3.2492 \(\int \frac{(2+3 x)^3}{\sqrt [3]{28+54 x+27 x^2}} \, dx\)

Optimal. Leaf size=589 \[ \frac{1}{30} \left (27 x^2+54 x+28\right )^{2/3} (3 x+2)^2-\frac{1}{35} (8 x+1) \left (27 x^2+54 x+28\right )^{2/3}+\frac{4 \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} F\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt{3}\right )}{63\ 3^{3/4} (x+1) \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}}}-\frac{\sqrt{2 \left (2+\sqrt{3}\right )} \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} E\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt{3}\right )}{63 \sqrt [4]{3} (x+1) \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}}}+\frac{72 (x+1)}{7 \left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )} \]

[Out]

((2 + 3*x)^2*(28 + 54*x + 27*x^2)^(2/3))/30 - ((1 + 8*x)*(28 + 54*x + 27*x^2)^(2
/3))/35 + (72*(1 + x))/(7*(6*(1 - Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3)
)) - (Sqrt[2*(2 + Sqrt[3])]*(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]*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]])/(63*3^(1/4)*(1 + x)*Sqrt[-((6 - 2^(1/3)*(108 + (54 + 54*
x)^2)^(1/3))/(6*(1 - Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))^2)]) + (4*(
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]])/(63*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 + 54*x)^2)^(1/3))^2)])

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Rubi [A]  time = 1.03769, antiderivative size = 589, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ \frac{1}{30} \left (27 x^2+54 x+28\right )^{2/3} (3 x+2)^2-\frac{1}{35} (8 x+1) \left (27 x^2+54 x+28\right )^{2/3}+\frac{4 \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} F\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt{3}\right )}{63\ 3^{3/4} (x+1) \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}}}-\frac{\sqrt{2 \left (2+\sqrt{3}\right )} \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} E\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt{3}\right )}{63 \sqrt [4]{3} (x+1) \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}}}+\frac{72 (x+1)}{7 \left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )} \]

Antiderivative was successfully verified.

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

[Out]

((2 + 3*x)^2*(28 + 54*x + 27*x^2)^(2/3))/30 - ((1 + 8*x)*(28 + 54*x + 27*x^2)^(2
/3))/35 + (72*(1 + x))/(7*(6*(1 - Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3)
)) - (Sqrt[2*(2 + Sqrt[3])]*(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]*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]])/(63*3^(1/4)*(1 + x)*Sqrt[-((6 - 2^(1/3)*(108 + (54 + 54*
x)^2)^(1/3))/(6*(1 - Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))^2)]) + (4*(
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]])/(63*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 + 54*x)^2)^(1/3))^2)])

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Rubi in Sympy [A]  time = 31.2053, size = 403, normalized size = 0.68 \[ \frac{\left (3 x + 2\right )^{2} \left (27 x^{2} + 54 x + 28\right )^{\frac{2}{3}}}{30} + \frac{2 \left (54 x + 54\right )}{63 \left (- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} - \sqrt{3} + 1\right )} - \frac{\left (46656 x + 5832\right ) \left (27 x^{2} + 54 x + 28\right )^{\frac{2}{3}}}{204120} - \frac{2 \sqrt [4]{3} \sqrt{\frac{\left (27 \left (x + 1\right )^{2} + 1\right )^{\frac{2}{3}} + \sqrt [3]{27 \left (x + 1\right )^{2} + 1} + 1}{\left (- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} + 1\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} + 1 + \sqrt{3}}{- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{63 \sqrt{\frac{\sqrt [3]{27 \left (x + 1\right )^{2} + 1} - 1}{\left (- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} - \sqrt{3} + 1\right )^{2}}} \left (x + 1\right )} + \frac{4 \sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{\left (27 \left (x + 1\right )^{2} + 1\right )^{\frac{2}{3}} + \sqrt [3]{27 \left (x + 1\right )^{2} + 1} + 1}{\left (- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} + 1\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} + 1 + \sqrt{3}}{- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{189 \sqrt{\frac{\sqrt [3]{27 \left (x + 1\right )^{2} + 1} - 1}{\left (- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} - \sqrt{3} + 1\right )^{2}}} \left (x + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(3*x + 2)**2*(27*x**2 + 54*x + 28)**(2/3)/30 + 2*(54*x + 54)/(63*(-(27*(x + 1)**
2 + 1)**(1/3) - sqrt(3) + 1)) - (46656*x + 5832)*(27*x**2 + 54*x + 28)**(2/3)/20
4120 - 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))/(63*sqrt(((27
*(x + 1)**2 + 1)**(1/3) - 1)/(-(27*(x + 1)**2 + 1)**(1/3) - sqrt(3) + 1)**2)*(x
+ 1)) + 4*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))/(-(2
7*(x + 1)**2 + 1)**(1/3) - sqrt(3) + 1)), -7 + 4*sqrt(3))/(189*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 = 0.188089, size = 120, normalized size = 0.2 \[ \frac{-10\ 2^{2/3} \sqrt [3]{3} \sqrt [3]{-9 i x+\sqrt{3}-9 i} \left (3 \sqrt{3} x+3 \sqrt{3}-i\right ) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{9 i x+\sqrt{3}+9 i}{2 \sqrt{3}}\right )+1701 x^4+4374 x^3+4302 x^2+2196 x+616}{210 \sqrt [3]{27 x^2+54 x+28}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(616 + 2196*x + 4302*x^2 + 4374*x^3 + 1701*x^4 - 10*2^(2/3)*3^(1/3)*(-9*I + Sqrt
[3] - (9*I)*x)^(1/3)*(-I + 3*Sqrt[3] + 3*Sqrt[3]*x)*Hypergeometric2F1[1/3, 2/3,
5/3, (9*I + Sqrt[3] + (9*I)*x)/(2*Sqrt[3])])/(210*(28 + 54*x + 27*x^2)^(1/3))

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Maple [F]  time = 0.253, size = 0, normalized size = 0. \[ \int{ \left ( 2+3\,x \right ) ^{3}{\frac{1}{\sqrt [3]{27\,{x}^{2}+54\,x+28}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac{1}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((27*x^3 + 54*x^2 + 36*x + 8)/(27*x^2 + 54*x + 28)^(1/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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