3.2498 \(\int \frac{(2+3 x)^2}{\sqrt [3]{52-54 x+27 x^2}} \, dx\)

Optimal. Leaf size=628 \[ \frac{5\ 5^{5/6} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right ),4 \sqrt{3}-7\right )}{63\ 3^{3/4} \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)}+\frac{1}{21} (3 x+2) \left (27 x^2-54 x+52\right )^{2/3}+\frac{25}{42} \left (27 x^2-54 x+52\right )^{2/3}+\frac{2700 \sqrt [3]{5} (1-x)}{7 \left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )}-\frac{5\ 5^{5/6} \sqrt{2+\sqrt{3}} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} E\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{126 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)} \]

[Out]

(25*(52 - 54*x + 27*x^2)^(2/3))/42 + ((2 + 3*x)*(52 - 54*x + 27*x^2)^(2/3))/21 + (2700*5^(1/3)*(1 - x))/(7*(30
*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))) - (5*5^(5/6)*Sqrt[2 + Sqrt[3]]*(30 - 10^(1/3)*(2700
+ (-54 + 54*x)^2)^(1/3))*Sqrt[(900 + 30*10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3) + 10^(2/3)*(2700 + (-54 + 54*x)
^2)^(2/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))^2]*EllipticE[ArcSin[(30*(1 + Sqrt[3]) -
10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))], -7 + 4*
Sqrt[3]])/(126*Sqrt[2]*3^(1/4)*(1 - x)*Sqrt[-((30 - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))/(30*(1 - Sqrt[3])
- 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))^2)]) + (5*5^(5/6)*(30 - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))*Sqrt
[(900 + 30*10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3) + 10^(2/3)*(2700 + (-54 + 54*x)^2)^(2/3))/(30*(1 - Sqrt[3])
- 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))^2]*EllipticF[ArcSin[(30*(1 + Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x
)^2)^(1/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))], -7 + 4*Sqrt[3]])/(63*3^(3/4)*(1 - x)
*Sqrt[-((30 - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/
3))^2)])

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Rubi [A]  time = 0.541443, antiderivative size = 628, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {742, 12, 640, 619, 235, 304, 219, 1879} \[ \frac{1}{21} (3 x+2) \left (27 x^2-54 x+52\right )^{2/3}+\frac{25}{42} \left (27 x^2-54 x+52\right )^{2/3}+\frac{2700 \sqrt [3]{5} (1-x)}{7 \left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )}+\frac{5\ 5^{5/6} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} F\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{63\ 3^{3/4} \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)}-\frac{5\ 5^{5/6} \sqrt{2+\sqrt{3}} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} E\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{126 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^2/(52 - 54*x + 27*x^2)^(1/3),x]

[Out]

(25*(52 - 54*x + 27*x^2)^(2/3))/42 + ((2 + 3*x)*(52 - 54*x + 27*x^2)^(2/3))/21 + (2700*5^(1/3)*(1 - x))/(7*(30
*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))) - (5*5^(5/6)*Sqrt[2 + Sqrt[3]]*(30 - 10^(1/3)*(2700
+ (-54 + 54*x)^2)^(1/3))*Sqrt[(900 + 30*10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3) + 10^(2/3)*(2700 + (-54 + 54*x)
^2)^(2/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))^2]*EllipticE[ArcSin[(30*(1 + Sqrt[3]) -
10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))], -7 + 4*
Sqrt[3]])/(126*Sqrt[2]*3^(1/4)*(1 - x)*Sqrt[-((30 - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))/(30*(1 - Sqrt[3])
- 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))^2)]) + (5*5^(5/6)*(30 - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))*Sqrt
[(900 + 30*10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3) + 10^(2/3)*(2700 + (-54 + 54*x)^2)^(2/3))/(30*(1 - Sqrt[3])
- 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))^2]*EllipticF[ArcSin[(30*(1 + Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x
)^2)^(1/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))], -7 + 4*Sqrt[3]])/(63*3^(3/4)*(1 - x)
*Sqrt[-((30 - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/
3))^2)])

Rule 742

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2
*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]*(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,
p, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 235

Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Dist[(3*Sqrt[b*x^2])/(2*b*x), Subst[Int[x/Sqrt[-a + x^3], x], x
, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b}, x]

Rule 304

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, -Dist[(S
qrt[2]*s)/(Sqrt[2 - Sqrt[3]]*r), Int[1/Sqrt[a + b*x^3], x], x] + Dist[1/r, Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a
+ b*x^3], x], x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 219

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 - Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 + Sqrt[3
])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[-((s*(s + r*x))/((1 - S
qrt[3])*s + r*x)^2)]), x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 1879

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[((1 + Sqrt[3])*d)/c]]
, s = Denom[Simplify[((1 + Sqrt[3])*d)/c]]}, Simp[(2*d*s^3*Sqrt[a + b*x^3])/(a*r^2*((1 - Sqrt[3])*s + r*x)), x
] + Simp[(3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]*Elli
pticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[-((s
*(s + r*x))/((1 - Sqrt[3])*s + r*x)^2)]), x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqr
t[3])*a*d^3, 0]

