Integrand size = 26, antiderivative size = 170 \[ \int \frac {e+f x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=\frac {2 \left (e-2^{2/3} f\right ) \text {arctanh}\left (\frac {\sqrt {3} \left (1+\sqrt [3]{2} x\right )}{\sqrt {-1-x^3}}\right )}{3 \sqrt {3}}+\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{2} e+f\right ) (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}} \]
2/9*(e-2^(2/3)*f)*arctanh((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)),2*I-I*3^(1/2)) *(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 = 20.39 (sec) , antiderivative size = 342, normalized size of antiderivative = 2.01 \[ \int \frac {e+f x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=\frac {2 \sqrt [6]{2} \sqrt {\frac {i (1+x)}{3 i+\sqrt {3}}} \left (f \sqrt {-i+\sqrt {3}+2 i x} \left (-6-3 \sqrt [3]{2}-2 i \sqrt {3}+i \sqrt [3]{2} \sqrt {3}+\left (3 \sqrt [3]{2}+4 i \sqrt {3}+i \sqrt [3]{2} \sqrt {3}\right ) x\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}-2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )-2 \sqrt {3} \left (\sqrt [3]{2} e-2 f\right ) \sqrt {i+\sqrt {3}-2 i x} \sqrt {1-x+x^2} \operatorname {EllipticPi}\left (\frac {2 \sqrt {3}}{i+2 i 2^{2/3}+\sqrt {3}},\arcsin \left (\frac {\sqrt {i+\sqrt {3}-2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )\right )}{\sqrt {3} \left (i+2 i 2^{2/3}+\sqrt {3}\right ) \sqrt {i+\sqrt {3}-2 i x} \sqrt {-1-x^3}} \]
(2*2^(1/6)*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*(f*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])] - 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]/(Sq rt[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 = 0.47 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.04, 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.
| \(\displaystyle \int \frac {e+f x}{\left (x+2^{2/3}\right ) \sqrt {-x^3-1}} \, dx\) |
| \(\Big \downarrow \) 2564 |
| \(\displaystyle \frac {1}{3} \left (\sqrt [3]{2} e+f\right ) \int \frac {1}{\sqrt {-x^3-1}}dx+\frac {1}{6} \left (\sqrt [3]{2} e-2 f\right ) \int \frac {2^{2/3}-2 x}{\left (x+2^{2/3}\right ) \sqrt {-x^3-1}}dx\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle \frac {1}{6} \left (\sqrt [3]{2} e-2 f\right ) \int \frac {2^{2/3}-2 x}{\left (x+2^{2/3}\right ) \sqrt {-x^3-1}}dx+\frac {2 \sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \left (\sqrt [3]{2} e+f\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}\) |
| \(\Big \downarrow \) 2562 |
| \(\displaystyle \frac {1}{3} 2^{2/3} \left (\sqrt [3]{2} e-2 f\right ) \int \frac {1}{1-\frac {3 \left (\sqrt [3]{2} x+1\right )^2}{-x^3-1}}d\frac {\sqrt [3]{2} x+1}{\sqrt {-x^3-1}}+\frac {2 \sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \left (\sqrt [3]{2} e+f\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 \sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \left (\sqrt [3]{2} e+f\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}+\frac {2^{2/3} \text {arctanh}\left (\frac {\sqrt {3} \left (\sqrt [3]{2} x+1\right )}{\sqrt {-x^3-1}}\right ) \left (\sqrt [3]{2} e-2 f\right )}{3 \sqrt {3}}\) |
(2^(2/3)*(2^(1/3)*e - 2*f)*ArcTanh[(Sqrt[3]*(1 + 2^(1/3)*x))/Sqrt[-1 - x^3 ]])/(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)] *Sqrt[-1 - x^3])
3.1.59.3.1 Defintions of rubi rules used
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 = 1.93 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.50
| method | result | size |
| default | \(-\frac {2 i f \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}}-\frac {2 i \left (e -2^{\frac {2}{3}} f \right ) \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{2^{\frac {2}{3}}+\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}\, \left (2^{\frac {2}{3}}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}\) | \(255\) |
| elliptic | \(-\frac {2 i f \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}}-\frac {2 i \left (e -2^{\frac {2}{3}} f \right ) \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{2^{\frac {2}{3}}+\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}\, \left (2^{\frac {2}{3}}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}\) | \(255\) |
-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))*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)/(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 = 1002, normalized size of antiderivative = 5.89 \[ \int \frac {e+f x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=\text {Too large to display} \]
[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 + 53 12*e*f*x^7 - 5408*e^2*x^6 + 9216*f^2*x^5 - 512*e*f*x^4 - 2560*e^2*x^3 + 46 08*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))*s qrt(-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...
\[ \int \frac {e+f x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=\int \frac {e + f x}{\sqrt {- \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x + 2^{\frac {2}{3}}\right )}\, dx \]
\[ \int \frac {e+f x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=\int { \frac {f x + e}{\sqrt {-x^{3} - 1} {\left (x + 2^{\frac {2}{3}}\right )}} \,d x } \]
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,[1]%%%} / %%%{%%{[1,0,0]:[1,0,0,-2]%%},[1]%%%} Error: Ba d Argumen
Timed out. \[ \int \frac {e+f x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=\int \frac {e+f\,x}{\sqrt {-x^3-1}\,\left (x+2^{2/3}\right )} \,d x \]