Integrand size = 26, antiderivative size = 169 \[ \int \frac {2+3 x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=\frac {2 \left (2-3\ 2^{2/3}\right ) \text {arctanh}\left (\frac {\sqrt {3} \left (1+\sqrt [3]{2} x\right )}{\sqrt {-1-x^3}}\right )}{3 \sqrt {3}}+\frac {2 \left (3+2 \sqrt [3]{2}\right ) \sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}} \] Output:
2/9*(2-3*2^(2/3))*arctanh(3^(1/2)*(1+2^(1/3)*x)/(-x^3-1)^(1/2))*3^(1/2)+2/ 9*(3+2*2^(1/3))*(1/2*6^(1/2)-1/2*2^(1/2))*(1+x)*((x^2-x+1)/(1+x-3^(1/2))^2 )^(1/2)*EllipticF((1+x+3^(1/2))/(1+x-3^(1/2)),2*I-I*3^(1/2))*3^(3/4)/(-(1+ x)/(1+x-3^(1/2))^2)^(1/2)/(-x^3-1)^(1/2)
Result contains complex when optimal does not.
Time = 20.61 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.00 \[ \int \frac {2+3 x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=\frac {2 \sqrt [6]{2} \sqrt {\frac {i (1+x)}{3 i+\sqrt {3}}} \left (3 \sqrt {-i+\sqrt {3}+2 i x} \left (-6-3 \sqrt [3]{2}-2 i \sqrt {3}+i \sqrt [3]{2} \sqrt {3}+\left (3 \sqrt [3]{2}+4 i \sqrt {3}+i \sqrt [3]{2} \sqrt {3}\right ) x\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}-2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )-4 \sqrt {3} \left (-3+\sqrt [3]{2}\right ) \sqrt {i+\sqrt {3}-2 i x} \sqrt {1-x+x^2} \operatorname {EllipticPi}\left (\frac {2 \sqrt {3}}{i+2 i 2^{2/3}+\sqrt {3}},\arcsin \left (\frac {\sqrt {i+\sqrt {3}-2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )\right )}{\sqrt {3} \left (i+2 i 2^{2/3}+\sqrt {3}\right ) \sqrt {i+\sqrt {3}-2 i x} \sqrt {-1-x^3}} \] Input:
Integrate[(2 + 3*x)/((2^(2/3) + x)*Sqrt[-1 - x^3]),x]
Output:
(2*2^(1/6)*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*(3*Sqrt[-I + Sqrt[3] + (2*I)* x]*(-6 - 3*2^(1/3) - (2*I)*Sqrt[3] + I*2^(1/3)*Sqrt[3] + (3*2^(1/3) + (4*I )*Sqrt[3] + I*2^(1/3)*Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (2*I )*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] - 4*Sqrt[3]*(-3 + 2^ (1/3))*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 - x + x^2]*EllipticPi[(2*Sqrt[3] )/(I + (2*I)*2^(2/3) + Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[ 2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/(Sqrt[3]*(I + (2*I)*2^(2/3) + Sqrt[3])*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[-1 - x^3])
Time = 0.75 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2564, 760, 2562, 219}
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.