Rubi steps

\begin{align*} \int \frac{(2+3 x)^2}{\sqrt [3]{52-54 x+27 x^2}} \, dx &=\frac{1}{21} (2+3 x) \left (52-54 x+27 x^2\right )^{2/3}+\frac{1}{63} \int \frac{1350 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx\\ &=\frac{1}{21} (2+3 x) \left (52-54 x+27 x^2\right )^{2/3}+\frac{150}{7} \int \frac{x}{\sqrt [3]{52-54 x+27 x^2}} \, dx\\ &=\frac{25}{42} \left (52-54 x+27 x^2\right )^{2/3}+\frac{1}{21} (2+3 x) \left (52-54 x+27 x^2\right )^{2/3}+\frac{150}{7} \int \frac{1}{\sqrt [3]{52-54 x+27 x^2}} \, dx\\ &=\frac{25}{42} \left (52-54 x+27 x^2\right )^{2/3}+\frac{1}{21} (2+3 x) \left (52-54 x+27 x^2\right )^{2/3}+\frac{1}{63} \left (5 \sqrt [3]{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1+\frac{x^2}{2700}}} \, dx,x,-54+54 x\right )\\ &=\frac{25}{42} \left (52-54 x+27 x^2\right )^{2/3}+\frac{1}{21} (2+3 x) \left (52-54 x+27 x^2\right )^{2/3}+\frac{\left (25 \sqrt{3} \sqrt [3]{5} \sqrt{(-54+54 x)^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+x^3}} \, dx,x,\frac{\sqrt [3]{2700+(-54+54 x)^2}}{3\ 10^{2/3}}\right )}{7 (-54+54 x)}\\ &=\frac{25}{42} \left (52-54 x+27 x^2\right )^{2/3}+\frac{1}{21} (2+3 x) \left (52-54 x+27 x^2\right )^{2/3}-\frac{\left (25 \sqrt{3} \sqrt [3]{5} \sqrt{(-54+54 x)^2}\right ) \operatorname{Subst}\left (\int \frac{1+\sqrt{3}-x}{\sqrt{-1+x^3}} \, dx,x,\frac{\sqrt [3]{2700+(-54+54 x)^2}}{3\ 10^{2/3}}\right )}{7 (-54+54 x)}+\frac{\left (25 \sqrt [3]{5} \sqrt{6 \left (2+\sqrt{3}\right )} \sqrt{(-54+54 x)^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x^3}} \, dx,x,\frac{\sqrt [3]{2700+(-54+54 x)^2}}{3\ 10^{2/3}}\right )}{7 (-54+54 x)}\\ &=\frac{25}{42} \left (52-54 x+27 x^2\right )^{2/3}+\frac{1}{21} (2+3 x) \left (52-54 x+27 x^2\right )^{2/3}+\frac{2700 \sqrt [3]{5} (1-x)}{7 \left (30-30 \sqrt{3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )}-\frac{5\ 5^{5/6} \sqrt{2+\sqrt{3}} \left (30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right ) \sqrt{\frac{900+30 \sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}+10^{2/3} \left (2700+(-54+54 x)^2\right )^{2/3}}{\left (30-30 \sqrt{3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{30+30 \sqrt{3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{30-30 \sqrt{3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}\right )|-7+4 \sqrt{3}\right )}{126 \sqrt{2} \sqrt [4]{3} (1-x) \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{\left (30-30 \sqrt{3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}}}+\frac{5\ 5^{5/6} \left (30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right ) \sqrt{\frac{900+30 \sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}+10^{2/3} \left (2700+(-54+54 x)^2\right )^{2/3}}{\left (30-30 \sqrt{3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{30+30 \sqrt{3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{30-30 \sqrt{3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}\right )|-7+4 \sqrt{3}\right )}{63\ 3^{3/4} (1-x) \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{\left (30-30 \sqrt{3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}}}\\ \end{align*}

Mathematica [C]  time = 0.0241067, size = 54, normalized size = 0.09 \[ \frac{1}{42} \left (180 \sqrt [3]{5} (x-1) \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};-\frac{27}{25} (x-1)^2\right )+\left (27 x^2-54 x+52\right )^{2/3} (6 x+29)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^2/(52 - 54*x + 27*x^2)^(1/3),x]

[Out]

((29 + 6*x)*(52 - 54*x + 27*x^2)^(2/3) + 180*5^(1/3)*(-1 + x)*Hypergeometric2F1[1/3, 1/2, 3/2, (-27*(-1 + x)^2
)/25])/42

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Maple [F]  time = 1.437, size = 0, normalized size = 0. \begin{align*} \int{ \left ( 2+3\,x \right ) ^{2}{\frac{1}{\sqrt [3]{27\,{x}^{2}-54\,x+52}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^2/(27*x^2-54*x+52)^(1/3),x)

[Out]

int((2+3*x)^2/(27*x^2-54*x+52)^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (3 \, x + 2\right )}^{2}}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^2/(27*x^2-54*x+52)^(1/3),x, algorithm="maxima")

[Out]

integrate((3*x + 2)^2/(27*x^2 - 54*x + 52)^(1/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{9 \, x^{2} + 12 \, x + 4}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac{1}{3}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^2/(27*x^2-54*x+52)^(1/3),x, algorithm="fricas")

[Out]

integral((9*x^2 + 12*x + 4)/(27*x^2 - 54*x + 52)^(1/3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (3 x + 2\right )^{2}}{\sqrt [3]{27 x^{2} - 54 x + 52}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**2/(27*x**2-54*x+52)**(1/3),x)

[Out]

Integral((3*x + 2)**2/(27*x**2 - 54*x + 52)**(1/3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (3 \, x + 2\right )}^{2}}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^2/(27*x^2-54*x+52)^(1/3),x, algorithm="giac")

[Out]

integrate((3*x + 2)^2/(27*x^2 - 54*x + 52)^(1/3), x)