\(\int \frac {e+f x}{(1+\sqrt {3}+x) \sqrt {1+x^3}} \, dx\) [173]

Optimal result

Integrand size = 25, antiderivative size = 173 \[ \int \frac {e+f x}{\left (1+\sqrt {3}+x\right ) \sqrt {1+x^3}} \, dx=\frac {\left (e-f-\sqrt {3} f\right ) \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} (1+x)}{\sqrt {1+x^3}}\right )}{\sqrt {3 \left (3+2 \sqrt {3}\right )}}+\frac {\sqrt {2+\sqrt {3}} \left (e-\left (1-\sqrt {3}\right ) f\right ) (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}} \] Output:

(e-f-3^(1/2)*f)*arctan((3+2*3^(1/2))^(1/2)*(1+x)/(x^3+1)^(1/2))/(9+6*3^(1/ 
2))^(1/2)+1/3*(1/2*6^(1/2)+1/2*2^(1/2))*(e-(1-3^(1/2))*f)*(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)),I*3^(1/2)+2* 
I)*3^(1/4)/((1+x)/(1+x+3^(1/2))^2)^(1/2)/(x^3+1)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 20.53 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.68 \[ \int \frac {e+f x}{\left (1+\sqrt {3}+x\right ) \sqrt {1+x^3}} \, dx=\frac {2 \sqrt {\frac {2}{3}} \sqrt {\frac {i (1+x)}{3 i+\sqrt {3}}} \left (3 f \sqrt {-i+\sqrt {3}+2 i x} \left ((-2-i)-\sqrt {3}+\left ((1+2 i)+i \sqrt {3}\right ) x\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}-2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )+2 \left (-\sqrt {3} e+\left (3+\sqrt {3}\right ) f\right ) \sqrt {i+\sqrt {3}-2 i x} \sqrt {1-x+x^2} \operatorname {EllipticPi}\left (\frac {2 \sqrt {3}}{3 i+(1+2 i) \sqrt {3}},\arcsin \left (\frac {\sqrt {i+\sqrt {3}-2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )\right )}{\left (3 i+(1+2 i) \sqrt {3}\right ) \sqrt {i+\sqrt {3}-2 i x} \sqrt {1+x^3}} \] Input:

Integrate[(e + f*x)/((1 + Sqrt[3] + x)*Sqrt[1 + x^3]),x]
 

Output:

(2*Sqrt[2/3]*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*(3*f*Sqrt[-I + Sqrt[3] + (2 
*I)*x]*((-2 - I) - Sqrt[3] + ((1 + 2*I) + I*Sqrt[3])*x)*EllipticF[ArcSin[S 
qrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3]) 
] + 2*(-(Sqrt[3]*e) + (3 + Sqrt[3])*f)*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 
- x + x^2]*EllipticPi[(2*Sqrt[3])/(3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[I 
 + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/( 
(3*I + (1 + 2*I)*Sqrt[3])*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 + x^3])
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2566, 27, 759, 2565, 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.

\(\displaystyle \int \frac {e+f x}{\left (x+\sqrt {3}+1\right ) \sqrt {x^3+1}} \, dx\)

\(\Big \downarrow \) 2566

\(\displaystyle \frac {\left (e-\left (1-\sqrt {3}\right ) f\right ) \int \frac {1}{\sqrt {x^3+1}}dx}{2 \sqrt {3}}-\frac {1}{12} \left (\frac {e-f}{\sqrt {3}}-f\right ) \int \frac {6 \left (x-\sqrt {3}+1\right )}{\left (x+\sqrt {3}+1\right ) \sqrt {x^3+1}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (e-\left (1-\sqrt {3}\right ) f\right ) \int \frac {1}{\sqrt {x^3+1}}dx}{2 \sqrt {3}}-\frac {1}{2} \left (\frac {e-f}{\sqrt {3}}-f\right ) \int \frac {x-\sqrt {3}+1}{\left (x+\sqrt {3}+1\right ) \sqrt {x^3+1}}dx\)

\(\Big \downarrow \) 759

\(\displaystyle \frac {\sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (e-\left (1-\sqrt {3}\right ) f\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {1}{2} \left (\frac {e-f}{\sqrt {3}}-f\right ) \int \frac {x-\sqrt {3}+1}{\left (x+\sqrt {3}+1\right ) \sqrt {x^3+1}}dx\)