| \(\displaystyle \int \frac {3 x+2}{\left (x+2^{2/3}\right ) \sqrt {-x^3-1}} \, dx\) |
| \(\Big \downarrow \) 2564 |
| \(\displaystyle \frac {1}{3} \left (3+2 \sqrt [3]{2}\right ) \int \frac {1}{\sqrt {-x^3-1}}dx-\frac {1}{3} \left (3-\sqrt [3]{2}\right ) \int \frac {2^{2/3}-2 x}{\left (x+2^{2/3}\right ) \sqrt {-x^3-1}}dx\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle \frac {2 \left (3+2 \sqrt [3]{2}\right ) \sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}-\frac {1}{3} \left (3-\sqrt [3]{2}\right ) \int \frac {2^{2/3}-2 x}{\left (x+2^{2/3}\right ) \sqrt {-x^3-1}}dx\) |
| \(\Big \downarrow \) 2562 |
| \(\displaystyle \frac {2 \left (3+2 \sqrt [3]{2}\right ) \sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}-\frac {2}{3} 2^{2/3} \left (3-\sqrt [3]{2}\right ) \int \frac {1}{1-\frac {3 \left (\sqrt [3]{2} x+1\right )^2}{-x^3-1}}d\frac {\sqrt [3]{2} x+1}{\sqrt {-x^3-1}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 \left (3+2 \sqrt [3]{2}\right ) \sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}-\frac {2\ 2^{2/3} \left (3-\sqrt [3]{2}\right ) \text {arctanh}\left (\frac {\sqrt {3} \left (\sqrt [3]{2} x+1\right )}{\sqrt {-x^3-1}}\right )}{3 \sqrt {3}}\) |
Input:
Int[(2 + 3*x)/((2^(2/3) + x)*Sqrt[-1 - x^3]),x]
Output:
(-2*2^(2/3)*(3 - 2^(1/3))*ArcTanh[(Sqrt[3]*(1 + 2^(1/3)*x))/Sqrt[-1 - x^3] ])/(3*Sqrt[3]) + (2*(3 + 2*2^(1/3))*Sqrt[2 - Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 - Sqrt[3] + x)^2]*EllipticF[ArcSin[(1 + Sqrt[3] + x)/(1 - Sqrt[3 ] + x)], -7 + 4*Sqrt[3]])/(3*3^(1/4)*Sqrt[-((1 + x)/(1 - Sqrt[3] + x)^2)]* Sqrt[-1 - x^3])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ Symbol] :> Simp[2*(e/d) Subst[Int[1/(1 + 3*a*x^2), x], x, (1 + 2*d*(x/c)) /Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b*c^3 - 4*a*d^3, 0] && EqQ[2*d*e + c*f, 0]
Int[((e_.) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x _Symbol] :> Simp[(2*d*e + c*f)/(3*c*d) Int[1/Sqrt[a + b*x^3], x], x] + Si mp[(d*e - c*f)/(3*c*d) Int[(c - 2*d*x)/((c + d*x)*Sqrt[a + b*x^3]), x], x ] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && (EqQ[b*c^3 - 4*a* d^3, 0] || EqQ[b*c^3 + 8*a*d^3, 0]) && NeQ[2*d*e + c*f, 0]
Time = 3.31 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.50
| method | result | size |
| default | \(-\frac {2 i \left (2-3 \,2^{\frac {2}{3}}\right ) \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}+2^{\frac {2}{3}}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}\, \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}+2^{\frac {2}{3}}\right )}-\frac {2 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-x^{3}-1}}\) | \(253\) |
| elliptic | \(-\frac {2 i \left (2-3 \,2^{\frac {2}{3}}\right ) \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}+2^{\frac {2}{3}}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}\, \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}+2^{\frac {2}{3}}\right )}-\frac {2 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-x^{3}-1}}\) | \(253\) |
Input:
int((2+3*x)/(2^(2/3)+x)/(-x^3-1)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/3*I*(2-3*2^(2/3))*3^(1/2)*(I*(x-1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*((x+1 )/(3/2+1/2*I*3^(1/2)))^(1/2)*(-I*(x-1/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(-x^ 3-1)^(1/2)/(1/2+1/2*I*3^(1/2)+2^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x-1/2-1/ 2*I*3^(1/2))*3^(1/2))^(1/2),I*3^(1/2)/(1/2+1/2*I*3^(1/2)+2^(2/3)),(I*3^(1/ 2)/(3/2+1/2*I*3^(1/2)))^(1/2))-2*I*3^(1/2)*(I*(x-1/2-1/2*I*3^(1/2))*3^(1/2 ))^(1/2)*((x+1)/(3/2+1/2*I*3^(1/2)))^(1/2)*(-I*(x-1/2+1/2*I*3^(1/2))*3^(1/ 2))^(1/2)/(-x^3-1)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x-1/2-1/2*I*3^(1/2))*3^ (1/2))^(1/2),(I*3^(1/2)/(3/2+1/2*I*3^(1/2)))^(1/2))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (130) = 260\).
