3.2493 \(\int \frac{(2+3 x)^2}{\sqrt [3]{28+54 x+27 x^2}} \, dx\)

Optimal. Leaf size=585 \[ \frac{1}{21} (3 x+2) \left (27 x^2+54 x+28\right )^{2/3}-\frac{5}{42} \left (27 x^2+54 x+28\right )^{2/3}-\frac{2 \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} F\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt{3}\right )}{21\ 3^{3/4} \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}+\frac{\sqrt{2+\sqrt{3}} \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} E\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt{3}\right )}{21 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}-\frac{108 (x+1)}{7 \left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )} \]

[Out]

(-5*(28 + 54*x + 27*x^2)^(2/3))/42 + ((2 + 3*x)*(28 + 54*x + 27*x^2)^(2/3))/21 -
(108*(1 + x))/(7*(6*(1 - Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))) + (Sq
rt[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]])/(21*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)]) - (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]])/(21*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)])

_______________________________________________________________________________________

Rubi [A]  time = 0.940337, antiderivative size = 585, 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 \[ \frac{1}{21} (3 x+2) \left (27 x^2+54 x+28\right )^{2/3}-\frac{5}{42} \left (27 x^2+54 x+28\right )^{2/3}-\frac{2 \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} F\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt{3}\right )}{21\ 3^{3/4} \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}+\frac{\sqrt{2+\sqrt{3}} \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} E\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt{3}\right )}{21 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}-\frac{108 (x+1)}{7 \left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-5*(28 + 54*x + 27*x^2)^(2/3))/42 + ((2 + 3*x)*(28 + 54*x + 27*x^2)^(2/3))/21 -
(108*(1 + x))/(7*(6*(1 - Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))) + (Sq
rt[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]])/(21*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)]) - (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]])/(21*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)])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 28.112, size = 396, normalized size = 0.68 \[ \frac{\left (9 x + 6\right ) \left (27 x^{2} + 54 x + 28\right )^{\frac{2}{3}}}{63} - \frac{54 x + 54}{21 \left (- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} - \sqrt{3} + 1\right )} - \frac{5 \left (27 x^{2} + 54 x + 28\right )^{\frac{2}{3}}}{42} + \frac{\sqrt [4]{3} \sqrt{\frac{\left (27 \left (x + 1\right )^{2} + 1\right )^{\frac{2}{3}} + \sqrt [3]{27 \left (x + 1\right )^{2} + 1} + 1}{\left (- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} + 1\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} + 1 + \sqrt{3}}{- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{21 \sqrt{\frac{\sqrt [3]{27 \left (x + 1\right )^{2} + 1} - 1}{\left (- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} - \sqrt{3} + 1\right )^{2}}} \left (x + 1\right )} - \frac{2 \sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{\left (27 \left (x + 1\right )^{2} + 1\right )^{\frac{2}{3}} + \sqrt [3]{27 \left (x + 1\right )^{2} + 1} + 1}{\left (- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} + 1\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} + 1 + \sqrt{3}}{- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{63 \sqrt{\frac{\sqrt [3]{27 \left (x + 1\right )^{2} + 1} - 1}{\left (- \sqrt [3]{27 \left (x + 1\right )^{2} + 1} - \sqrt{3} + 1\right )^{2}}} \left (x + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(9*x + 6)*(27*x**2 + 54*x + 28)**(2/3)/63 - (54*x + 54)/(21*(-(27*(x + 1)**2 + 1
)**(1/3) - sqrt(3) + 1)) - 5*(27*x**2 + 54*x + 28)**(2/3)/42 + 3**(1/4)*sqrt(((2
7*(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)*e
lliptic_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))/(21*sqrt(((27*(x + 1)**2 + 1)**(1/3) -
1)/(-(27*(x + 1)**2 + 1)**(1/3) - sqrt(3) + 1)**2)*(x + 1)) - 2*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))/(63*sqrt(((27*(x + 1)**2 + 1)**(1/3) - 1)/(-(27*
(x + 1)**2 + 1)**(1/3) - sqrt(3) + 1)**2)*(x + 1))

_______________________________________________________________________________________

Mathematica [C]  time = 0.155545, size = 115, normalized size = 0.2 \[ \frac{3\ 2^{2/3} \sqrt [3]{3} \sqrt [3]{-9 i x+\sqrt{3}-9 i} \left (3 \sqrt{3} x+3 \sqrt{3}-i\right ) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{9 i x+\sqrt{3}+9 i}{2 \sqrt{3}}\right )+162 x^3+297 x^2+114 x-28}{42 \sqrt [3]{27 x^2+54 x+28}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-28 + 114*x + 297*x^2 + 162*x^3 + 3*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 + Sqr
t[3] + (9*I)*x)/(2*Sqrt[3])])/(42*(28 + 54*x + 27*x^2)^(1/3))

_______________________________________________________________________________________

Maple [F]  time = 0.095, size = 0, normalized size = 0. \[ \int{ \left ( 2+3\,x \right ) ^{2}{\frac{1}{\sqrt [3]{27\,{x}^{2}+54\,x+28}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{9 \, x^{2} + 12 \, x + 4}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac{1}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((9*x^2 + 12*x + 4)/(27*x^2 + 54*x + 28)^(1/3), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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