3.2488 \(\int \frac{2+3 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx\)

Optimal. Leaf size=603 \[ \frac{1}{12} \left (27 x^2-54 x+52\right )^{2/3}+\frac{90 \sqrt [3]{5} (1-x)}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}+\frac{5^{5/6} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} F\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{54\ 3^{3/4} \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)}-\frac{5^{5/6} \sqrt{2+\sqrt{3}} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} E\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{108 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)} \]

[Out]

(52 - 54*x + 27*x^2)^(2/3)/12 + (90*5^(1/3)*(1 - x))/(30*(1 - Sqrt[3]) - 10^(1/3
)*(2700 + (-54 + 54*x)^2)^(1/3)) - (5^(5/6)*Sqrt[2 + Sqrt[3]]*(30 - 10^(1/3)*(27
00 + (-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]*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]])/(108*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)]) + (5^(5/6)*(30 - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))*Sqrt[(900 + 3
0*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[ArcS
in[(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]])/(54*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 = 0.992301, 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 \[ \frac{1}{12} \left (27 x^2-54 x+52\right )^{2/3}+\frac{90 \sqrt [3]{5} (1-x)}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}+\frac{5^{5/6} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} F\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{54\ 3^{3/4} \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)}-\frac{5^{5/6} \sqrt{2+\sqrt{3}} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} E\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{108 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)} \]

Antiderivative was successfully verified.

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

[Out]

(52 - 54*x + 27*x^2)^(2/3)/12 + (90*5^(1/3)*(1 - x))/(30*(1 - Sqrt[3]) - 10^(1/3
)*(2700 + (-54 + 54*x)^2)^(1/3)) - (5^(5/6)*Sqrt[2 + Sqrt[3]]*(30 - 10^(1/3)*(27
00 + (-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]*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]])/(108*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)]) + (5^(5/6)*(30 - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))*Sqrt[(900 + 3
0*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[ArcS
in[(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]])/(54*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 = 22.1089, size = 420, normalized size = 0.7 \[ \frac{\sqrt [3]{5} \left (- 54 x + 54\right )}{18 \left (- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} - \sqrt{3} + 1\right )} + \frac{\left (27 x^{2} - 54 x + 52\right )^{\frac{2}{3}}}{12} - \frac{75 \sqrt [4]{3} \sqrt [3]{5} \sqrt{\frac{\left (\frac{\left (54 x - 54\right )^{2}}{2700} + 1\right )^{\frac{2}{3}} + \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} + 1}{\left (- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} + 1\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} + 1 + \sqrt{3}}{- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{\sqrt{\frac{\sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} - 1}{\left (- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} - \sqrt{3} + 1\right )^{2}}} \left (- 54 x + 54\right )} + \frac{50 \sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt [3]{5} \sqrt{\frac{\left (\frac{\left (54 x - 54\right )^{2}}{2700} + 1\right )^{\frac{2}{3}} + \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} + 1}{\left (- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} + 1\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} + 1 + \sqrt{3}}{- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{\sqrt{\frac{\sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} - 1}{\left (- \sqrt [3]{\frac{\left (54 x - 54\right )^{2}}{2700} + 1} - \sqrt{3} + 1\right )^{2}}} \left (- 54 x + 54\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5**(1/3)*(-54*x + 54)/(18*(-((54*x - 54)**2/2700 + 1)**(1/3) - sqrt(3) + 1)) + (
27*x**2 - 54*x + 52)**(2/3)/12 - 75*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))/(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)) + 50*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))/(sqrt((((54*x - 54)**2/2700 + 1)**(1/3) - 1)/(-((5
4*x - 54)**2/2700 + 1)**(1/3) - sqrt(3) + 1)**2)*(-54*x + 54))

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Mathematica [C]  time = 0.178617, size = 113, normalized size = 0.19 \[ \frac{\sqrt [3]{3} 10^{2/3} \sqrt [3]{-9 i x+5 \sqrt{3}+9 i} \left (3 \sqrt{3} x-3 \sqrt{3}-5 i\right ) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{9 i x+5 \sqrt{3}-9 i}{10 \sqrt{3}}\right )+27 x^2-54 x+52}{12 \sqrt [3]{27 x^2-54 x+52}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(52 - 54*x + 27*x^2 + 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])])/(12*(52 - 54*x + 27*x^2)^(1/3))

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Maple [F]  time = 0.094, size = 0, normalized size = 0. \[ \int{(2+3\,x){\frac{1}{\sqrt [3]{27\,{x}^{2}-54\,x+52}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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