3.690 \(\int \frac{(2+3 x)^2}{\sqrt [3]{4+27 x^2}} \, dx\)

Optimal. Leaf size=551 \[ -\frac{72 x}{7 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )}+\frac{1}{21} (3 x+2) \left (27 x^2+4\right )^{2/3}+\frac{5}{21} \left (27 x^2+4\right )^{2/3}-\frac{8\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt{\frac{\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt{3}\right )}{21 \sqrt [4]{3} \sqrt{-\frac{2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}+\frac{4 \sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt{\frac{\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt{3}\right )}{7\ 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x} \]

[Out]

(5*(4 + 27*x^2)^(2/3))/21 + ((2 + 3*x)*(4 + 27*x^2)^(2/3))/21 - (72*x)/(7*(2^(2/
3)*(1 - Sqrt[3]) - (4 + 27*x^2)^(1/3))) + (4*2^(1/3)*Sqrt[2 + Sqrt[3]]*(2^(2/3)
- (4 + 27*x^2)^(1/3))*Sqrt[(2*2^(1/3) + 2^(2/3)*(4 + 27*x^2)^(1/3) + (4 + 27*x^2
)^(2/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 + 27*x^2)^(1/3))^2]*EllipticE[ArcSin[(2^(2/
3)*(1 + Sqrt[3]) - (4 + 27*x^2)^(1/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 + 27*x^2)^(1/
3))], -7 + 4*Sqrt[3]])/(7*3^(3/4)*x*Sqrt[-((2^(2/3) - (4 + 27*x^2)^(1/3))/(2^(2/
3)*(1 - Sqrt[3]) - (4 + 27*x^2)^(1/3))^2)]) - (8*2^(5/6)*(2^(2/3) - (4 + 27*x^2)
^(1/3))*Sqrt[(2*2^(1/3) + 2^(2/3)*(4 + 27*x^2)^(1/3) + (4 + 27*x^2)^(2/3))/(2^(2
/3)*(1 - Sqrt[3]) - (4 + 27*x^2)^(1/3))^2]*EllipticF[ArcSin[(2^(2/3)*(1 + Sqrt[3
]) - (4 + 27*x^2)^(1/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 + 27*x^2)^(1/3))], -7 + 4*S
qrt[3]])/(21*3^(1/4)*x*Sqrt[-((2^(2/3) - (4 + 27*x^2)^(1/3))/(2^(2/3)*(1 - Sqrt[
3]) - (4 + 27*x^2)^(1/3))^2)])

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Rubi [A]  time = 0.69981, antiderivative size = 551, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ -\frac{72 x}{7 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )}+\frac{1}{21} (3 x+2) \left (27 x^2+4\right )^{2/3}+\frac{5}{21} \left (27 x^2+4\right )^{2/3}-\frac{8\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt{\frac{\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt{3}\right )}{21 \sqrt [4]{3} \sqrt{-\frac{2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}+\frac{4 \sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt{\frac{\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt{3}\right )}{7\ 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x} \]

Warning: Unable to verify antiderivative.

[In]  Int[(2 + 3*x)^2/(4 + 27*x^2)^(1/3),x]

[Out]

