### 3.2504 $$\int \frac{(2+3 x)^2}{\sqrt [3]{28+54 x+27 x^2}} \, dx$$

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

[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))) + (Sqrt[2 + Sqrt[3]]*(6 - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))*Sq
rt[(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)])

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Rubi [A]  time = 0.485929, 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, Rules used = {742, 640, 619, 235, 304, 219, 1879} $\frac{1}{21} (3 x+2) \left (27 x^2+54 x+28\right )^{2/3}-\frac{5}{42} \left (27 x^2+54 x+28\right )^{2/3}-\frac{2 \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} F\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt{3}\right )}{21\ 3^{3/4} \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}+\frac{\sqrt{2+\sqrt{3}} \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} E\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt{3}\right )}{21 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}-\frac{108 (x+1)}{7 \left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )}$

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))) + (Sqrt[2 + Sqrt[3]]*(6 - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))*Sq
rt[(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)])

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 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]{28+54 x+27 x^2}} \, dx &=\frac{1}{21} (2+3 x) \left (28+54 x+27 x^2\right )^{2/3}+\frac{1}{63} \int \frac{-216-270 x}{\sqrt [3]{28+54 x+27 x^2}} \, dx\\ &=-\frac{5}{42} \left (28+54 x+27 x^2\right )^{2/3}+\frac{1}{21} (2+3 x) \left (28+54 x+27 x^2\right )^{2/3}+\frac{6}{7} \int \frac{1}{\sqrt [3]{28+54 x+27 x^2}} \, dx\\ &=-\frac{5}{42} \left (28+54 x+27 x^2\right )^{2/3}+\frac{1}{21} (2+3 x) \left (28+54 x+27 x^2\right )^{2/3}+\frac{1}{63} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1+\frac{x^2}{108}}} \, dx,x,54+54 x\right )\\ &=-\frac{5}{42} \left (28+54 x+27 x^2\right )^{2/3}+\frac{1}{21} (2+3 x) \left (28+54 x+27 x^2\right )^{2/3}+\frac{\left (\sqrt{3} \sqrt{(54+54 x)^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{28+54 x+27 x^2}\right )}{7 (54+54 x)}\\ &=-\frac{5}{42} \left (28+54 x+27 x^2\right )^{2/3}+\frac{1}{21} (2+3 x) \left (28+54 x+27 x^2\right )^{2/3}-\frac{\left (\sqrt{3} \sqrt{(54+54 x)^2}\right ) \operatorname{Subst}\left (\int \frac{1+\sqrt{3}-x}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{28+54 x+27 x^2}\right )}{7 (54+54 x)}+\frac{\left (\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,\sqrt [3]{28+54 x+27 x^2}\right )}{7 (54+54 x)}\\ &=-\frac{5}{42} \left (28+54 x+27 x^2\right )^{2/3}+\frac{1}{21} (2+3 x) \left (28+54 x+27 x^2\right )^{2/3}-\frac{18 (1+x)}{7 \left (1-\sqrt{3}-\sqrt [3]{28+54 x+27 x^2}\right )}+\frac{\sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{28+54 x+27 x^2}\right ) \sqrt{\frac{1+\sqrt [3]{28+54 x+27 x^2}+\left (28+54 x+27 x^2\right )^{2/3}}{\left (1-\sqrt{3}-\sqrt [3]{28+54 x+27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-\sqrt [3]{28+54 x+27 x^2}}{1-\sqrt{3}-\sqrt [3]{28+54 x+27 x^2}}\right )|-7+4 \sqrt{3}\right )}{7\ 3^{3/4} (1+x) \sqrt{-\frac{1-\sqrt [3]{28+54 x+27 x^2}}{\left (1-\sqrt{3}-\sqrt [3]{28+54 x+27 x^2}\right )^2}}}-\frac{2 \sqrt{2} \left (1-\sqrt [3]{28+54 x+27 x^2}\right ) \sqrt{\frac{1+\sqrt [3]{28+54 x+27 x^2}+\left (28+54 x+27 x^2\right )^{2/3}}{\left (1-\sqrt{3}-\sqrt [3]{28+54 x+27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-\sqrt [3]{28+54 x+27 x^2}}{1-\sqrt{3}-\sqrt [3]{28+54 x+27 x^2}}\right )|-7+4 \sqrt{3}\right )}{21 \sqrt [4]{3} (1+x) \sqrt{-\frac{1-\sqrt [3]{28+54 x+27 x^2}}{\left (1-\sqrt{3}-\sqrt [3]{28+54 x+27 x^2}\right )^2}}}\\ \end{align*}

Mathematica [C]  time = 0.0184156, size = 47, normalized size = 0.08 $\frac{1}{42} \left (36 (x+1) \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};-27 (x+1)^2\right )+\left (27 x^2+54 x+28\right )^{2/3} (6 x-1)\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((-1 + 6*x)*(28 + 54*x + 27*x^2)^(2/3) + 36*(1 + x)*Hypergeometric2F1[1/3, 1/2, 3/2, -27*(1 + x)^2])/42

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Maple [F]  time = 1.506, size = 0, normalized size = 0. \begin{align*} \int{ \left ( 2+3\,x \right ) ^{2}{\frac{1}{\sqrt [3]{27\,{x}^{2}+54\,x+28}}}}\, dx \end{align*}

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)

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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 + 28\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+28)^(1/3),x, algorithm="maxima")

[Out]

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

[Out]

integral((9*x^2 + 12*x + 4)/(27*x^2 + 54*x + 28)^(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 + 28}}\, dx \end{align*}

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)

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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 + 28\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+28)^(1/3),x, algorithm="giac")

[Out]

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