Integrand size = 21, antiderivative size = 557 \[ \int \frac {(2+3 i x)^2}{\sqrt [3]{4-27 x^2}} \, dx=-\frac {5}{21} i \left (4-27 x^2\right )^{2/3}-\frac {1}{21} i (2+3 i x) \left (4-27 x^2\right )^{2/3}-\frac {72 x}{7 \left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )}-\frac {4 \sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4-27 x^2}+\left (4-27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt {3}\right )}{7\ 3^{3/4} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}}+\frac {8\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4-27 x^2}+\left (4-27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right ),-7+4 \sqrt {3}\right )}{21 \sqrt [4]{3} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}} \]
-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)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.04 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.09 \[ \int \frac {(2+3 i x)^2}{\sqrt [3]{4-27 x^2}} \, dx=\left (-\frac {i}{3}+\frac {x}{7}\right ) \left (4-27 x^2\right )^{2/3}+\frac {12}{7} \sqrt [3]{2} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},\frac {27 x^2}{4}\right ) \]
(-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
Time = 0.51 (sec) , antiderivative size = 625, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {497, 27, 455, 233, 833, 760, 2418}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(2+3 i x)^2}{\sqrt [3]{4-27 x^2}} \, dx\) |
\(\Big \downarrow \) 497 |
\(\displaystyle -\frac {1}{63} \int -\frac {108 (5 i x+2)}{\sqrt [3]{4-27 x^2}}dx-\frac {1}{21} i \left (4-27 x^2\right )^{2/3} (2+3 i x)\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {12}{7} \int \frac {5 i x+2}{\sqrt [3]{4-27 x^2}}dx-\frac {1}{21} i (2+3 i x) \left (4-27 x^2\right )^{2/3}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle \frac {12}{7} \left (2 \int \frac {1}{\sqrt [3]{4-27 x^2}}dx-\frac {5}{36} i \left (4-27 x^2\right )^{2/3}\right )-\frac {1}{21} i (2+3 i x) \left (4-27 x^2\right )^{2/3}\) |
\(\Big \downarrow \) 233 |
\(\displaystyle \frac {12}{7} \left (-\frac {\sqrt {-x^2} \int \frac {\sqrt [3]{4-27 x^2}}{3 \sqrt {3} \sqrt {-x^2}}d\sqrt [3]{4-27 x^2}}{\sqrt {3} x}-\frac {5}{36} i \left (4-27 x^2\right )^{2/3}\right )-\frac {1}{21} i (2+3 i x) \left (4-27 x^2\right )^{2/3}\) |
\(\Big \downarrow \) 833 |
\(\displaystyle \frac {12}{7} \left (-\frac {\sqrt {-x^2} \left (2^{2/3} \left (1+\sqrt {3}\right ) \int \frac {1}{3 \sqrt {3} \sqrt {-x^2}}d\sqrt [3]{4-27 x^2}-\int \frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{3 \sqrt {3} \sqrt {-x^2}}d\sqrt [3]{4-27 x^2}\right )}{\sqrt {3} x}-\frac {5}{36} i \left (4-27 x^2\right )^{2/3}\right )-\frac {1}{21} i (2+3 i x) \left (4-27 x^2\right )^{2/3}\) |
\(\Big \downarrow \) 760 |
\(\displaystyle \frac {12}{7} \left (-\frac {\sqrt {-x^2} \left (-\int \frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{3 \sqrt {3} \sqrt {-x^2}}d\sqrt [3]{4-27 x^2}-\frac {2 \sqrt [3]{2} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right ),-7+4 \sqrt {3}\right )}{3\ 3^{3/4} \sqrt {-x^2} \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}}\right )}{\sqrt {3} x}-\frac {5}{36} i \left (4-27 x^2\right )^{2/3}\right )-\frac {1}{21} i (2+3 i x) \left (4-27 x^2\right )^{2/3}\) |
\(\Big \downarrow \) 2418 |
\(\displaystyle \frac {12}{7} \left (-\frac {\sqrt {-x^2} \left (-\frac {2 \sqrt [3]{2} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right ),-7+4 \sqrt {3}\right )}{3\ 3^{3/4} \sqrt {-x^2} \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}}+\frac {\sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-x^2} \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}}-\frac {6 \sqrt {3} \sqrt {-x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right )}{\sqrt {3} x}-\frac {5}{36} i \left (4-27 x^2\right )^{2/3}\right )-\frac {1}{21} i (2+3 i x) \left (4-27 x^2\right )^{2/3}\) |
