Integrand size = 24, antiderivative size = 159 \[ \int \frac {e+f x}{\left (2^{2/3}+x\right ) \sqrt {1+x^3}} \, dx=\frac {2 \left (e-2^{2/3} f\right ) \arctan \left (\frac {\sqrt {3} \left (1+\sqrt [3]{2} x\right )}{\sqrt {1+x^3}}\right )}{3 \sqrt {3}}+\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{2} e+f\right ) (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}} \]
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 = 20.37 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.14 \[ \int \frac {e+f x}{\left (2^{2/3}+x\right ) \sqrt {1+x^3}} \, dx=\frac {2 \sqrt [6]{2} \sqrt {\frac {i (1+x)}{3 i+\sqrt {3}}} \left (f \sqrt {-i+\sqrt {3}+2 i x} \left (-6-3 \sqrt [3]{2}-2 i \sqrt {3}+i \sqrt [3]{2} \sqrt {3}+\left (3 \sqrt [3]{2}+4 i \sqrt {3}+i \sqrt [3]{2} \sqrt {3}\right ) x\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}-2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )-2 \sqrt {3} \left (\sqrt [3]{2} e-2 f\right ) \sqrt {i+\sqrt {3}-2 i x} \sqrt {1-x+x^2} \operatorname {EllipticPi}\left (\frac {2 \sqrt {3}}{i+2 i 2^{2/3}+\sqrt {3}},\arcsin \left (\frac {\sqrt {i+\sqrt {3}-2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )\right )}{\sqrt {3} \left (i+2 i 2^{2/3}+\sqrt {3}\right ) \sqrt {i+\sqrt {3}-2 i x} \sqrt {1+x^3}} \]
(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 = 0.46 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2564, 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.
| \(\displaystyle \int \frac {e+f x}{\left (x+2^{2/3}\right ) \sqrt {x^3+1}} \, dx\) |
| \(\Big \downarrow \) 2564 |
| \(\displaystyle \frac {1}{3} \left (\sqrt [3]{2} e+f\right ) \int \frac {1}{\sqrt {x^3+1}}dx+\frac {1}{6} \left (\sqrt [3]{2} e-2 f\right ) \int \frac {2^{2/3}-2 x}{\left (x+2^{2/3}\right ) \sqrt {x^3+1}}dx\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {1}{6} \left (\sqrt [3]{2} e-2 f\right ) \int \frac {2^{2/3}-2 x}{\left (x+2^{2/3}\right ) \sqrt {x^3+1}}dx+\frac {2 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (\sqrt [3]{2} e+f\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 2562 |
| \(\displaystyle \frac {1}{3} 2^{2/3} \left (\sqrt [3]{2} e-2 f\right ) \int \frac {1}{\frac {3 \left (\sqrt [3]{2} x+1\right )^2}{x^3+1}+1}d\frac {\sqrt [3]{2} x+1}{\sqrt {x^3+1}}+\frac {2 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (\sqrt [3]{2} e+f\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {2 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (\sqrt [3]{2} e+f\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {2^{2/3} \arctan \left (\frac {\sqrt {3} \left (\sqrt [3]{2} x+1\right )}{\sqrt {x^3+1}}\right ) \left (\sqrt [3]{2} e-2 f\right )}{3 \sqrt {3}}\) |
(2^(2/3)*(2^(1/3)*e - 2*f)*ArcTan[(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])
3.1.56.3.1 Defintions of rubi rules used
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (126 ) = 252\).
Time = 1.96 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.66
| method | result | size |
| default | \(\frac {2 f \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\frac {2 \left (e -2^{\frac {2}{3}} f \right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \Pi \left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{2^{\frac {2}{3}}-1}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}\, \left (2^{\frac {2}{3}}-1\right )}\) | \(264\) |
| elliptic | \(\frac {2 f \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\frac {2 \left (e -2^{\frac {2}{3}} f \right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \Pi \left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{2^{\frac {2}{3}}-1}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}\, \left (2^{\frac {2}{3}}-1\right )}\) | \(264\) |
2*f*(3/2-1/2*I*3^(1/2))*((x+1)/(3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2-1/2*I*3^ (1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2+1/2*I*3^(1/2))/(-3/2+1/2*I*3^(1 /2)))^(1/2)/(x^3+1)^(1/2)*EllipticF(((x+1)/(3/2-1/2*I*3^(1/2)))^(1/2),((-3 /2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))+2*(e-2^(2/3)*f)*(3/2-1/2*I* 3^(1/2))*((x+1)/(3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2-1/2*I*3^(1/2))/(-3/2-1/ 2*I*3^(1/2)))^(1/2)*((x-1/2+1/2*I*3^(1/2))/(-3/2+1/2*I*3^(1/2)))^(1/2)/(x^ 3+1)^(1/2)/(2^(2/3)-1)*EllipticPi(((x+1)/(3/2-1/2*I*3^(1/2)))^(1/2),(-3/2+ 1/2*I*3^(1/2))/(2^(2/3)-1),((-3/2+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 = 0.24 (sec) , antiderivative size = 992, normalized size of antiderivative = 6.24 \[ \int \frac {e+f x}{\left (2^{2/3}+x\right ) \sqrt {1+x^3}} \, dx=\text {Too large to display} \]
[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^1 8 + 24*x^15 + 240*x^12 + 1280*x^9 + 3840*x^6 + 6144*x^3 + 4096)) + 2/3*(2^ (1/3)*e + f)*weierstrassPInverse(0, -4, x), 1/9*sqrt(3)*sqrt(-2*2^(2/3)...
\[ \int \frac {e+f x}{\left (2^{2/3}+x\right ) \sqrt {1+x^3}} \, dx=\int \frac {e + f x}{\sqrt {\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x + 2^{\frac {2}{3}}\right )}\, dx \]
\[ \int \frac {e+f x}{\left (2^{2/3}+x\right ) \sqrt {1+x^3}} \, dx=\int { \frac {f x + e}{\sqrt {x^{3} + 1} {\left (x + 2^{\frac {2}{3}}\right )}} \,d x } \]
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. \[ \int \frac {e+f x}{\left (2^{2/3}+x\right ) \sqrt {1+x^3}} \, dx=\int \frac {e+f\,x}{\sqrt {x^3+1}\,\left (x+2^{2/3}\right )} \,d x \]