3.2496 \(\int \frac{1}{(2+3 x)^2 \sqrt [3]{28+54 x+27 x^2}} \, dx\)

Optimal. Leaf size=671 \[ -\frac{\left (27 x^2+54 x+28\right )^{2/3}}{12 (3 x+2)}-\frac{\log \left (27 \sqrt [3]{2} \sqrt [3]{27 x^2+54 x+28}-81 x-108\right )}{12\ 2^{2/3}}+\frac{\tan ^{-1}\left (\frac{2^{2/3} (3 x+4)}{\sqrt{3} \sqrt [3]{27 x^2+54 x+28}}+\frac{1}{\sqrt{3}}\right )}{6\ 2^{2/3} \sqrt{3}}-\frac{\left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} F\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt{3}\right )}{36\ 3^{3/4} \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}+\frac{\sqrt{2+\sqrt{3}} \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} E\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt{3}\right )}{72 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}-\frac{9 (x+1)}{2 \left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )}+\frac{\log (3 x+2)}{12\ 2^{2/3}} \]

[Out]

-(28 + 54*x + 27*x^2)^(2/3)/(12*(2 + 3*x)) - (9*(1 + x))/(2*(6*(1 - Sqrt[3]) - 2
^(1/3)*(108 + (54 + 54*x)^2)^(1/3))) + ArcTan[1/Sqrt[3] + (2^(2/3)*(4 + 3*x))/(S
qrt[3]*(28 + 54*x + 27*x^2)^(1/3))]/(6*2^(2/3)*Sqrt[3]) + (Sqrt[2 + Sqrt[3]]*(6
- 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))*Sqrt[(1 + (28 + 54*x + 27*x^2)^(1/3) + (2
8 + 54*x + 27*x^2)^(2/3))/(6*(1 - Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3)
)^2]*EllipticE[ArcSin[(6*(1 + Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))/(6
*(1 - Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))], -7 + 4*Sqrt[3]])/(72*Sqr
t[2]*3^(1/4)*(1 + x)*Sqrt[-((6 - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))/(6*(1 - Sq
rt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))^2)]) - ((6 - 2^(1/3)*(108 + (54 +
54*x)^2)^(1/3))*Sqrt[(1 + (28 + 54*x + 27*x^2)^(1/3) + (28 + 54*x + 27*x^2)^(2/3
))/(6*(1 - Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))^2]*EllipticF[ArcSin[(
6*(1 + Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))/(6*(1 - Sqrt[3]) - 2^(1/3
)*(108 + (54 + 54*x)^2)^(1/3))], -7 + 4*Sqrt[3]])/(36*3^(3/4)*(1 + x)*Sqrt[-((6
- 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))/(6*(1 - Sqrt[3]) - 2^(1/3)*(108 + (54 + 5
4*x)^2)^(1/3))^2)]) + Log[2 + 3*x]/(12*2^(2/3)) - Log[-108 - 81*x + 27*2^(1/3)*(
28 + 54*x + 27*x^2)^(1/3)]/(12*2^(2/3))

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Rubi [A]  time = 1.12032, antiderivative size = 671, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409 \[ -\frac{\left (27 x^2+54 x+28\right )^{2/3}}{12 (3 x+2)}-\frac{\log \left (27 \sqrt [3]{2} \sqrt [3]{27 x^2+54 x+28}-81 x-108\right )}{12\ 2^{2/3}}+\frac{\tan ^{-1}\left (\frac{2^{2/3} (3 x+4)}{\sqrt{3} \sqrt [3]{27 x^2+54 x+28}}+\frac{1}{\sqrt{3}}\right )}{6\ 2^{2/3} \sqrt{3}}-\frac{\left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} F\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt{3}\right )}{36\ 3^{3/4} \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}+\frac{\sqrt{2+\sqrt{3}} \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} E\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt{3}\right )}{72 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}-\frac{9 (x+1)}{2 \left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )}+\frac{\log (3 x+2)}{12\ 2^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]  Int[1/((2 + 3*x)^2*(28 + 54*x + 27*x^2)^(1/3)),x]

[Out]

