/6} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} F\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{189\ 3^{3/4} (1-x) \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}}}-\frac{25\ 5^{5/6} \sqrt{2+\sqrt{3}} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} E\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{189 \sqrt{2} \sqrt [4]{3} (1-x) \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}}} \]

[Out]

((2 + 3*x)^2*(52 - 54*x + 27*x^2)^(2/3))/30 + ((27 + 8*x)*(52 - 54*x + 27*x^2)^(
2/3))/7 + (9000*5^(1/3)*(1 - x))/(7*(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 +
54*x)^2)^(1/3))) - (25*5^(5/6)*Sqrt[2 + Sqrt[3]]*(30 - 10^(1/3)*(2700 + (-54 + 5
4*x)^2)^(1/3))*Sqrt[(900 + 30*10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3) + 10^(2/3)*
(2700 + (-54 + 54*x)^2)^(2/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)
^2)^(1/3))^2]*EllipticE[ArcSin[(30*(1 + Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)
^2)^(1/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))], -7 + 4*
Sqrt[3]])/(189*Sqrt[2]*3^(1/4)*(1 - x)*Sqrt[-((30 - 10^(1/3)*(2700 + (-54 + 54*x
)^2)^(1/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))^2)]) + (
50*5^(5/6)*(30 - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))*Sqrt[(900 + 30*10^(1/3)
*(2700 + (-54 + 54*x)^2)^(1/3) + 10^(2/3)*(2700 + (-54 + 54*x)^2)^(2/3))/(30*(1
- Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))^2]*EllipticF[ArcSin[(30*(1
+ Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))/(30*(1 - Sqrt[3]) - 10^(1/3
)*(2700 + (-54 + 54*x)^2)^(1/3))], -7 + 4*Sqrt[3]])/(189*3^(3/4)*(1 - x)*Sqrt[-(
(30 - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700
+ (-54 + 54*x)^2)^(1/3))^2)])

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Rubi [A]  time = 1.52478, antiderivative size = 635, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ \frac{1}{30} \left (27 x^2-54 x+52\right )^{2/3} (3 x+2)^2+\frac{1}{7} (8 x+27) \left (27 x^2-54 x+52\right )^{2/3}+\frac{9000 \sqrt [3]{5} (1-x)}{7 \left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )}+\frac{50\ 5^{5/6} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} F\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{189\ 3^{3/4} (1-x) \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}}}-\frac{25\ 5^{5/6} \sqrt{2+\sqrt{3}} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} E\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{189 \sqrt{2} \sqrt [4]{3} (1-x) \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*x)^3/(52 - 54*x + 27*x^2)^(1/3),x]

[Out]

((2 + 3*x)^2*(52 - 54*x + 27*x^2)^(2/3))/30 + ((27 + 8*x)*(52 - 54*x + 27*x^2)^(
2/3))/7 + (9000*5^(1/3)*(1 - x))/(7*(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 +
54*x)^2)^(1/3))) - (25*5^(5/6)*Sqrt[2 + Sqrt[3]]*(30 - 10^(1/3)*(2700 + (-54 + 5
4*x)^2)^(1/3))*Sqrt[(900 + 30*10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3) + 10^(2/3)*
(2700 + (-54 + 54*x)^2)^(2/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)
^2)^(1/3))^2]*EllipticE[ArcSin[(30*(1 + Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)
^2)^(1/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))], -7 + 4*
Sqrt[3]])/(189*Sqrt[2]*3^(1/4)*(1 - x)*Sqrt[-((30 - 10^(1/3)*(2700 + (-54 + 54*x
)^2)^(1/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))^2)]) + (
50*5^(5/6)*(30 - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))*Sqrt[(900 + 30*10^(1/3)
*(2700 + (-54 + 54*x)^2)^(1/3) + 10^(2/3)*(2700 + (-54 + 54*x)^2)^(2/3))/(30*(1
- Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))^2]*EllipticF[ArcSin[(30*(1
+ Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))/(30*(1 - Sqrt[3]) - 10^(1/3
)*(2700 + (-54 + 54*x)^2)^(1/3))], -7 + 4*Sqrt[3]])/(189*3^(3/4)*(1 - x)*Sqrt[-(
(30 - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700
+ (-54 + 54*x)^2)^(1/3))^2)])

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Rubi in Sympy [A]  time = 32.7735, size = 452, normalized size = 0.71 \[ \frac{50 \sqrt [3]{5} \left (- 54 x + 54\right )}{63 \left (- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} - \sqrt{3} + 1\right )} + \frac{\left (3 x + 2\right )^{2} \left (27 x^{2} - 54 x + 52\right )^{\frac{2}{3}}}{30} + \frac{\left (233280 x + 787320\right ) \left (27 x^{2} - 54 x + 52\right )^{\frac{2}{3}}}{204120} - \frac{7500 \sqrt [4]{3} \sqrt [3]{5} \sqrt{\frac{\left (\frac{\left (54 x - 54\right )^{2}}{2700} + 1\right )^{\frac{2}{3}} + \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} + 1}{\left (- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} + 1\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} + 1 + \sqrt{3}}{- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{7 \sqrt{\frac{\sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} - 1}{\left (- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} - \sqrt{3} + 1\right )^{2}}} \left (- 54 x + 54\right )} + \frac{5000 \sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt [3]{5} \sqrt{\frac{\left (\frac{\left (54 x - 54\right )^{2}}{2700} + 1\right )^{\frac{2}{3}} + \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} + 1}{\left (- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} + 1\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} + 1 + \sqrt{3}}{- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{7 \sqrt{\frac{\sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} - 1}{\left (- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} - \sqrt{3} + 1\right )^{2}}} \left (- 54 x + 54\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**3/(27*x**2-54*x+52)**(1/3),x)

