Integrand size = 19, antiderivative size = 531 \[ \int \frac {2+3 i x}{\sqrt [3]{4-27 x^2}} \, dx=-\frac {1}{12} i \left (4-27 x^2\right )^{2/3}-\frac {6 x}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}-\frac {\sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4-27 x^2}+\left (4-27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt {3}\right )}{3\ 3^{3/4} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}}+\frac {2\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4-27 x^2}+\left (4-27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right ),-7+4 \sqrt {3}\right )}{9 \sqrt [4]{3} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}} \]
-1/12*I*(-27*x^2+4)^(2/3)-6*x/(-(-27*x^2+4)^(1/3)+2^(2/3)*(1-3^(1/2)))+2/2 7*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)-(-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 = 9.88 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.08 \[ \int \frac {2+3 i x}{\sqrt [3]{4-27 x^2}} \, dx=-\frac {1}{12} i \left (4-27 x^2\right )^{2/3}+\sqrt [3]{2} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},\frac {27 x^2}{4}\right ) \]
Time = 0.46 (sec) , antiderivative size = 596, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {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.
| \(\displaystyle \int \frac {2+3 i x}{\sqrt [3]{4-27 x^2}} \, dx\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle 2 \int \frac {1}{\sqrt [3]{4-27 x^2}}dx-\frac {1}{12} i \left (4-27 x^2\right )^{2/3}\) |
| \(\Big \downarrow \) 233 |
| \(\displaystyle -\frac {\sqrt {-x^2} \int \frac {\sqrt [3]{4-27 x^2}}{3 \sqrt {3} \sqrt {-x^2}}d\sqrt [3]{4-27 x^2}}{\sqrt {3} x}-\frac {1}{12} i \left (4-27 x^2\right )^{2/3}\) |
| \(\Big \downarrow \) 833 |
| \(\displaystyle -\frac {\sqrt {-x^2} \left (2^{2/3} \left (1+\sqrt {3}\right ) \int \frac {1}{3 \sqrt {3} \sqrt {-x^2}}d\sqrt [3]{4-27 x^2}-\int \frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{3 \sqrt {3} \sqrt {-x^2}}d\sqrt [3]{4-27 x^2}\right )}{\sqrt {3} x}-\frac {1}{12} i \left (4-27 x^2\right )^{2/3}\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle -\frac {\sqrt {-x^2} \left (-\int \frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{3 \sqrt {3} \sqrt {-x^2}}d\sqrt [3]{4-27 x^2}-\frac {2 \sqrt [3]{2} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right ),-7+4 \sqrt {3}\right )}{3\ 3^{3/4} \sqrt {-x^2} \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}}\right )}{\sqrt {3} x}-\frac {1}{12} i \left (4-27 x^2\right )^{2/3}\) |
| \(\Big \downarrow \) 2418 |
| \(\displaystyle -\frac {\sqrt {-x^2} \left (-\frac {2 \sqrt [3]{2} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right ),-7+4 \sqrt {3}\right )}{3\ 3^{3/4} \sqrt {-x^2} \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}}+\frac {\sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-x^2} \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}}-\frac {6 \sqrt {3} \sqrt {-x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right )}{\sqrt {3} x}-\frac {1}{12} i \left (4-27 x^2\right )^{2/3}\) |
(-1/12*I)*(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]*Elliptic E[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)*Sqrt[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)
3.8.8.3.1 Defintions of rubi rules used
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[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 = 2.49 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.07
| method | result | size |
| risch | \(\frac {i \left (27 x^{2}-4\right )}{12 \left (-27 x^{2}+4\right )^{\frac {1}{3}}}+2^{\frac {1}{3}} x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};\frac {27 x^{2}}{4}\right )\) | \(37\) |
| meijerg | \(2^{\frac {1}{3}} x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};\frac {27 x^{2}}{4}\right )+\frac {3 i 2^{\frac {1}{3}} x^{2} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},1;2;\frac {27 x^{2}}{4}\right )}{4}\) | \(38\) |
\[ \int \frac {2+3 i x}{\sqrt [3]{4-27 x^2}} \, dx=\int { \frac {3 i \, x + 2}{{\left (-27 \, x^{2} + 4\right )}^{\frac {1}{3}}} \,d x } \]
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
Time = 1.39 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.07 \[ \int \frac {2+3 i x}{\sqrt [3]{4-27 x^2}} \, dx=\sqrt [3]{2} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {27 x^{2} e^{2 i \pi }}{4}} \right )} - \frac {i \left (4 - 27 x^{2}\right )^{\frac {2}{3}}}{12} \]
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
\[ \int \frac {2+3 i x}{\sqrt [3]{4-27 x^2}} \, dx=\int { \frac {3 i \, x + 2}{{\left (-27 \, x^{2} + 4\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {2+3 i x}{\sqrt [3]{4-27 x^2}} \, dx=\int { \frac {3 i \, x + 2}{{\left (-27 \, x^{2} + 4\right )}^{\frac {1}{3}}} \,d x } \]
Time = 9.93 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.05 \[ \int \frac {2+3 i x}{\sqrt [3]{4-27 x^2}} \, dx=2^{1/3}\,x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {1}{2};\ \frac {3}{2};\ \frac {27\,x^2}{4}\right )-\frac {{\left (4-27\,x^2\right )}^{2/3}\,1{}\mathrm {i}}{12} \]