\(\Big \downarrow \) 2565

\(\displaystyle \left (\frac {e-f}{\sqrt {3}}-f\right ) \int \frac {1}{\frac {\left (3+2 \sqrt {3}\right ) (x+1)^2}{x^3+1}+1}d\frac {x+1}{\sqrt {x^3+1}}+\frac {\sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (e-\left (1-\sqrt {3}\right ) f\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (e-\left (1-\sqrt {3}\right ) f\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {\arctan \left (\frac {\sqrt {3+2 \sqrt {3}} (x+1)}{\sqrt {x^3+1}}\right ) \left (\frac {e-f}{\sqrt {3}}-f\right )}{\sqrt {3+2 \sqrt {3}}}\)

Input:

Int[(e + f*x)/((1 + Sqrt[3] + x)*Sqrt[1 + x^3]),x]
 

Output:

(((e - f)/Sqrt[3] - f)*ArcTan[(Sqrt[3 + 2*Sqrt[3]]*(1 + x))/Sqrt[1 + x^3]] 
)/Sqrt[3 + 2*Sqrt[3]] + (Sqrt[2 + Sqrt[3]]*(e - (1 - Sqrt[3])*f)*(1 + x)*S 
qrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)/ 
(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(3^(3/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + 
x)^2]*Sqrt[1 + x^3])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 216

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])
 

rule 759

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]
 

rule 2565

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ 
Symbol] :> With[{k = Simplify[(d*e + 2*c*f)/(c*f)]}, Simp[(1 + k)*(e/d)   S 
ubst[Int[1/(1 + (3 + 2*k)*a*x^2), x], x, (1 + (1 + k)*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^2*c 
^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0] && EqQ[6*a*d^4*e - c*f*(b*c^3 - 22*a*d^ 
3), 0]
 

rule 2566

Int[((e_.) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x 
_Symbol] :> Simp[-(6*a*d^4*e - c*f*(b*c^3 - 22*a*d^3))/(c*d*(b*c^3 - 28*a*d 
^3))   Int[1/Sqrt[a + b*x^3], x], x] + Simp[(d*e - c*f)/(c*d*(b*c^3 - 28*a* 
d^3))   Int[(c*(b*c^3 - 22*a*d^3) + 6*a*d^4*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^2*c^6 
 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0] && NeQ[6*a*d^4*e - c*f*(b*c^3 - 22*a*d^3) 
, 0]
 

Maple [A] (verified)

Time = 1.08 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.50

method	result	size
default	\(\frac {2 f \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\frac {2 \left (e -f -\sqrt {3}\, f \right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}+1}}\)	\(260\)
elliptic	\(\frac {2 f \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\frac {2 \left (e -f -\sqrt {3}\, f \right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}+1}}\)	\(260\)

Input:

int((f*x+e)/(1+3^(1/2)+x)/(x^3+1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

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))+2/3*(e-f-3^(1/2)*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)*3^(1/2)*EllipticPi(((x+1)/(3/2-1/2*I*3^(1/2)))^(1/2),1/3*(- 
3/2+1/2*I*3^(1/2))*3^(1/2),((-3/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/ 
2))
 

Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 716, normalized size of antiderivative = 4.14 \[ \int \frac {e+f x}{\left (1+\sqrt {3}+x\right ) \sqrt {1+x^3}} \, dx =\text {Too large to display} \] Input:

integrate((f*x+e)/(1+3^(1/2)+x)/(x^3+1)^(1/2),x, algorithm="fricas")
 

Output:

