Integrand size = 28, antiderivative size = 175 \[ \int \frac {e+f x}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}} \, dx=-\frac {2 \left (e+2^{2/3} f\right ) \arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {1-x^3}}\right )}{3 \sqrt {3}}-\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{2} e-f\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}-x}{1+\sqrt {3}-x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {1-x^3}} \]
-2/9*(e+2^(2/3)*f)*arctan((1-2^(1/3)*x)*3^(1/2)/(-x^3+1)^(1/2))*3^(1/2)-2/ 9*(2^(1/3)*e-f)*(1-x)*EllipticF((1-x-3^(1/2))/(1-x+3^(1/2)),I*3^(1/2)+2*I) *(1/2*6^(1/2)+1/2*2^(1/2))*((x^2+x+1)/(1-x+3^(1/2))^2)^(1/2)*3^(3/4)/(-x^3 +1)^(1/2)/((1-x)/(1-x+3^(1/2))^2)^(1/2)
Result contains complex when optimal does not.
Time = 20.42 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.94 \[ \int \frac {e+f x}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}} \, dx=\frac {2 \sqrt [6]{2} \sqrt {-\frac {i (-1+x)}{3 i+\sqrt {3}}} \left (-i f \sqrt {-i+\sqrt {3}-2 i x} \left (-6 i-3 i \sqrt [3]{2}+2 \sqrt {3}-\sqrt [3]{2} \sqrt {3}+\left (-3 i \sqrt [3]{2}+4 \sqrt {3}+\sqrt [3]{2} \sqrt {3}\right ) x\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}+2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )+2 \sqrt {3} \left (\sqrt [3]{2} e+2 f\right ) \sqrt {i+\sqrt {3}+2 i x} \sqrt {1+x+x^2} \operatorname {EllipticPi}\left (\frac {2 \sqrt {3}}{i+2 i 2^{2/3}+\sqrt {3}},\arcsin \left (\frac {\sqrt {i+\sqrt {3}+2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )\right )}{\sqrt {3} \left (i+2 i 2^{2/3}+\sqrt {3}\right ) \sqrt {i+\sqrt {3}+2 i x} \sqrt {1-x^3}} \]
(2*2^(1/6)*Sqrt[((-I)*(-1 + x))/(3*I + Sqrt[3])]*((-I)*f*Sqrt[-I + Sqrt[3] - (2*I)*x]*(-6*I - (3*I)*2^(1/3) + 2*Sqrt[3] - 2^(1/3)*Sqrt[3] + ((-3*I)* 2^(1/3) + 4*Sqrt[3] + 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])] + 2*Sqrt[3]* (2^(1/3)*e + 2*f)*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 = 0.48 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2564, 27, 759, 2562, 216}
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.
| \(\displaystyle \int \frac {e+f x}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}} \, dx\) |
| \(\Big \downarrow \) 2564 |
| \(\displaystyle \frac {1}{3} \left (\sqrt [3]{2} e-f\right ) \int \frac {1}{\sqrt {1-x^3}}dx+\frac {1}{6} \left (\sqrt [3]{2} e+2 f\right ) \int \frac {2^{2/3} \left (\sqrt [3]{2} x+1\right )}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\sqrt [3]{2} e-f\right ) \int \frac {1}{\sqrt {1-x^3}}dx+\frac {\left (\sqrt [3]{2} e+2 f\right ) \int \frac {\sqrt [3]{2} x+1}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}}dx}{3 \sqrt [3]{2}}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {\left (\sqrt [3]{2} e+2 f\right ) \int \frac {\sqrt [3]{2} x+1}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}}dx}{3 \sqrt [3]{2}}-\frac {2 \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (\sqrt [3]{2} e-f\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}\) |
| \(\Big \downarrow \) 2562 |
| \(\displaystyle -\frac {1}{3} 2^{2/3} \left (\sqrt [3]{2} e+2 f\right ) \int \frac {1}{\frac {3 \left (1-\sqrt [3]{2} x\right )^2}{1-x^3}+1}d\frac {1-\sqrt [3]{2} x}{\sqrt {1-x^3}}-\frac {2 \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (\sqrt [3]{2} e-f\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {2 \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (\sqrt [3]{2} e-f\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}-\frac {2^{2/3} \arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {1-x^3}}\right ) \left (\sqrt [3]{2} e+2 f\right )}{3 \sqrt {3}}\) |
-1/3*(2^(2/3)*(2^(1/3)*e + 2*f)*ArcTan[(Sqrt[3]*(1 - 2^(1/3)*x))/Sqrt[1 - x^3]])/Sqrt[3] - (2*Sqrt[2 + Sqrt[3]]*(2^(1/3)*e - f)*(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]*Sqr t[1 - x^3])