[Out]

50*5**(1/3)*(-54*x + 54)/(63*(-((54*x - 54)**2/2700 + 1)**(1/3) - sqrt(3) + 1))
+ (3*x + 2)**2*(27*x**2 - 54*x + 52)**(2/3)/30 + (233280*x + 787320)*(27*x**2 -
54*x + 52)**(2/3)/204120 - 7500*3**(1/4)*5**(1/3)*sqrt((((54*x - 54)**2/2700 + 1
)**(2/3) + ((54*x - 54)**2/2700 + 1)**(1/3) + 1)/(-((54*x - 54)**2/2700 + 1)**(1
/3) - sqrt(3) + 1)**2)*sqrt(sqrt(3) + 2)*(-((54*x - 54)**2/2700 + 1)**(1/3) + 1)
*elliptic_e(asin((-((54*x - 54)**2/2700 + 1)**(1/3) + 1 + sqrt(3))/(-((54*x - 54
)**2/2700 + 1)**(1/3) - sqrt(3) + 1)), -7 + 4*sqrt(3))/(7*sqrt((((54*x - 54)**2/
2700 + 1)**(1/3) - 1)/(-((54*x - 54)**2/2700 + 1)**(1/3) - sqrt(3) + 1)**2)*(-54
*x + 54)) + 5000*sqrt(2)*3**(3/4)*5**(1/3)*sqrt((((54*x - 54)**2/2700 + 1)**(2/3
) + ((54*x - 54)**2/2700 + 1)**(1/3) + 1)/(-((54*x - 54)**2/2700 + 1)**(1/3) - s
qrt(3) + 1)**2)*(-((54*x - 54)**2/2700 + 1)**(1/3) + 1)*elliptic_f(asin((-((54*x
- 54)**2/2700 + 1)**(1/3) + 1 + sqrt(3))/(-((54*x - 54)**2/2700 + 1)**(1/3) - s
qrt(3) + 1)), -7 + 4*sqrt(3))/(7*sqrt((((54*x - 54)**2/2700 + 1)**(1/3) - 1)/(-(
(54*x - 54)**2/2700 + 1)**(1/3) - sqrt(3) + 1)**2)*(-54*x + 54))

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Mathematica [C]  time = 0.270299, size = 124, normalized size = 0.2 \[ \frac{250 \sqrt [3]{3} 10^{2/3} \sqrt [3]{-9 i x+5 \sqrt{3}+9 i} \left (3 \sqrt{3} x-3 \sqrt{3}-5 i\right ) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{9 i x+5 \sqrt{3}-9 i}{10 \sqrt{3}}\right )+1701 x^4+5346 x^3+8406 x^2-28404 x+43576}{210 \sqrt [3]{27 x^2-54 x+52}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(2 + 3*x)^3/(52 - 54*x + 27*x^2)^(1/3),x]

[Out]

(43576 - 28404*x + 8406*x^2 + 5346*x^3 + 1701*x^4 + 250*3^(1/3)*10^(2/3)*(9*I +
5*Sqrt[3] - (9*I)*x)^(1/3)*(-5*I - 3*Sqrt[3] + 3*Sqrt[3]*x)*Hypergeometric2F1[1/
3, 2/3, 5/3, (-9*I + 5*Sqrt[3] + (9*I)*x)/(10*Sqrt[3])])/(210*(52 - 54*x + 27*x^
2)^(1/3))

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Maple [F]  time = 0.24, size = 0, normalized size = 0. \[ \int{ \left ( 2+3\,x \right ) ^{3}{\frac{1}{\sqrt [3]{27\,{x}^{2}-54\,x+52}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^3/(27*x^2-54*x+52)^(1/3),x)

[Out]

int((2+3*x)^3/(27*x^2-54*x+52)^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{3}}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^3/(27*x^2 - 54*x + 52)^(1/3),x, algorithm="maxima")

[Out]

integrate((3*x + 2)^3/(27*x^2 - 54*x + 52)^(1/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac{1}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^3/(27*x^2 - 54*x + 52)^(1/3),x, algorithm="fricas")

[Out]

integral((27*x^3 + 54*x^2 + 36*x + 8)/(27*x^2 - 54*x + 52)^(1/3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x + 2\right )^{3}}{\sqrt [3]{27 x^{2} - 54 x + 52}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+3*x)**3/(27*x**2-54*x+52)**(1/3),x)

[Out]

Integral((3*x + 2)**3/(27*x**2 - 54*x + 52)**(1/3), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{3}}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^3/(27*x^2 - 54*x + 52)^(1/3),x, algorithm="giac")

[Out]

integrate((3*x + 2)^3/(27*x^2 - 54*x + 52)^(1/3), x)