3.2494 \(\int \frac{2+3 x}{\sqrt [3]{28+54 x+27 x^2}} \, dx\)

Optimal. Leaf size=560 \[ \frac{1}{12} \left (27 x^2+54 x+28\right )^{2/3}+\frac{\left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} F\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt{3}\right )}{9\ 3^{3/4} \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}-\frac{\sqrt{2+\sqrt{3}} \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} E\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt{3}\right )}{18 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}+\frac{18 (x+1)}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}} \]

[Out]

(28 + 54*x + 27*x^2)^(2/3)/12 + (18*(1 + x))/(6*(1 - Sqrt[3]) - 2^(1/3)*(108 + (
54 + 54*x)^2)^(1/3)) - (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) + (28 + 54*x + 27*x^2)^(2/3))/(6*(1 - S
qrt[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]])/(18*Sqrt[2]*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)]) + ((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 + (5
4 + 54*x)^2)^(1/3))/(6*(1 - Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))], -7
+ 4*Sqrt[3]])/(9*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 = 0.753029, antiderivative size = 560, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{1}{12} \left (27 x^2+54 x+28\right )^{2/3}+\frac{\left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} F\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt{3}\right )}{9\ 3^{3/4} \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}-\frac{\sqrt{2+\sqrt{3}} \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} E\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt{3}\right )}{18 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}+\frac{18 (x+1)}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}} \]

Antiderivative was successfully verified.

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

[Out]

(28 + 54*x + 27*x^2)^(2/3)/12 + (18*(1 + x))/(6*(1 - Sqrt[3]) - 2^(1/3)*(108 + (
54 + 54*x)^2)^(1/3)) - (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) + (28 + 54*x + 27*x^2)^(2/3))/(6*(1 - S
qrt[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]])/(18*Sqrt[2]*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)]) + ((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 + (5
4 + 54*x)^2)^(1/3))/(6*(1 - Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))], -7
+ 4*Sqrt[3]])/(9*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 = 18.8148, size = 372, normalized size = 0.66 \[ \frac{54 x + 54}{18 \left (- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} - \sqrt{3} + 1\right )} + \frac{\left (27 x^{2} + 54 x + 28\right )^{\frac{2}{3}}}{12} - \frac{\sqrt [4]{3} \sqrt{\frac{\left (27 \left (x + 1\right )^{2} + 1\right )^{\frac{2}{3}} + \sqrt [3]{27 \left (x + 1\right )^{2} + 1} + 1}{\left (- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} + 1\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} + 1 + \sqrt{3}}{- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{18 \sqrt{\frac{\sqrt [3]{27 \left (x + 1\right )^{2} + 1} - 1}{\left (- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} - \sqrt{3} + 1\right )^{2}}} \left (x + 1\right )} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{\left (27 \left (x + 1\right )^{2} + 1\right )^{\frac{2}{3}} + \sqrt [3]{27 \left (x + 1\right )^{2} + 1} + 1}{\left (- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} + 1\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} + 1 + \sqrt{3}}{- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{27 \sqrt{\frac{\sqrt [3]{27 \left (x + 1\right )^{2} + 1} - 1}{\left (- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} - \sqrt{3} + 1\right )^{2}}} \left (x + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(54*x + 54)/(18*(-(27*(x + 1)**2 + 1)**(1/3) - sqrt(3) + 1)) + (27*x**2 + 54*x +
28)**(2/3)/12 - 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))/(18*s
qrt(((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))/(27*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 = 0.155018, size = 109, normalized size = 0.19 \[ \frac{2^{2/3} \sqrt [3]{3} \sqrt [3]{-9 i x+\sqrt{3}-9 i} \left (-3 \sqrt{3} x-3 \sqrt{3}+i\right ) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{9 i x+\sqrt{3}+9 i}{2 \sqrt{3}}\right )+27 x^2+54 x+28}{12 \sqrt [3]{27 x^2+54 x+28}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(28 + 54*x + 27*x^2 + 2^(2/3)*3^(1/3)*(-9*I + Sqrt[3] - (9*I)*x)^(1/3)*(I - 3*Sq
rt[3] - 3*Sqrt[3]*x)*Hypergeometric2F1[1/3, 2/3, 5/3, (9*I + Sqrt[3] + (9*I)*x)/
(2*Sqrt[3])])/(12*(28 + 54*x + 27*x^2)^(1/3))

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Maple [F]  time = 0.092, size = 0, normalized size = 0. \[ \int{(2+3\,x){\frac{1}{\sqrt [3]{27\,{x}^{2}+54\,x+28}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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