Time = 0.13 (sec) , antiderivative size = 423, normalized size of antiderivative = 2.50 \[ \int \frac {2+3 x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=-\frac {2}{3} \, {\left (2 i \cdot 2^{\frac {1}{3}} + 3 i\right )} {\rm weierstrassPInverse}\left (0, -4, x\right ) + \frac {1}{6} \, \sqrt {-4 \cdot 2^{\frac {2}{3}} + 6 \cdot 2^{\frac {1}{3}} + \frac {4}{3}} \log \left (\frac {25 \, x^{18} - 36000 \, x^{15} + 435000 \, x^{12} + 526400 \, x^{9} - 259200 \, x^{6} - 384000 \, x^{3} + 6 \, {\left (6 \, x^{16} - 34 \, x^{15} + 1134 \, x^{14} - 1860 \, x^{13} + 2116 \, x^{12} - 23976 \, x^{11} + 13992 \, x^{10} - 5056 \, x^{9} + 15936 \, x^{7} - 10816 \, x^{6} + 41472 \, x^{5} - 1536 \, x^{4} - 5120 \, x^{3} + 20736 \, x^{2} + 3 \cdot 2^{\frac {2}{3}} {\left (3 \, x^{16} - 17 \, x^{15} + 42 \, x^{14} - 930 \, x^{13} + 1058 \, x^{12} - 888 \, x^{11} + 6996 \, x^{10} - 2528 \, x^{9} + 7968 \, x^{7} - 5408 \, x^{6} + 1536 \, x^{5} - 768 \, x^{4} - 2560 \, x^{3} + 768 \, x^{2} - 1536 \, x - 512\right )} + 2^{\frac {1}{3}} {\left (2 \, x^{16} - 153 \, x^{15} + 378 \, x^{14} - 620 \, x^{13} + 9522 \, x^{12} - 7992 \, x^{11} + 4664 \, x^{10} - 22752 \, x^{9} + 5312 \, x^{7} - 48672 \, x^{6} + 13824 \, x^{5} - 512 \, x^{4} - 23040 \, x^{3} + 6912 \, x^{2} - 1024 \, x - 4608\right )} - 3072 \, x - 1024\right )} \sqrt {-x^{3} - 1} \sqrt {-4 \cdot 2^{\frac {2}{3}} + 6 \cdot 2^{\frac {1}{3}} + \frac {4}{3}} - 600 \cdot 2^{\frac {2}{3}} {\left (x^{17} - 121 \, x^{14} + 478 \, x^{11} + 1144 \, x^{8} + 608 \, x^{5} + 64 \, x^{2}\right )} + 1200 \cdot 2^{\frac {1}{3}} {\left (5 \, x^{16} - 176 \, x^{13} + 83 \, x^{10} + 680 \, x^{7} + 544 \, x^{4} + 128 \, x\right )} - 51200}{x^{18} + 24 \, x^{15} + 240 \, x^{12} + 1280 \, x^{9} + 3840 \, x^{6} + 6144 \, x^{3} + 4096}\right ) \] Input:
integrate((2+3*x)/(2^(2/3)+x)/(-x^3-1)^(1/2),x, algorithm="fricas")
Output:
-2/3*(2*I*2^(1/3) + 3*I)*weierstrassPInverse(0, -4, x) + 1/6*sqrt(-4*2^(2/ 3) + 6*2^(1/3) + 4/3)*log((25*x^18 - 36000*x^15 + 435000*x^12 + 526400*x^9 - 259200*x^6 - 384000*x^3 + 6*(6*x^16 - 34*x^15 + 1134*x^14 - 1860*x^13 + 2116*x^12 - 23976*x^11 + 13992*x^10 - 5056*x^9 + 15936*x^7 - 10816*x^6 + 41472*x^5 - 1536*x^4 - 5120*x^3 + 20736*x^2 + 3*2^(2/3)*(3*x^16 - 17*x^15 + 42*x^14 - 930*x^13 + 