Integrand size = 28, antiderivative size = 175 \[ \int \frac {e+f x}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}} \, dx=-\frac {2 \left (e+2^{2/3} f\right ) \arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {1-x^3}}\right )}{3 \sqrt {3}}-\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{2} e-f\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}-x}{1+\sqrt {3}-x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {1-x^3}} \] Output:
-2/9*(e+2^(2/3)*f)*arctan(3^(1/2)*(1-2^(1/3)*x)/(-x^3+1)^(1/2))*3^(1/2)-2/ 9*(1/2*6^(1/2)+1/2*2^(1/2))*(2^(1/3)*e-f)*(1-x)*((x^2+x+1)/(1+3^(1/2)-x)^2 )^(1/2)*EllipticF((1-3^(1/2)-x)/(1+3^(1/2)-x),I*3^(1/2)+2*I)*3^(3/4)/((1-x )/(1+3^(1/2)-x)^2)^(1/2)/(-x^3+1)^(1/2)
Result contains complex when optimal does not.
Time = 20.76 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.94 \[ \int \frac {e+f x}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}} \, dx=\frac {2 \sqrt [6]{2} \sqrt {-\frac {i (-1+x)}{3 i+\sqrt {3}}} \left (-i f \sqrt {-i+\sqrt {3}-2 i x} \left (-6 i-3 i \sqrt [3]{2}+2 \sqrt {3}-\sqrt [3]{2} \sqrt {3}+\left (-3 i \sqrt [3]{2}+4 \sqrt {3}+\sqrt [3]{2} \sqrt {3}\right ) x\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}+2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )+2 \sqrt {3} \left (\sqrt [3]{2} e+2 f\right ) \sqrt {i+\sqrt {3}+2 i x} \sqrt {1+x+x^2} \operatorname {EllipticPi}\left (\frac {2 \sqrt {3}}{i+2 i 2^{2/3}+\sqrt {3}},\arcsin \left (\frac {\sqrt {i+\sqrt {3}+2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )\right )}{\sqrt {3} \left (i+2 i 2^{2/3}+\sqrt {3}\right ) \sqrt {i+\sqrt {3}+2 i x} \sqrt {1-x^3}} \] Input:
Integrate[(e + f*x)/((2^(2/3) - x)*Sqrt[1 - x^3]),x]
Output:
(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 = 0.80 (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 |
\(\displaystyle -\frac {2 \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (\sqrt [3]{2} e-f\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}-\frac {2^{2/3} \arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {1-x^3}}\right ) \left (\sqrt [3]{2} e+2 f\right )}{3 \sqrt {3}}\) |
Input:
Int[(e + f*x)/((2^(2/3) - x)*Sqrt[1 - x^3]),x]
Output:
-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])
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 = 1.25 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.47
method | result | size |
default | \(\frac {2 i \left (e +2^{\frac {2}{3}} f \right ) \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}-2^{\frac {2}{3}}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}\, \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}-2^{\frac {2}{3}}\right )}+\frac {2 i f \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}\) | \(258\) |
elliptic | \(\frac {2 i f \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}-\frac {2 i \left (-e -2^{\frac {2}{3}} f \right ) \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}-2^{\frac {2}{3}}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}\, \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}-2^{\frac {2}{3}}\right )}\) | \(261\) |
Input:
int((f*x+e)/(2^(2/3)-x)/(-x^3+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
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)/(-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))+2/3*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)*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))
Time = 0.21 (sec) , antiderivative size = 997, normalized size of antiderivative = 5.70 \[ \int \frac {e+f x}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)/(2^(2/3)-x)/(-x^3+1)^(1/2),x, algorithm="fricas")
Output:
[-2/3*(I*2^(1/3)*e - I*f)*weierstrassPInverse(0, 4, x) + 1/6*sqrt(-2/3*2^( 2/3)*e*f - 2/3*2^(1/3)*f^2 - 1/3*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 - 12*(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 - 53 28*f^2*x^11 + 4664*e*f*x^10 - 2528*e^2*x^9 - 5312*e*f*x^7 + 5408*e^2*x^6 + 9216*f^2*x^5 - 512*e*f*x^4 - 2560*e^2*x^3 - 4608*f^2*x^2 + 1024*e*f*x + 5 12*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))*sqrt(-x^3 + 1)*sqrt(-2/3*2^( 2/3)*e*f - 2/3*2^(1/3)*f^2 - 1/3*e^2) + 24*2^(2/3)*((e^3 + 4*f^3)*x^17 + 1 21*(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 + 24 0*x^12 - 1280*x^9 + 3840*x^6 - 6144*x^3 + 4096)), -2/3*(I*2^(1/3)*e - I...
\[ \int \frac {e+f x}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}} \, dx=- \int \frac {e}{x \sqrt {1 - x^{3}} - 2^{\frac {2}{3}} \sqrt {1 - x^{3}}}\, dx - \int \frac {f x}{x \sqrt {1 - x^{3}} - 2^{\frac {2}{3}} \sqrt {1 - x^{3}}}\, dx \] Input:
integrate((f*x+e)/(2**(2/3)-x)/(-x**3+1)**(1/2),x)
Output:
-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)
\[ \int \frac {e+f x}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}} \, dx=\int { -\frac {f x + e}{\sqrt {-x^{3} + 1} {\left (x - 2^{\frac {2}{3}}\right )}} \,d x } \] Input:
integrate((f*x+e)/(2^(2/3)-x)/(-x^3+1)^(1/2),x, algorithm="maxima")
Output:
-integrate((f*x + e)/(sqrt(-x^3 + 1)*(x - 2^(2/3))), x)
Exception generated. \[ \int \frac {e+f x}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((f*x+e)/(2^(2/3)-x)/(-x^3+1)^(1/2),x, algorithm="giac")
Output:
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. \[ \int \frac {e+f x}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}} \, dx=\int -\frac {e+f\,x}{\sqrt {1-x^3}\,\left (x-2^{2/3}\right )} \,d x \] Input:
int(-(e + f*x)/((1 - x^3)^(1/2)*(x - 2^(2/3))),x)
Output:
int(-(e + f*x)/((1 - x^3)^(1/2)*(x - 2^(2/3))), x)
\[ \int \frac {e+f x}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}} \, dx=\left (\int \frac {x}{\sqrt {-x^{3}+1}\, 2^{\frac {2}{3}}-\sqrt {-x^{3}+1}\, x}d x \right ) f +\left (\int \frac {1}{\sqrt {-x^{3}+1}\, 2^{\frac {2}{3}}-\sqrt {-x^{3}+1}\, x}d x \right ) e \] Input:
int((f*x+e)/(2^(2/3)-x)/(-x^3+1)^(1/2),x)
Output:
int(x/(sqrt( - x**3 + 1)*2**(2/3) - sqrt( - x**3 + 1)*x),x)*f + int(1/(sqr t( - x**3 + 1)*2**(2/3) - sqrt( - x**3 + 1)*x),x)*e