-(28 + 54*x + 27*x^2)^(2/3)/(12*(2 + 3*x)) - (9*(1 + x))/(2*(6*(1 - Sqrt[3]) - 2
^(1/3)*(108 + (54 + 54*x)^2)^(1/3))) + ArcTan[1/Sqrt[3] + (2^(2/3)*(4 + 3*x))/(S
qrt[3]*(28 + 54*x + 27*x^2)^(1/3))]/(6*2^(2/3)*Sqrt[3]) + (Sqrt[2 + Sqrt[3]]*(6
- 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))*Sqrt[(1 + (28 + 54*x + 27*x^2)^(1/3) + (2
8 + 54*x + 27*x^2)^(2/3))/(6*(1 - Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3)
)^2]*EllipticE[ArcSin[(6*(1 + Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))/(6
*(1 - Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))], -7 + 4*Sqrt[3]])/(72*Sqr
t[2]*3^(1/4)*(1 + x)*Sqrt[-((6 - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))/(6*(1 - Sq
rt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))^2)]) - ((6 - 2^(1/3)*(108 + (54 +
54*x)^2)^(1/3))*Sqrt[(1 + (28 + 54*x + 27*x^2)^(1/3) + (28 + 54*x + 27*x^2)^(2/3
))/(6*(1 - Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))^2]*EllipticF[ArcSin[(
6*(1 + Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))/(6*(1 - Sqrt[3]) - 2^(1/3
)*(108 + (54 + 54*x)^2)^(1/3))], -7 + 4*Sqrt[3]])/(36*3^(3/4)*(1 + x)*Sqrt[-((6
- 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))/(6*(1 - Sqrt[3]) - 2^(1/3)*(108 + (54 + 5
4*x)^2)^(1/3))^2)]) + Log[2 + 3*x]/(12*2^(2/3)) - Log[-108 - 81*x + 27*2^(1/3)*(
28 + 54*x + 27*x^2)^(1/3)]/(12*2^(2/3))

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Rubi in Sympy [A]  time = 40.3514, size = 476, normalized size = 0.71 \[ - \frac{54 x + 54}{72 \left (- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} - \sqrt{3} + 1\right )} + \frac{\sqrt [3]{2} \log{\left (3 x + 2 \right )}}{24} - \frac{\sqrt [3]{2} \log{\left (- 81 x + 27 \sqrt [3]{2} \sqrt [3]{27 x^{2} + 54 x + 28} - 108 \right )}}{24} + \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (- \frac{2^{\frac{2}{3}} \sqrt{3} \left (- 81 x - 108\right )}{81 \sqrt [3]{27 x^{2} + 54 x + 28}} + \frac{\sqrt{3}}{3} \right )}}{36} - \frac{\left (27 x^{2} + 54 x + 28\right )^{\frac{2}{3}}}{12 \left (3 x + 2\right )} + \frac{\sqrt [4]{3} \sqrt{\frac{\left (27 \left (x + 1\right )^{2} + 1\right )^{\frac{2}{3}} + \sqrt [3]{27 \left (x + 1\right )^{2} + 1} + 1}{\left (- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} + 1\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} + 1 + \sqrt{3}}{- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{72 \sqrt{\frac{\sqrt [3]{27 \left (x + 1\right )^{2} + 1} - 1}{\left (- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} - \sqrt{3} + 1\right )^{2}}} \left (x + 1\right )} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{\left (27 \left (x + 1\right )^{2} + 1\right )^{\frac{2}{3}} + \sqrt [3]{27 \left (x + 1\right )^{2} + 1} + 1}{\left (- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} + 1\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} + 1 + \sqrt{3}}{- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{108 \sqrt{\frac{\sqrt [3]{27 \left (x + 1\right )^{2} + 1} - 1}{\left (- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} - \sqrt{3} + 1\right )^{2}}} \left (x + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2+3*x)**2/(27*x**2+54*x+28)**(1/3),x)

[Out]

-(54*x + 54)/(72*(-(27*(x + 1)**2 + 1)**(1/3) - sqrt(3) + 1)) + 2**(1/3)*log(3*x
+ 2)/24 - 2**(1/3)*log(-81*x + 27*2**(1/3)*(27*x**2 + 54*x + 28)**(1/3) - 108)/
24 + 2**(1/3)*sqrt(3)*atan(-2**(2/3)*sqrt(3)*(-81*x - 108)/(81*(27*x**2 + 54*x +
28)**(1/3)) + sqrt(3)/3)/36 - (27*x**2 + 54*x + 28)**(2/3)/(12*(3*x + 2)) + 3**
(1/4)*sqrt(((27*(x + 1)**2 + 1)**(2/3) + (27*(x + 1)**2 + 1)**(1/3) + 1)/(-(27*(
x + 1)**2 + 1)**(1/3) - sqrt(3) + 1)**2)*sqrt(sqrt(3) + 2)*(-(27*(x + 1)**2 + 1)
**(1/3) + 1)*elliptic_e(asin((-(27*(x + 1)**2 + 1)**(1/3) + 1 + sqrt(3))/(-(27*(
x + 1)**2 + 1)**(1/3) - sqrt(3) + 1)), -7 + 4*sqrt(3))/(72*sqrt(((27*(x + 1)**2
+ 1)**(1/3) - 1)/(-(27*(x + 1)**2 + 1)**(1/3) - sqrt(3) + 1)**2)*(x + 1)) - sqrt
(2)*3**(3/4)*sqrt(((27*(x + 1)**2 + 1)**(2/3) + (27*(x + 1)**2 + 1)**(1/3) + 1)/
(-(27*(x + 1)**2 + 1)**(1/3) - sqrt(3) + 1)**2)*(-(27*(x + 1)**2 + 1)**(1/3) + 1
)*elliptic_f(asin((-(27*(x + 1)**2 + 1)**(1/3) + 1 + sqrt(3))/(-(27*(x + 1)**2 +
1)**(1/3) - sqrt(3) + 1)), -7 + 4*sqrt(3))/(108*sqrt(((27*(x + 1)**2 + 1)**(1/3
) - 1)/(-(27*(x + 1)**2 + 1)**(1/3) - sqrt(3) + 1)**2)*(x + 1))