1058*x^12 - 888*x^11 + 6996*x^10 - 2528*x^9 + 7968* x^7 - 5408*x^6 + 1536*x^5 - 768*x^4 - 2560*x^3 + 768*x^2 - 1536*x - 512) + 2^(1/3)*(2*x^16 - 153*x^15 + 378*x^14 - 620*x^13 + 9522*x^12 - 7992*x^11 + 4664*x^10 - 22752*x^9 + 5312*x^7 - 48672*x^6 + 13824*x^5 - 512*x^4 - 230 40*x^3 + 6912*x^2 - 1024*x - 4608) - 3072*x - 1024)*sqrt(-x^3 - 1)*sqrt(-4 *2^(2/3) + 6*2^(1/3) + 4/3) - 600*2^(2/3)*(x^17 - 121*x^14 + 478*x^11 + 11 44*x^8 + 608*x^5 + 64*x^2) + 1200*2^(1/3)*(5*x^16 - 176*x^13 + 83*x^10 + 6 80*x^7 + 544*x^4 + 128*x) - 51200)/(x^18 + 24*x^15 + 240*x^12 + 1280*x^9 + 3840*x^6 + 6144*x^3 + 4096))
\[ \int \frac {2+3 x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=\int \frac {3 x + 2}{\sqrt {- \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x + 2^{\frac {2}{3}}\right )}\, dx \] Input:
integrate((2+3*x)/(2**(2/3)+x)/(-x**3-1)**(1/2),x)
Output:
Integral((3*x + 2)/(sqrt(-(x + 1)*(x**2 - x + 1))*(x + 2**(2/3))), x)
\[ \int \frac {2+3 x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=\int { \frac {3 \, x + 2}{\sqrt {-x^{3} - 1} {\left (x + 2^{\frac {2}{3}}\right )}} \,d x } \] Input:
integrate((2+3*x)/(2^(2/3)+x)/(-x^3-1)^(1/2),x, algorithm="maxima")
Output:
integrate((3*x + 2)/(sqrt(-x^3 - 1)*(x + 2^(2/3))), x)
\[ \int \frac {2+3 x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=\int { \frac {3 \, x + 2}{\sqrt {-x^{3} - 1} {\left (x + 2^{\frac {2}{3}}\right )}} \,d x } \] Input:
integrate((2+3*x)/(2^(2/3)+x)/(-x^3-1)^(1/2),x, algorithm="giac")
Output:
integrate((3*x + 2)/(sqrt(-x^3 - 1)*(x + 2^(2/3))), x)
Timed out. \[ \int \frac {2+3 x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=\int \frac {3\,x+2}{\sqrt {-x^3-1}\,\left (x+2^{2/3}\right )} \,d x \] Input:
int((3*x + 2)/((- x^3 - 1)^(1/2)*(x + 2^(2/3))),x)
Output:
int((3*x + 2)/((- x^3 - 1)^(1/2)*(x + 2^(2/3))), x)
\[ \int \frac {2+3 x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=i \left (-3 \left (\int \frac {x}{\sqrt {x^{3}+1}\, 2^{\frac {2}{3}}+\sqrt {x^{3}+1}\, x}d x \right )-2 \left (\int \frac {1}{\sqrt {x^{3}+1}\, 2^{\frac {2}{3}}+\sqrt {x^{3}+1}\, x}d x \right )\right ) \] Input:
int((2+3*x)/(2^(2/3)+x)/(-x^3-1)^(1/2),x)
Output:
i*( - 3*int(x/(sqrt(x**3 + 1)*2**(2/3) + sqrt(x**3 + 1)*x),x) - 2*int(1/(s qrt(x**3 + 1)*2**(2/3) + sqrt(x**3 + 1)*x),x))