3.8.7 \(\int \frac {(2+3 i x)^2}{\sqrt [3]{4-27 x^2}} \, dx\) [707]

Optimal. Leaf size=557 \[ -\frac {5}{21} i \left (4-27 x^2\right )^{2/3}-\frac {1}{21} i (2+3 i x) \left (4-27 x^2\right )^{2/3}-\frac {72 x}{7 \left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )}-\frac {4 \sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4-27 x^2}+\left (4-27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt {3}\right )}{7\ 3^{3/4} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}}+\frac {8\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4-27 x^2}+\left (4-27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt {3}\right )}{21 \sqrt [4]{3} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}} \]

[Out]

-5/21*I*(-27*x^2+4)^(2/3)-1/21*I*(2+3*I*x)*(-27*x^2+4)^(2/3)-72/7*x/(-(-27*x^2+4)^(1/3)+2^(2/3)*(1-3^(1/2)))+8
/63*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)-4/21*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)

________________________________________________________________________________________

Rubi [A]
time = 0.22, antiderivative size = 557, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {757, 655, 241, 310, 225, 1893} \begin {gather*} \frac {8\ 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 (\text {ArcSin}\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 )}{21 \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}}}-\frac {4 \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 (\text {ArcSin}\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 )}{7\ 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 {1}{21} i \left (4-27 x^2\right )^{2/3} (2+3 i x)-\frac {5}{21} i \left (4-27 x^2\right )^{2/3}-\frac {72 x}{7 \left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-5*I)/21)*(4 - 27*x^2)^(2/3) - (I/21)*(2 + (3*I)*x)*(4 - 27*x^2)^(2/3) - (72*x)/(7*(2^(2/3)*(1 - Sqrt[3]) -
(4 - 27*x^2)^(1/3))) - (4*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]])/(7*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)]) + (8*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]])/(21*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 757

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

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 i x)^2}{\sqrt [3]{4-27 x^2}} \, dx &=-\frac {1}{21} i (2+3 i x) \left (4-27 x^2\right )^{2/3}-\frac {1}{63} \int \frac {-216-540 i x}{\sqrt [3]{4-27 x^2}} \, dx\\ &=-\frac {5}{21} i \left (4-27 x^2\right )^{2/3}-\frac {1}{21} i (2+3 i x) \left (4-27 x^2\right )^{2/3}+\frac {24}{7} \int \frac {1}{\sqrt [3]{4-27 x^2}} \, dx\\ &=-\frac {5}{21} i \left (4-27 x^2\right )^{2/3}-\frac {1}{21} i (2+3 i x) \left (4-27 x^2\right )^{2/3}-\frac {\left (4 \sqrt {3} \sqrt {-x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-4+x^3}} \, dx,x,\sqrt [3]{4-27 x^2}\right )}{7 x}\\ &=-\frac {5}{21} i \left (4-27 x^2\right )^{2/3}-\frac {1}{21} i (2+3 i x) \left (4-27 x^2\right )^{2/3}+\frac {\left (4 \sqrt {3} \sqrt {-x^2}\right ) \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 )}{7 x}-\frac {\left (8 \sqrt [6]{2} \sqrt {3} \sqrt {-x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-4+x^3}} \, dx,x,\sqrt [3]{4-27 x^2}\right )}{7 \sqrt {2-\sqrt {3}} x}\\ &=-\frac {5}{21} i \left (4-27 x^2\right )^{2/3}-\frac {1}{21} i (2+3 i x) \left (4-27 x^2\right )^{2/3}-\frac {72 x}{7 \left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )}-\frac {4 \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 )}{7\ 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 {8\ 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 )}{21 \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*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.03, size = 51, normalized size = 0.09 \begin {gather*} \left (-\frac {i}{3}+\frac {x}{7}\right ) \left (4-27 x^2\right )^{2/3}+\frac {12}{7} \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*I)*x)^2/(4 - 27*x^2)^(1/3),x]

[Out]

(-1/3*I + x/7)*(4 - 27*x^2)^(2/3) + (12*2^(1/3)*x*Hypergeometric2F1[1/3, 1/2, 3/2, (27*x^2)/4])/7

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 4.
time = 0.49, size = 43, normalized size = 0.08

method	result	size
risch	\(-\frac {\left (-7 i+3 x \right ) \left (27 x^{2}-4\right )}{21 \left (-27 x^{2}+4\right )^{\frac {1}{3}}}+\frac {12 \,2^{\frac {1}{3}} x \hypergeom \left (\left [\frac {1}{3}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], \frac {27 x^{2}}{4}\right )}{7}\)	\(43\)
meijerg	\(2 \,2^{\frac {1}{3}} x \hypergeom \left (\left [\frac {1}{3}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], \frac {27 x^{2}}{4}\right )+3 i 2^{\frac {1}{3}} x^{2} \hypergeom \left (\left [\frac {1}{3}, 1\right ], \left [2\right ], \frac {27 x^{2}}{4}\right )-\frac {3 \,2^{\frac {1}{3}} x^{3} \hypergeom \left (\left [\frac {1}{3}, \frac {3}{2}\right ], \left [\frac {5}{2}\right ], \frac {27 x^{2}}{4}\right )}{2}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*(-7*I+3*x)*(27*x^2-4)/(-27*x^2+4)^(1/3)+12/7*2^(1/3)*x*hypergeom([1/3,1/2],[3/2],27/4*x^2)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

1/21*(21*x*integral(32/21*(-27*x^2 + 4)^(2/3)/(27*x^4 - 4*x^2), x) + (3*x^2 - 7*I*x - 8)*(-27*x^2 + 4)^(2/3))/
x

________________________________________________________________________________________

Sympy [A]
time = 2.47, size = 73, normalized size = 0.13 \begin {gather*} - \frac {3 \cdot \sqrt [3]{2} x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {27 x^{2} e^{2 i \pi }}{4}} \right )}}{2} + 2 \cdot \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}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*2**(1/3)*x**3*hyper((1/3, 3/2), (5/2,), 27*x**2*exp_polar(2*I*pi)/4)/2 + 2*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)/3

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (2+x\,3{}\mathrm {i}\right )}^2}{{\left (4-27\,x^2\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________