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Mathematica [C]  time = 0.56723, size = 405, normalized size = 0.6 \[ \frac{\frac{540 (3 x+2) \left (9 x-i \sqrt{3}+9\right ) \left (9 x+i \sqrt{3}+9\right ) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};-\frac{3+i \sqrt{3}}{9 x+6},\frac{-3+i \sqrt{3}}{9 x+6}\right )}{15 (3 x+2) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};-\frac{3+i \sqrt{3}}{9 x+6},\frac{-3+i \sqrt{3}}{9 x+6}\right )+i \left (\sqrt{3}+3 i\right ) F_1\left (\frac{5}{3};\frac{1}{3},\frac{4}{3};\frac{8}{3};-\frac{3+i \sqrt{3}}{9 x+6},\frac{-3+i \sqrt{3}}{9 x+6}\right )+\left (-3-i \sqrt{3}\right ) F_1\left (\frac{5}{3};\frac{4}{3},\frac{1}{3};\frac{8}{3};-\frac{3+i \sqrt{3}}{9 x+6},\frac{-3+i \sqrt{3}}{9 x+6}\right )}+3\ 2^{2/3} 3^{5/6} \sqrt [3]{-9 i x+\sqrt{3}-9 i} \left (9 x-i \sqrt{3}+9\right ) \left (27 x^2+54 x+28\right ) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{9 i x+\sqrt{3}+9 i}{2 \sqrt{3}}\right )-\frac{36 \left (27 x^2+54 x+28\right )^2}{3 x+2}}{432 \left (27 x^2+54 x+28\right )^{4/3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((2 + 3*x)^2*(28 + 54*x + 27*x^2)^(1/3)),x]

[Out]

((-36*(28 + 54*x + 27*x^2)^2)/(2 + 3*x) + (540*(2 + 3*x)*(9 - I*Sqrt[3] + 9*x)*(
9 + I*Sqrt[3] + 9*x)*AppellF1[2/3, 1/3, 1/3, 5/3, -((3 + I*Sqrt[3])/(6 + 9*x)),
(-3 + I*Sqrt[3])/(6 + 9*x)])/(15*(2 + 3*x)*AppellF1[2/3, 1/3, 1/3, 5/3, -((3 + I
*Sqrt[3])/(6 + 9*x)), (-3 + I*Sqrt[3])/(6 + 9*x)] + I*(3*I + Sqrt[3])*AppellF1[5
/3, 1/3, 4/3, 8/3, -((3 + I*Sqrt[3])/(6 + 9*x)), (-3 + I*Sqrt[3])/(6 + 9*x)] + (
-3 - I*Sqrt[3])*AppellF1[5/3, 4/3, 1/3, 8/3, -((3 + I*Sqrt[3])/(6 + 9*x)), (-3 +
I*Sqrt[3])/(6 + 9*x)]) + 3*2^(2/3)*3^(5/6)*(-9*I + Sqrt[3] - (9*I)*x)^(1/3)*(9
- I*Sqrt[3] + 9*x)*(28 + 54*x + 27*x^2)*Hypergeometric2F1[1/3, 2/3, 5/3, (9*I +
Sqrt[3] + (9*I)*x)/(2*Sqrt[3])])/(432*(28 + 54*x + 27*x^2)^(4/3))

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Maple [F]  time = 0.116, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( 2+3\,x \right ) ^{2}}{\frac{1}{\sqrt [3]{27\,{x}^{2}+54\,x+28}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2+3*x)^2/(27*x^2+54*x+28)^(1/3),x)

[Out]

int(1/(2+3*x)^2/(27*x^2+54*x+28)^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac{1}{3}}{\left (3 \, x + 2\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((27*x^2 + 54*x + 28)^(1/3)*(3*x + 2)^2),x, algorithm="maxima")

[Out]

integrate(1/((27*x^2 + 54*x + 28)^(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 + 54*x + 28)^(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. \[ \int \frac{1}{\left (3 x + 2\right )^{2} \sqrt [3]{27 x^{2} + 54 x + 28}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2+3*x)**2/(27*x**2+54*x+28)**(1/3),x)

[Out]

Integral(1/((3*x + 2)**2*(27*x**2 + 54*x + 28)**(1/3)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac{1}{3}}{\left (3 \, x + 2\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((27*x^2 + 54*x + 28)^(1/3)*(3*x + 2)^2),x, algorithm="giac")

[Out]

integrate(1/((27*x^2 + 54*x + 28)^(1/3)*(3*x + 2)^2), x)