3.8.2 \(\int \frac {2+3 x}{\sqrt [3]{4+27 x^2}} \, dx\) [702]

Optimal. Leaf size=529 \[ \frac {1}{12} \left (4+27 x^2\right )^{2/3}-\frac {6 x}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}+\frac {\sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{4+27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4+27 x^2}+\left (4+27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}\right )|-7+4 \sqrt {3}\right )}{3\ 3^{3/4} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4+27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}}}-\frac {2\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{4+27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4+27 x^2}+\left (4+27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}\right )|-7+4 \sqrt {3}\right )}{9 \sqrt [4]{3} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4+27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}}} \]

[Out]

1/12*(27*x^2+4)^(2/3)-6*x/(-(27*x^2+4)^(1/3)+2^(2/3)*(1-3^(1/2)))-2/27*2^(5/6)*(2^(2/3)-(27*x^2+4)^(1/3))*Elli
pticF((-(27*x^2+4)^(1/3)+2^(2/3)*(1+3^(1/2)))/(-(27*x^2+4)^(1/3)+2^(2/3)*(1-3^(1/2))),2*I-I*3^(1/2))*((2*2^(1/
3)+2^(2/3)*(27*x^2+4)^(1/3)+(27*x^2+4)^(2/3))/(-(27*x^2+4)^(1/3)+2^(2/3)*(1-3^(1/2)))^2)^(1/2)*3^(3/4)/x/((-2^
(2/3)+(27*x^2+4)^(1/3))/(-(27*x^2+4)^(1/3)+2^(2/3)*(1-3^(1/2)))^2)^(1/2)+1/9*2^(1/3)*(2^(2/3)-(27*x^2+4)^(1/3)
)*EllipticE((-(27*x^2+4)^(1/3)+2^(2/3)*(1+3^(1/2)))/(-(27*x^2+4)^(1/3)+2^(2/3)*(1-3^(1/2))),2*I-I*3^(1/2))*((2
*2^(1/3)+2^(2/3)*(27*x^2+4)^(1/3)+(27*x^2+4)^(2/3))/(-(27*x^2+4)^(1/3)+2^(2/3)*(1-3^(1/2)))^2)^(1/2)*(1/2*6^(1
/2)+1/2*2^(1/2))*3^(1/4)/x/((-2^(2/3)+(27*x^2+4)^(1/3))/(-(27*x^2+4)^(1/3)+2^(2/3)*(1-3^(1/2)))^2)^(1/2)

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Rubi [A]
time = 0.41, antiderivative size = 529, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {655, 241, 310, 225, 1893} \begin {gather*} -\frac {2\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt {\frac {\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} F\left (\text {ArcSin}\left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt {3}\right )}{9 \sqrt [4]{3} \sqrt {-\frac {2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}+\frac {\sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt {\frac {\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} E\left (\text {ArcSin}\left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt {3}\right )}{3\ 3^{3/4} \sqrt {-\frac {2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}-\frac {6 x}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}+\frac {1}{12} \left (27 x^2+4\right )^{2/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)/(4 + 27*x^2)^(1/3),x]

[Out]

(4 + 27*x^2)^(2/3)/12 - (6*x)/(2^(2/3)*(1 - Sqrt[3]) - (4 + 27*x^2)^(1/3)) + (2^(1/3)*Sqrt[2 + Sqrt[3]]*(2^(2/
3) - (4 + 27*x^2)^(1/3))*Sqrt[(2*2^(1/3) + 2^(2/3)*(4 + 27*x^2)^(1/3) + (4 + 27*x^2)^(2/3))/(2^(2/3)*(1 - Sqrt
[3]) - (4 + 27*x^2)^(1/3))^2]*EllipticE[ArcSin[(2^(2/3)*(1 + Sqrt[3]) - (4 + 27*x^2)^(1/3))/(2^(2/3)*(1 - Sqrt
[3]) - (4 + 27*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(3*3^(3/4)*x*Sqrt[-((2^(2/3) - (4 + 27*x^2)^(1/3))/(2^(2/3)*(1 -
Sqrt[3]) - (4 + 27*x^2)^(1/3))^2)]) - (2*2^(5/6)*(2^(2/3) - (4 + 27*x^2)^(1/3))*Sqrt[(2*2^(1/3) + 2^(2/3)*(4
+ 27*x^2)^(1/3) + (4 + 27*x^2)^(2/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 + 27*x^2)^(1/3))^2]*EllipticF[ArcSin[(2^(2/3
)*(1 + Sqrt[3]) - (4 + 27*x^2)^(1/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 + 27*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(9*3^(1/
4)*x*Sqrt[-((2^(2/3) - (4 + 27*x^2)^(1/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 + 27*x^2)^(1/3))^2)])