(-1/21*I)*(2 + (3*I)*x)*(4 - 27*x^2)^(2/3) + (12*(((-5*I)/36)*(4 - 27*x^2) ^(2/3) - (Sqrt[-x^2]*((-6*Sqrt[3]*Sqrt[-x^2])/(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^(1/4)*Sqrt[-x^2]*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^(1/3)*Sqr t[2 - Sqrt[3]]*(1 + 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]*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]])/(3*3^(3/4)*Sqrt[-x^2]*Sqrt[-((2^(2/3) - (4 - 27*x^2)^(1/3))/(2^(2/3)* (1 - Sqrt[3]) - (4 - 27*x^2)^(1/3))^2)])))/(Sqrt[3]*x)))/7
3.8.7.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Simp[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]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[1/(b *(n + 2*p + 1)) Int[(c + d*x)^(n - 2)*(a + b*x^2)^p*Simp[b*c^2*(n + 2*p + 1) - a*d^2*(n - 1) + 2*b*c*d*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, n , p}, x] && If[RationalQ[n], GtQ[n, 1], SumSimplerQ[n, -2]] && NeQ[n + 2*p + 1, 0] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
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]*Sqrt[(- 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]
Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] ], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 + Sqrt[3]))*(s/r) Int[1/Sqrt[a + b*x ^3], x], x] + Simp[1/r Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x ]] /; FreeQ[{a, b}, x] && NegQ[a]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[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] + S imp[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 - S qrt[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*Sqrt[3])*a*d^3, 0]
Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 2.37 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.08
method | result | size |
risch | \(-\frac {\left (-7 i+3 x \right ) \left (27 x^{2}-4\right )}{21 \left (-27 x^{2}+4\right )^{\frac {1}{3}}}+\frac {12 \,2^{\frac {1}{3}} x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};\frac {27 x^{2}}{4}\right )}{7}\) | \(43\) |
meijerg | \(2 \,2^{\frac {1}{3}} x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};\frac {27 x^{2}}{4}\right )+3 i 2^{\frac {1}{3}} x^{2} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},1;2;\frac {27 x^{2}}{4}\right )-\frac {3 \,2^{\frac {1}{3}} x^{3} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},\frac {3}{2};\frac {5}{2};\frac {27 x^{2}}{4}\right )}{2}\) | \(58\) |
-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)
\[ \int \frac {(2+3 i x)^2}{\sqrt [3]{4-27 x^2}} \, dx=\int { \frac {{\left (3 i \, x + 2\right )}^{2}}{{\left (-27 \, x^{2} + 4\right )}^{\frac {1}{3}}} \,d x } \]
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
Time = 2.35 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.13 \[ \int \frac {(2+3 i x)^2}{\sqrt [3]{4-27 x^2}} \, dx=- \frac {3 \cdot \sqrt [3]{2} x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {27 x^{2} e^{2 i \pi }}{4}} \right )}}{2} + 2 \cdot \sqrt [3]{2} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {27 x^{2} e^{2 i \pi }}{4}} \right )} - \frac {i \left (4 - 27 x^{2}\right )^{\frac {2}{3}}}{3} \]
-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
\[ \int \frac {(2+3 i x)^2}{\sqrt [3]{4-27 x^2}} \, dx=\int { \frac {{\left (3 i \, x + 2\right )}^{2}}{{\left (-27 \, x^{2} + 4\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {(2+3 i x)^2}{\sqrt [3]{4-27 x^2}} \, dx=\int { \frac {{\left (3 i \, x + 2\right )}^{2}}{{\left (-27 \, x^{2} + 4\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {(2+3 i x)^2}{\sqrt [3]{4-27 x^2}} \, dx=\int \frac {{\left (2+x\,3{}\mathrm {i}\right )}^2}{{\left (4-27\,x^2\right )}^{1/3}} \,d x \]