(5*(4 + 27*x^2)^(2/3))/21 + ((2 + 3*x)*(4 + 27*x^2)^(2/3))/21 - (72*x)/(7*(2^(2/
3)*(1 - Sqrt[3]) - (4 + 27*x^2)^(1/3))) + (4*2^(1/3)*Sqrt[2 + Sqrt[3]]*(2^(2/3)
- (4 + 27*x^2)^(1/3))*Sqrt[(2*2^(1/3) + 2^(2/3)*(4 + 27*x^2)^(1/3) + (4 + 27*x^2
)^(2/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 + 27*x^2)^(1/3))^2]*EllipticE[ArcSin[(2^(2/
3)*(1 + Sqrt[3]) - (4 + 27*x^2)^(1/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 + 27*x^2)^(1/
3))], -7 + 4*Sqrt[3]])/(7*3^(3/4)*x*Sqrt[-((2^(2/3) - (4 + 27*x^2)^(1/3))/(2^(2/
3)*(1 - Sqrt[3]) - (4 + 27*x^2)^(1/3))^2)]) - (8*2^(5/6)*(2^(2/3) - (4 + 27*x^2)
^(1/3))*Sqrt[(2*2^(1/3) + 2^(2/3)*(4 + 27*x^2)^(1/3) + (4 + 27*x^2)^(2/3))/(2^(2
/3)*(1 - Sqrt[3]) - (4 + 27*x^2)^(1/3))^2]*EllipticF[ArcSin[(2^(2/3)*(1 + Sqrt[3
]) - (4 + 27*x^2)^(1/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 + 27*x^2)^(1/3))], -7 + 4*S
qrt[3]])/(21*3^(1/4)*x*Sqrt[-((2^(2/3) - (4 + 27*x^2)^(1/3))/(2^(2/3)*(1 - Sqrt[
3]) - (4 + 27*x^2)^(1/3))^2)])

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Rubi in Sympy [A]  time = 24.4984, size = 478, normalized size = 0.87 \[ - \frac{72 \sqrt [3]{2} x}{7 \left (- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 2 \sqrt{3} + 2\right )} + \frac{\left (3 x + 2\right ) \left (27 x^{2} + 4\right )^{\frac{2}{3}}}{21} + \frac{5 \left (27 x^{2} + 4\right )^{\frac{2}{3}}}{21} + \frac{2 \cdot 2^{\frac{2}{3}} \sqrt [4]{3} \sqrt{\frac{2^{\frac{2}{3}} \left (27 x^{2} + 4\right )^{\frac{2}{3}} + 2 \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} + 4}{\left (- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- 2 \sqrt [3]{27 x^{2} + 4} + 2 \cdot 2^{\frac{2}{3}}\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} + 2 + 2 \sqrt{3}}{- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 2 \sqrt{3} + 2} \right )}\middle | -7 + 4 \sqrt{3}\right )}{21 x \sqrt{\frac{2 \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 4}{\left (- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}}} - \frac{8 \sqrt [6]{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{2^{\frac{2}{3}} \left (27 x^{2} + 4\right )^{\frac{2}{3}} + 2 \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} + 4}{\left (- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}} \left (- 2 \sqrt [3]{27 x^{2} + 4} + 2 \cdot 2^{\frac{2}{3}}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} + 2 + 2 \sqrt{3}}{- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 2 \sqrt{3} + 2} \right )}\middle | -7 + 4 \sqrt{3}\right )}{63 x \sqrt{\frac{2 \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 4}{\left (- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**2/(27*x**2+4)**(1/3),x)

[Out]

-72*2**(1/3)*x/(7*(-2**(1/3)*(27*x**2 + 4)**(1/3) - 2*sqrt(3) + 2)) + (3*x + 2)*
(27*x**2 + 4)**(2/3)/21 + 5*(27*x**2 + 4)**(2/3)/21 + 2*2**(2/3)*3**(1/4)*sqrt((
2**(2/3)*(27*x**2 + 4)**(2/3) + 2*2**(1/3)*(27*x**2 + 4)**(1/3) + 4)/(-2**(1/3)*
(27*x**2 + 4)**(1/3) - 2*sqrt(3) + 2)**2)*sqrt(sqrt(3) + 2)*(-2*(27*x**2 + 4)**(
1/3) + 2*2**(2/3))*elliptic_e(asin((-2**(1/3)*(27*x**2 + 4)**(1/3) + 2 + 2*sqrt(
3))/(-2**(1/3)*(27*x**2 + 4)**(1/3) - 2*sqrt(3) + 2)), -7 + 4*sqrt(3))/(21*x*sqr
t((2*2**(1/3)*(27*x**2 + 4)**(1/3) - 4)/(-2**(1/3)*(27*x**2 + 4)**(1/3) - 2*sqrt
(3) + 2)**2)) - 8*2**(1/6)*3**(3/4)*sqrt((2**(2/3)*(27*x**2 + 4)**(2/3) + 2*2**(
1/3)*(27*x**2 + 4)**(1/3) + 4)/(-2**(1/3)*(27*x**2 + 4)**(1/3) - 2*sqrt(3) + 2)*
*2)*(-2*(27*x**2 + 4)**(1/3) + 2*2**(2/3))*elliptic_f(asin((-2**(1/3)*(27*x**2 +
4)**(1/3) + 2 + 2*sqrt(3))/(-2**(1/3)*(27*x**2 + 4)**(1/3) - 2*sqrt(3) + 2)), -
7 + 4*sqrt(3))/(63*x*sqrt((2*2**(1/3)*(27*x**2 + 4)**(1/3) - 4)/(-2**(1/3)*(27*x
**2 + 4)**(1/3) - 2*sqrt(3) + 2)**2))