Rule 225

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

Rule 241

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 310

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(-(
1 + Sqrt[3]))*(s/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 655

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

Rule 1893

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]/(r^2
*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/(
(1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], 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]{4+27 x^2}} \, dx &=\frac {1}{12} \left (4+27 x^2\right )^{2/3}+2 \int \frac {1}{\sqrt [3]{4+27 x^2}} \, dx\\ &=\frac {1}{12} \left (4+27 x^2\right )^{2/3}+\frac {\sqrt {x^2} \text {Subst}\left (\int \frac {x}{\sqrt {-4+x^3}} \, dx,x,\sqrt [3]{4+27 x^2}\right )}{\sqrt {3} x}\\ &=\frac {1}{12} \left (4+27 x^2\right )^{2/3}-\frac {\sqrt {x^2} \text {Subst}\left (\int \frac {2^{2/3} \left (1+\sqrt {3}\right )-x}{\sqrt {-4+x^3}} \, dx,x,\sqrt [3]{4+27 x^2}\right )}{\sqrt {3} x}+\frac {\left (2 \sqrt [6]{2} \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-4+x^3}} \, dx,x,\sqrt [3]{4+27 x^2}\right )}{\sqrt {3 \left (2-\sqrt {3}\right )} x}\\ &=\frac {1}{12} \left (4+27 x^2\right )^{2/3}-\frac {6 x}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}+\frac {\sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{4+27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4+27 x^2}+\left (4+27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}\right )|-7+4 \sqrt {3}\right )}{3\ 3^{3/4} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4+27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}}}-\frac {2\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{4+27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4+27 x^2}+\left (4+27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}\right )|-7+4 \sqrt {3}\right )}{9 \sqrt [4]{3} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4+27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 7.37, size = 40, normalized size = 0.08 \begin {gather*} \frac {1}{12} \left (4+27 x^2\right )^{2/3}+\sqrt [3]{2} x \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};-\frac {27 x^2}{4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)/(4 + 27*x^2)^(1/3),x]

[Out]

(4 + 27*x^2)^(2/3)/12 + 2^(1/3)*x*Hypergeometric2F1[1/3, 1/2, 3/2, (-27*x^2)/4]

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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 4.
time = 0.43, size = 29, normalized size = 0.05

method	result	size
risch	\(\frac {\left (27 x^{2}+4\right )^{\frac {2}{3}}}{12}+2^{\frac {1}{3}} x \hypergeom \left (\left [\frac {1}{3}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -\frac {27 x^{2}}{4}\right )\)	\(29\)
meijerg	\(2^{\frac {1}{3}} x \hypergeom \left (\left [\frac {1}{3}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -\frac {27 x^{2}}{4}\right )+\frac {3 \,2^{\frac {1}{3}} x^{2} \hypergeom \left (\left [\frac {1}{3}, 1\right ], \left [2\right ], -\frac {27 x^{2}}{4}\right )}{4}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)/(27*x^2+4)^(1/3),x,method=_RETURNVERBOSE)

[Out]

1/12*(27*x^2+4)^(2/3)+2^(1/3)*x*hypergeom([1/3,1/2],[3/2],-27/4*x^2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)/(27*x^2+4)^(1/3),x, algorithm="maxima")

[Out]

integrate((3*x + 2)/(27*x^2 + 4)^(1/3), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)/(27*x^2+4)^(1/3),x, algorithm="fricas")

[Out]

integral((3*x + 2)/(27*x^2 + 4)^(1/3), x)

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Sympy [A]
time = 1.43, size = 36, normalized size = 0.07 \begin {gather*} \sqrt [3]{2} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {27 x^{2} e^{i \pi }}{4}} \right )} + \frac {\left (27 x^{2} + 4\right )^{\frac {2}{3}}}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)/(27*x**2+4)**(1/3),x)

[Out]

2**(1/3)*x*hyper((1/3, 1/2), (3/2,), 27*x**2*exp_polar(I*pi)/4) + (27*x**2 + 4)**(2/3)/12

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)/(27*x^2+4)^(1/3),x, algorithm="giac")

[Out]

integrate((3*x + 2)/(27*x^2 + 4)^(1/3), x)

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Mupad [B]
time = 0.25, size = 27, normalized size = 0.05 \begin {gather*} \frac {{\left (27\,x^2+4\right )}^{2/3}}{12}+2^{1/3}\,x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {1}{2};\ \frac {3}{2};\ -\frac {27\,x^2}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x + 2)/(27*x^2 + 4)^(1/3),x)

[Out]

(27*x^2 + 4)^(2/3)/12 + 2^(1/3)*x*hypergeom([1/3, 1/2], 3/2, -(27*x^2)/4)

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