3.693 \(\int \frac{1}{(2+3 x)^2 \sqrt [3]{4+27 x^2}} \, dx\)

Optimal. Leaf size=634 \[ -\frac{3 x}{16 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )}-\frac{\left (27 x^2+4\right )^{2/3}}{48 (3 x+2)}+\frac{\log \left (-27\ 2^{2/3} \sqrt [3]{27 x^2+4}-81 x+54\right )}{48 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2} (2-3 x)}{\sqrt{3} \sqrt [3]{27 x^2+4}}+\frac{1}{\sqrt{3}}\right )}{24 \sqrt [3]{2} \sqrt{3}}-\frac{\left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt{\frac{\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt{3}\right )}{72 \sqrt [6]{2} \sqrt [4]{3} \sqrt{-\frac{2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}+\frac{\sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt{\frac{\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt{3}\right )}{48\ 2^{2/3} 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}-\frac{\log (3 x+2)}{48 \sqrt [3]{2}} \]

[Out]

-(4 + 27*x^2)^(2/3)/(48*(2 + 3*x)) - (3*x)/(16*(2^(2/3)*(1 - Sqrt[3]) - (4 + 27*
x^2)^(1/3))) - ArcTan[1/Sqrt[3] + (2^(1/3)*(2 - 3*x))/(Sqrt[3]*(4 + 27*x^2)^(1/3
))]/(24*2^(1/3)*Sqrt[3]) + (Sqrt[2 + Sqrt[3]]*(2^(2/3) - (4 + 27*x^2)^(1/3))*Sqr
t[(2*2^(1/3) + 2^(2/3)*(4 + 27*x^2)^(1/3) + (4 + 27*x^2)^(2/3))/(2^(2/3)*(1 - Sq
rt[3]) - (4 + 27*x^2)^(1/3))^2]*EllipticE[ArcSin[(2^(2/3)*(1 + Sqrt[3]) - (4 + 2
7*x^2)^(1/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 + 27*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(4
8*2^(2/3)*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^(2/3) - (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 + 2
7*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*Sqrt[3]])/(72*2^(1/6)*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)]) - Log[2 + 3*x]/(48*2^(1/3)) + Log[54 - 81*x - 27*2^(2/3)*(4 + 27*x^
2)^(1/3)]/(48*2^(1/3))

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Rubi [A]  time = 0.801756, antiderivative size = 634, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ -\frac{3 x}{16 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )}-\frac{\left (27 x^2+4\right )^{2/3}}{48 (3 x+2)}+\frac{\log \left (-27\ 2^{2/3} \sqrt [3]{27 x^2+4}-81 x+54\right )}{48 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2} (2-3 x)}{\sqrt{3} \sqrt [3]{27 x^2+4}}+\frac{1}{\sqrt{3}}\right )}{24 \sqrt [3]{2} \sqrt{3}}-\frac{\left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt{\frac{\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt{3}\right )}{72 \sqrt [6]{2} \sqrt [4]{3} \sqrt{-\frac{2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}+\frac{\sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt{\frac{\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt{3}\right )}{48\ 2^{2/3} 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}-\frac{\log (3 x+2)}{48 \sqrt [3]{2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

-(4 + 27*x^2)^(2/3)/(48*(2 + 3*x)) - (3*x)/(16*(2^(2/3)*(1 - Sqrt[3]) - (4 + 27*
x^2)^(1/3))) - ArcTan[1/Sqrt[3] + (2^(1/3)*(2 - 3*x))/(Sqrt[3]*(4 + 27*x^2)^(1/3
))]/(24*2^(1/3)*Sqrt[3]) + (Sqrt[2 + Sqrt[3]]*(2^(2/3) - (4 + 27*x^2)^(1/3))*Sqr
t[(2*2^(1/3) + 2^(2/3)*(4 + 27*x^2)^(1/3) + (4 + 27*x^2)^(2/3))/(2^(2/3)*(1 - Sq
rt[3]) - (4 + 27*x^2)^(1/3))^2]*EllipticE[ArcSin[(2^(2/3)*(1 + Sqrt[3]) - (4 + 2
7*x^2)^(1/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 + 27*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(4
8*2^(2/3)*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^(2/3) - (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 + 2
7*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*Sqrt[3]])/(72*2^(1/6)*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)]) - Log[2 + 3*x]/(48*2^(1/3)) + Log[54 - 81*x - 27*2^(2/3)*(4 + 27*x^
2)^(1/3)]/(48*2^(1/3))

