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

Optimal. Leaf size=529 \[ -\frac{6 x}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}+\frac{1}{12} \left (27 x^2+4\right )^{2/3}-\frac{2\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt{\frac{\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt{3}\right )}{9 \sqrt [4]{3} \sqrt{-\frac{2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}+\frac{\sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt{\frac{\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt{3}\right )}{3\ 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x} \]

[Out]

(4 + 27*x^2)^(2/3)/12 - (6*x)/(2^(2/3)*(1 - Sqrt[3]) - (4 + 27*x^2)^(1/3)) + (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]])/(3*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)]) - (
2*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]*Elli
pticF[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]])/(9*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)])

_______________________________________________________________________________________

Rubi [A]  time = 0.607796, antiderivative size = 529, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ -\frac{6 x}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}+\frac{1}{12} \left (27 x^2+4\right )^{2/3}-\frac{2\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt{\frac{\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt{3}\right )}{9 \sqrt [4]{3} \sqrt{-\frac{2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}+\frac{\sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt{\frac{\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt{3}\right )}{3\ 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x} \]

Warning: Unable to verify antiderivative.

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

[Out]

(4 + 27*x^2)^(2/3)/12 - (6*x)/(2^(2/3)*(1 - Sqrt[3]) - (4 + 27*x^2)^(1/3)) + (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]])/(3*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)]) - (
2*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]*Elli
pticF[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]])/(9*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)])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 20.5892, size = 456, normalized size = 0.86 \[ - \frac{6 \sqrt [3]{2} x}{- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 2 \sqrt{3} + 2} + \frac{\left (27 x^{2} + 4\right )^{\frac{2}{3}}}{12} + \frac{2^{\frac{2}{3}} \sqrt [4]{3} \sqrt{\frac{2^{\frac{2}{3}} \left (27 x^{2} + 4\right )^{\frac{2}{3}} + 2 \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} + 4}{\left (- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- 2 \sqrt [3]{27 x^{2} + 4} + 2 \cdot 2^{\frac{2}{3}}\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} + 2 + 2 \sqrt{3}}{- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 2 \sqrt{3} + 2} \right )}\middle | -7 + 4 \sqrt{3}\right )}{18 x \sqrt{\frac{2 \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 4}{\left (- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}}} - \frac{2 \sqrt [6]{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{2^{\frac{2}{3}} \left (27 x^{2} + 4\right )^{\frac{2}{3}} + 2 \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} + 4}{\left (- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}} \left (- 2 \sqrt [3]{27 x^{2} + 4} + 2 \cdot 2^{\frac{2}{3}}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} + 2 + 2 \sqrt{3}}{- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 2 \sqrt{3} + 2} \right )}\middle | -7 + 4 \sqrt{3}\right )}{27 x \sqrt{\frac{2 \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 4}{\left (- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6*2**(1/3)*x/(-2**(1/3)*(27*x**2 + 4)**(1/3) - 2*sqrt(3) + 2) + (27*x**2 + 4)**
(2/3)/12 + 2**(2/3)*3**(1/4)*sqrt((2**(2/3)*(27*x**2 + 4)**(2/3) + 2*2**(1/3)*(2
7*x**2 + 4)**(1/3) + 4)/(-2**(1/3)*(27*x**2 + 4)**(1/3) - 2*sqrt(3) + 2)**2)*sqr
t(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))/(18*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)) - 2*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))*el
liptic_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))/(27*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))

_______________________________________________________________________________________

Mathematica [C]  time = 0.0270018, size = 40, normalized size = 0.08 \[ \sqrt [3]{2} x \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};-\frac{27 x^2}{4}\right )+\frac{1}{12} \left (27 x^2+4\right )^{2/3} \]

Antiderivative was successfully verified.

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

[Out]

(4 + 27*x^2)^(2/3)/12 + 2^(1/3)*x*Hypergeometric2F1[1/3, 1/2, 3/2, (-27*x^2)/4]

_______________________________________________________________________________________

Maple [C]  time = 0.032, size = 29, normalized size = 0.1 \[{\frac{1}{12} \left ( 27\,{x}^{2}+4 \right ) ^{{\frac{2}{3}}}}+\sqrt [3]{2}x{\mbox{$_2$F$_1$}({\frac{1}{3}},{\frac{1}{2}};\,{\frac{3}{2}};\,-{\frac{27\,{x}^{2}}{4}})} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)/(27*x^2+4)^(1/3),x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 4.47535, size = 36, normalized size = 0.07 \[ \sqrt [3]{2} x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{27 x^{2} e^{i \pi }}{4}} \right )} + \frac{\left (27 x^{2} + 4\right )^{\frac{2}{3}}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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)/12

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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