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Mathematica [C]  time = 0.14154, size = 92, normalized size = 0.17 \[ \frac{6 \sqrt [3]{6} \sqrt [3]{2 \sqrt{3}-9 i x} \left (3 \sqrt{3} x-2 i\right ) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{3}{4} i \sqrt{3} x+\frac{1}{2}\right )+81 x^3+189 x^2+12 x+28}{21 \sqrt [3]{27 x^2+4}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(2 + 3*x)^2/(4 + 27*x^2)^(1/3),x]

[Out]

(28 + 12*x + 189*x^2 + 81*x^3 + 6*6^(1/3)*(2*Sqrt[3] - (9*I)*x)^(1/3)*(-2*I + 3*
Sqrt[3]*x)*Hypergeometric2F1[1/3, 2/3, 5/3, 1/2 + ((3*I)/4)*Sqrt[3]*x])/(21*(4 +
27*x^2)^(1/3))

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Maple [C]  time = 0.034, size = 35, normalized size = 0.1 \[{\frac{7+3\,x}{21} \left ( 27\,{x}^{2}+4 \right ) ^{{\frac{2}{3}}}}+{\frac{12\,\sqrt [3]{2}x}{7}{\mbox{$_2$F$_1$}({\frac{1}{3}},{\frac{1}{2}};\,{\frac{3}{2}};\,-{\frac{27\,{x}^{2}}{4}})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^2/(27*x^2+4)^(1/3),x)

[Out]

1/21*(7+3*x)*(27*x^2+4)^(2/3)+12/7*2^(1/3)*x*hypergeom([1/3,1/2],[3/2],-27/4*x^2
)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{2}}{{\left (27 \, x^{2} + 4\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^2/(27*x^2 + 4)^(1/3),x, algorithm="maxima")

[Out]

integrate((3*x + 2)^2/(27*x^2 + 4)^(1/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{9 \, x^{2} + 12 \, x + 4}{{\left (27 \, x^{2} + 4\right )}^{\frac{1}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^2/(27*x^2 + 4)^(1/3),x, algorithm="fricas")

[Out]

integral((9*x^2 + 12*x + 4)/(27*x^2 + 4)^(1/3), x)

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Sympy [A]  time = 6.7543, size = 68, normalized size = 0.12 \[ \frac{3 \sqrt [3]{2} x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{27 x^{2} e^{i \pi }}{4}} \right )}}{2} + 2 \sqrt [3]{2} x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{27 x^{2} e^{i \pi }}{4}} \right )} + \frac{\left (27 x^{2} + 4\right )^{\frac{2}{3}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+3*x)**2/(27*x**2+4)**(1/3),x)

[Out]

3*2**(1/3)*x**3*hyper((1/3, 3/2), (5/2,), 27*x**2*exp_polar(I*pi)/4)/2 + 2*2**(1
/3)*x*hyper((1/3, 1/2), (3/2,), 27*x**2*exp_polar(I*pi)/4) + (27*x**2 + 4)**(2/3
)/3

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{2}}{{\left (27 \, x^{2} + 4\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^2/(27*x^2 + 4)^(1/3),x, algorithm="giac")

[Out]

integrate((3*x + 2)^2/(27*x^2 + 4)^(1/3), x)