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Rubi in Sympy [A]  time = 29.9399, size = 551, normalized size = 0.87 \[ - \frac{3 \sqrt [3]{2} x}{16 \left (- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 2 \sqrt{3} + 2\right )} - \frac{2^{\frac{2}{3}} \log{\left (3 x + 2 \right )}}{96} + \frac{2^{\frac{2}{3}} \log{\left (- 81 x - 27 \cdot 2^{\frac{2}{3}} \sqrt [3]{27 x^{2} + 4} + 54 \right )}}{96} - \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt [3]{2} \sqrt{3} \left (- 3 x + 2\right )}{3 \sqrt [3]{27 x^{2} + 4}} + \frac{\sqrt{3}}{3} \right )}}{144} - \frac{\left (27 x^{2} + 4\right )^{\frac{2}{3}}}{48 \left (3 x + 2\right )} + \frac{2^{\frac{2}{3}} \sqrt [4]{3} \sqrt{\frac{2^{\frac{2}{3}} \left (27 x^{2} + 4\right )^{\frac{2}{3}} + 2 \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} + 4}{\left (- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- 2 \sqrt [3]{27 x^{2} + 4} + 2 \cdot 2^{\frac{2}{3}}\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} + 2 + 2 \sqrt{3}}{- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 2 \sqrt{3} + 2} \right )}\middle | -7 + 4 \sqrt{3}\right )}{576 x \sqrt{\frac{2 \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 4}{\left (- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}}} - \frac{\sqrt [6]{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{2^{\frac{2}{3}} \left (27 x^{2} + 4\right )^{\frac{2}{3}} + 2 \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} + 4}{\left (- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}} \left (- 2 \sqrt [3]{27 x^{2} + 4} + 2 \cdot 2^{\frac{2}{3}}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} + 2 + 2 \sqrt{3}}{- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 2 \sqrt{3} + 2} \right )}\middle | -7 + 4 \sqrt{3}\right )}{432 x \sqrt{\frac{2 \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 4}{\left (- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*2**(1/3)*x/(16*(-2**(1/3)*(27*x**2 + 4)**(1/3) - 2*sqrt(3) + 2)) - 2**(2/3)*l
og(3*x + 2)/96 + 2**(2/3)*log(-81*x - 27*2**(2/3)*(27*x**2 + 4)**(1/3) + 54)/96
- 2**(2/3)*sqrt(3)*atan(2**(1/3)*sqrt(3)*(-3*x + 2)/(3*(27*x**2 + 4)**(1/3)) + s
qrt(3)/3)/144 - (27*x**2 + 4)**(2/3)/(48*(3*x + 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)*(2
7*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))/(576*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**(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))/(432*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 = 0.447731, size = 376, normalized size = 0.59 \[ \frac{-\frac{1080 (3 x+2) \left (9 x-2 i \sqrt{3}\right ) \left (9 x+2 i \sqrt{3}\right ) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{6-2 i \sqrt{3}}{9 x+6},\frac{6+2 i \sqrt{3}}{9 x+6}\right )}{15 (3 x+2) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{6-2 i \sqrt{3}}{9 x+6},\frac{6+2 i \sqrt{3}}{9 x+6}\right )+\left (6+2 i \sqrt{3}\right ) F_1\left (\frac{5}{3};\frac{1}{3},\frac{4}{3};\frac{8}{3};\frac{6-2 i \sqrt{3}}{9 x+6},\frac{6+2 i \sqrt{3}}{9 x+6}\right )+2 \left (3-i \sqrt{3}\right ) F_1\left (\frac{5}{3};\frac{4}{3},\frac{1}{3};\frac{8}{3};\frac{6-2 i \sqrt{3}}{9 x+6},\frac{6+2 i \sqrt{3}}{9 x+6}\right )}+3\ 3^{5/6} \sqrt [3]{4 \sqrt{3}-18 i x} \left (9 x-2 i \sqrt{3}\right ) \left (27 x^2+4\right ) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{3}{4} i \sqrt{3} x+\frac{1}{2}\right )-\frac{36 \left (27 x^2+4\right )^2}{3 x+2}}{1728 \left (27 x^2+4\right )^{4/3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((-36*(4 + 27*x^2)^2)/(2 + 3*x) - (1080*(2 + 3*x)*((-2*I)*Sqrt[3] + 9*x)*((2*I)*
Sqrt[3] + 9*x)*AppellF1[2/3, 1/3, 1/3, 5/3, (6 - (2*I)*Sqrt[3])/(6 + 9*x), (6 +
(2*I)*Sqrt[3])/(6 + 9*x)])/(15*(2 + 3*x)*AppellF1[2/3, 1/3, 1/3, 5/3, (6 - (2*I)
*Sqrt[3])/(6 + 9*x), (6 + (2*I)*Sqrt[3])/(6 + 9*x)] + (6 + (2*I)*Sqrt[3])*Appell
F1[5/3, 1/3, 4/3, 8/3, (6 - (2*I)*Sqrt[3])/(6 + 9*x), (6 + (2*I)*Sqrt[3])/(6 + 9
*x)] + 2*(3 - I*Sqrt[3])*AppellF1[5/3, 4/3, 1/3, 8/3, (6 - (2*I)*Sqrt[3])/(6 + 9
*x), (6 + (2*I)*Sqrt[3])/(6 + 9*x)]) + 3*3^(5/6)*(4*Sqrt[3] - (18*I)*x)^(1/3)*((
-2*I)*Sqrt[3] + 9*x)*(4 + 27*x^2)*Hypergeometric2F1[1/3, 2/3, 5/3, 1/2 + ((3*I)/
4)*Sqrt[3]*x])/(1728*(4 + 27*x^2)^(4/3))

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( 2+3\,x \right ) ^{2}}{\frac{1}{\sqrt [3]{27\,{x}^{2}+4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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 + 4)^(1/3)*(3*x + 2)^2),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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