3.1.57.3.1 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), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[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] & & PosQ[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 = 1.96 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.49
| method | result | size |
| default | \(\frac {2 i f \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}-\frac {2 i \left (-e -2^{\frac {2}{3}} f \right ) \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-2^{\frac {2}{3}}-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}\, \left (-2^{\frac {2}{3}}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}\) | \(261\) |
| elliptic | \(\frac {2 i f \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}-\frac {2 i \left (-e -2^{\frac {2}{3}} f \right ) \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-2^{\frac {2}{3}}-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}\, \left (-2^{\frac {2}{3}}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}\) | \(261\) |
2/3*I*f*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)*E llipticF(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))-2/3*I*(-e-2^(2/3)*f)*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)/(-2^(2/3)-1/2+1/2*I*3^(1/2))*Ellipt icPi(1/3*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),I*3^(1/2)/(-2^(2/ 3)-1/2+1/2*I*3^(1/2)),(I*3^(1/2)/(-3/2+1/2*I*3^(1/2)))^(1/2))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.22 (sec) , antiderivative size = 1005, normalized size of antiderivative = 5.74 \[ \int \frac {e+f x}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}} \, dx=\text {Too large to display} \]
[1/18*sqrt(3)*sqrt(-2*2^(2/3)*e*f - 2*2^(1/3)*f^2 - e^2)*log(((e^3 + 4*f^3 )*x^18 + 1440*(e^3 + 4*f^3)*x^15 + 17400*(e^3 + 4*f^3)*x^12 - 21056*(e^3 + 4*f^3)*x^9 - 10368*(e^3 + 4*f^3)*x^6 + 15360*(e^3 + 4*f^3)*x^3 - 2048*e^3 - 8192*f^3 - 4*sqrt(3)*(2*e*f*x^16 - 17*e^2*x^15 - 252*f^2*x^14 + 620*e*f *x^13 - 1058*e^2*x^12 - 5328*f^2*x^11 + 4664*e*f*x^10 - 2528*e^2*x^9 - 531 2*e*f*x^7 + 5408*e^2*x^6 + 9216*f^2*x^5 - 512*e*f*x^4 - 2560*e^2*x^3 - 460 8*f^2*x^2 + 1024*e*f*x + 512*e^2 - 2^(2/3)*(2*f^2*x^16 - 17*e*f*x^15 + 63* e^2*x^14 + 620*f^2*x^13 - 1058*e*f*x^12 + 1332*e^2*x^11 + 4664*f^2*x^10 - 2528*e*f*x^9 - 5312*f^2*x^7 + 5408*e*f*x^6 - 2304*e^2*x^5 - 512*f^2*x^4 - 2560*e*f*x^3 + 1152*e^2*x^2 + 1024*f^2*x + 512*e*f) - 2^(1/3)*(e^2*x^16 + 34*f^2*x^15 - 126*e*f*x^14 + 310*e^2*x^13 + 2116*f^2*x^12 - 2664*e*f*x^11 + 2332*e^2*x^10 + 5056*f^2*x^9 - 2656*e^2*x^7 - 10816*f^2*x^6 + 4608*e*f*x ^5 - 256*e^2*x^4 + 5120*f^2*x^3 - 2304*e*f*x^2 + 512*e^2*x - 1024*f^2))*sq rt(-x^3 + 1)*sqrt(-2*2^(2/3)*e*f - 2*2^(1/3)*f^2 - e^2) + 24*2^(2/3)*((e^3 + 4*f^3)*x^17 + 121*(e^3 + 4*f^3)*x^14 + 478*(e^3 + 4*f^3)*x^11 - 1144*(e ^3 + 4*f^3)*x^8 + 608*(e^3 + 4*f^3)*x^5 - 64*(e^3 + 4*f^3)*x^2) + 48*2^(1/ 3)*(5*(e^3 + 4*f^3)*x^16 + 176*(e^3 + 4*f^3)*x^13 + 83*(e^3 + 4*f^3)*x^10 - 680*(e^3 + 4*f^3)*x^7 + 544*(e^3 + 4*f^3)*x^4 - 128*(e^3 + 4*f^3)*x))/(x ^18 - 24*x^15 + 240*x^12 - 1280*x^9 + 3840*x^6 - 6144*x^3 + 4096)) - 2/3*( I*2^(1/3)*e - I*f)*weierstrassPInverse(0, 4, x), -1/9*sqrt(3)*sqrt(2*2^...
\[ \int \frac {e+f x}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}} \, dx=- \int \frac {e}{x \sqrt {1 - x^{3}} - 2^{\frac {2}{3}} \sqrt {1 - x^{3}}}\, dx - \int \frac {f x}{x \sqrt {1 - x^{3}} - 2^{\frac {2}{3}} \sqrt {1 - x^{3}}}\, dx \]
-Integral(e/(x*sqrt(1 - x**3) - 2**(2/3)*sqrt(1 - x**3)), x) - Integral(f* x/(x*sqrt(1 - x**3) - 2**(2/3)*sqrt(1 - x**3)), x)
\[ \int \frac {e+f x}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}} \, dx=\int { -\frac {f x + e}{\sqrt {-x^{3} + 1} {\left (x - 2^{\frac {2}{3}}\right )}} \,d x } \]
Exception generated. \[ \int \frac {e+f x}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{1,[2]%%%} / %%%{%%{[2,0]:[1,0,0,-2]%%},[2]%%%} Error: Bad Argument
Timed out. \[ \int \frac {e+f x}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}} \, dx=\int -\frac {e+f\,x}{\sqrt {1-x^3}\,\left (x-2^{2/3}\right )} \,d x \]