3.708 \(\int \frac {2+3 i x}{\sqrt [3]{4-27 x^2}} \, dx\)

Optimal. Leaf size=531 \[ -\frac {6 x}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}-\frac {1}{12} i \left (4-27 x^2\right )^{2/3}+\frac {2\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt {3}\right )}{9 \sqrt [4]{3} \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} x}-\frac {\sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt {3}\right )}{3\ 3^{3/4} \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} x} \]

[Out]

-1/12*I*(-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)
)*EllipticF((-(-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)-(-2
7*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 = 0.26, antiderivative size = 531, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {641, 235, 304, 219, 1879} \[ -\frac {6 x}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}-\frac {1}{12} i \left (4-27 x^2\right )^{2/3}+\frac {2\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt {3}\right )}{9 \sqrt [4]{3} \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} x}-\frac {\sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt {3}\right )}{3\ 3^{3/4} \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-I/12)*(4 - 27*x^2)^(2/3) - (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 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 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 641

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 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 i x}{\sqrt [3]{4-27 x^2}} \, dx &=-\frac {1}{12} i \left (4-27 x^2\right )^{2/3}+2 \int \frac {1}{\sqrt [3]{4-27 x^2}} \, dx\\ &=-\frac {1}{12} i \left (4-27 x^2\right )^{2/3}-\frac {\sqrt {-x^2} \operatorname {Subst}\left (\int \frac {x}{\sqrt {-4+x^3}} \, dx,x,\sqrt [3]{4-27 x^2}\right )}{\sqrt {3} x}\\ &=-\frac {1}{12} i \left (4-27 x^2\right )^{2/3}+\frac {\sqrt {-x^2} \operatorname {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 ) \operatorname {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} i \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]  time = 0.02, size = 42, normalized size = 0.08 \[ \sqrt [3]{2} x \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};\frac {27 x^2}{4}\right )-\frac {1}{12} i \left (4-27 x^2\right )^{2/3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.95, size = 0, normalized size = 0.00 \[ \frac {36 \, x {\rm integral}\left (\frac {8 \, {\left (-27 \, x^{2} + 4\right )}^{\frac {2}{3}}}{9 \, {\left (27 \, x^{4} - 4 \, x^{2}\right )}}, x\right ) + {\left (-27 \, x^{2} + 4\right )}^{\frac {2}{3}} {\left (-3 i \, x - 8\right )}}{36 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*(36*x*integral(8/9*(-27*x^2 + 4)^(2/3)/(27*x^4 - 4*x^2), x) + (-27*x^2 + 4)^(2/3)*(-3*I*x - 8))/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {3 i \, x + 2}{{\left (-27 \, x^{2} + 4\right )}^{\frac {1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.55, size = 37, normalized size = 0.07 \[ 2^{\frac {1}{3}} x \hypergeom \left (\left [\frac {1}{3}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], \frac {27 x^{2}}{4}\right )+\frac {i \left (27 x^{2}-4\right )}{12 \left (-27 x^{2}+4\right )^{\frac {1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*I*x)/(-27*x^2+4)^(1/3),x)

[Out]

1/12*I*(27*x^2-4)/(-27*x^2+4)^(1/3)+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 \[ \int \frac {3 i \, x + 2}{{\left (-27 \, x^{2} + 4\right )}^{\frac {1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.77, size = 28, normalized size = 0.05 \[ 2^{1/3}\,x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {1}{2};\ \frac {3}{2};\ \frac {27\,x^2}{4}\right )-\frac {{\left (4-27\,x^2\right )}^{2/3}\,1{}\mathrm {i}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*3i + 2)/(4 - 27*x^2)^(1/3),x)

[Out]

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

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sympy [A]  time = 2.56, size = 39, normalized size = 0.07 \[ \sqrt [3]{2} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {27 x^{2} e^{2 i \pi }}{4}} \right )} - \frac {i \left (4 - 27 x^{2}\right )^{\frac {2}{3}}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*I*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(2*I*pi)/4) - I*(4 - 27*x**2)**(2/3)/12

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