3.2505 \(\int \frac{2+3 x}{\sqrt [3]{28+54 x+27 x^2}} \, dx\)

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

[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 - 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]])/(1
8*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 + (54 + 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 = 0.378452, 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, Rules used = {640, 619, 235, 304, 219, 1879} \[ \frac{1}{12} \left (27 x^2+54 x+28\right )^{2/3}+\frac{\left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} F\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt{3}\right )}{9\ 3^{3/4} \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}-\frac{\sqrt{2+\sqrt{3}} \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} E\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt{3}\right )}{18 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}+\frac{18 (x+1)}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}} \]

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 - 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]])/(1
8*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 + (54 + 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)])

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}{\sqrt [3]{28+54 x+27 x^2}} \, dx &=\frac{1}{12} \left (28+54 x+27 x^2\right )^{2/3}-\int \frac{1}{\sqrt [3]{28+54 x+27 x^2}} \, dx\\ &=\frac{1}{12} \left (28+54 x+27 x^2\right )^{2/3}-\frac{1}{54} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1+\frac{x^2}{108}}} \, dx,x,54+54 x\right )\\ &=\frac{1}{12} \left (28+54 x+27 x^2\right )^{2/3}-\frac{\sqrt{(54+54 x)^2} \operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{28+54 x+27 x^2}\right )}{2 \sqrt{3} (54+54 x)}\\ &=\frac{1}{12} \left (28+54 x+27 x^2\right )^{2/3}+\frac{\sqrt{(54+54 x)^2} \operatorname{Subst}\left (\int \frac{1+\sqrt{3}-x}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{28+54 x+27 x^2}\right )}{2 \sqrt{3} (54+54 x)}-\frac{\left (\sqrt{\frac{1}{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 )}{54+54 x}\\ &=\frac{1}{12} \left (28+54 x+27 x^2\right )^{2/3}+\frac{3 (1+x)}{1-\sqrt{3}-\sqrt [3]{28+54 x+27 x^2}}-\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 )}{6\ 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{\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 )}{9 \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 = 0.007192, size = 41, normalized size = 0.07 \[ \frac{1}{12} \left (27 x^2+54 x+28\right )^{2/3}-(x+1) \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};-27 (x+1)^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(28 + 54*x + 27*x^2)^(2/3)/12 - (1 + x)*Hypergeometric2F1[1/3, 1/2, 3/2, -27*(1 + x)^2]

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Maple [F]  time = 1.478, size = 0, normalized size = 0. \begin{align*} \int{(2+3\,x){\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)/(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. \begin{align*} \int \frac{3 \, x + 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)/(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. \begin{align*}{\rm integral}\left (\frac{3 \, x + 2}{{\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)/(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. \begin{align*} \int \frac{3 x + 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)/(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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{3 \, x + 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)/(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)