[1/3*(sqrt(3)*(e - f) + 3*f)*weierstrassPInverse(0, -4, x) + 1/12*sqrt(3*e 
^2 + 6*e*f - 2*sqrt(3)*(e^2 + e*f + f^2))*log(-((e^2 - 2*e*f - 2*f^2)*x^8 
- 16*(e^2 - 2*e*f - 2*f^2)*x^7 + 112*(e^2 - 2*e*f - 2*f^2)*x^6 - 16*(e^2 - 
 2*e*f - 2*f^2)*x^5 + 112*(e^2 - 2*e*f - 2*f^2)*x^4 + 224*(e^2 - 2*e*f - 2 
*f^2)*x^3 + 64*(e^2 - 2*e*f - 2*f^2)*x^2 - 4*((2*e + f)*x^6 - 18*(e + f)*x 
^5 + 6*(7*e + 2*f)*x^4 - 8*(e + 5*f)*x^3 - 36*f*x^2 + 24*(e - f)*x + sqrt( 
3)*((e + f)*x^6 - 6*(2*e + f)*x^5 + 6*(3*e + 4*f)*x^4 - 8*(2*e - f)*x^3 - 
12*(e - f)*x^2 + 24*f*x - 8*e + 16*f) + 8*e - 32*f)*sqrt(x^3 + 1)*sqrt(3*e 
^2 + 6*e*f - 2*sqrt(3)*(e^2 + e*f + f^2)) + 112*e^2 - 224*e*f - 224*f^2 + 
128*(e^2 - 2*e*f - 2*f^2)*x - 16*sqrt(3)*((e^2 - 2*e*f - 2*f^2)*x^7 - 2*(e 
^2 - 2*e*f - 2*f^2)*x^6 + 6*(e^2 - 2*e*f - 2*f^2)*x^5 + 5*(e^2 - 2*e*f - 2 
*f^2)*x^4 + 2*(e^2 - 2*e*f - 2*f^2)*x^3 + 6*(e^2 - 2*e*f - 2*f^2)*x^2 + 4* 
e^2 - 8*e*f - 8*f^2 + 4*(e^2 - 2*e*f - 2*f^2)*x))/(x^8 + 8*x^7 + 16*x^6 - 
16*x^5 - 56*x^4 + 32*x^3 + 64*x^2 - 64*x + 16)), 1/3*(sqrt(3)*(e - f) + 3* 
f)*weierstrassPInverse(0, -4, x) - 1/6*sqrt(-3*e^2 - 6*e*f + 2*sqrt(3)*(e^ 
2 + e*f + f^2))*arctan(1/6*(3*f*x^2 - 6*(e + f)*x + sqrt(3)*((e - f)*x^2 - 
 2*(2*e + f)*x - 2*e - 4*f) - 6*e)*sqrt(x^3 + 1)*sqrt(-3*e^2 - 6*e*f + 2*s 
qrt(3)*(e^2 + e*f + f^2))/((e^2 - 2*e*f - 2*f^2)*x^3 + e^2 - 2*e*f - 2*f^2 
))]
 

Sympy [F]

\[ \int \frac {e+f x}{\left (1+\sqrt {3}+x\right ) \sqrt {1+x^3}} \, dx=\int \frac {e + f x}{\sqrt {\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x + 1 + \sqrt {3}\right )}\, dx \] Input:

integrate((f*x+e)/(1+3**(1/2)+x)/(x**3+1)**(1/2),x)
 

Output:

Integral((e + f*x)/(sqrt((x + 1)*(x**2 - x + 1))*(x + 1 + sqrt(3))), x)
 

Maxima [F]

\[ \int \frac {e+f x}{\left (1+\sqrt {3}+x\right ) \sqrt {1+x^3}} \, dx=\int { \frac {f x + e}{\sqrt {x^{3} + 1} {\left (x + \sqrt {3} + 1\right )}} \,d x } \] Input:

integrate((f*x+e)/(1+3^(1/2)+x)/(x^3+1)^(1/2),x, algorithm="maxima")
 

Output:

integrate((f*x + e)/(sqrt(x^3 + 1)*(x + sqrt(3) + 1)), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {e+f x}{\left (1+\sqrt {3}+x\right ) \sqrt {1+x^3}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((f*x+e)/(1+3^(1/2)+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]%%%} / %%%{%%{[2,4]:[1,0,-3]%%},[2]%%%} Error: Bad Ar 
gument Va
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e+f x}{\left (1+\sqrt {3}+x\right ) \sqrt {1+x^3}} \, dx=\text {Hanged} \] Input:

int((e + f*x)/((x^3 + 1)^(1/2)*(x + 3^(1/2) + 1)),x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {e+f x}{\left (1+\sqrt {3}+x\right ) \sqrt {1+x^3}} \, dx=\frac {\sqrt {3}\, \left (\int \frac {\sqrt {x^{3}+1}}{x^{5}+2 x^{4}-2 x^{3}+x^{2}+2 x -2}d x \right ) e +\sqrt {3}\, \left (\int \frac {\sqrt {x^{3}+1}\, x^{2}}{x^{5}+2 x^{4}-2 x^{3}+x^{2}+2 x -2}d x \right ) f +\sqrt {3}\, \left (\int \frac {\sqrt {x^{3}+1}\, x}{x^{5}+2 x^{4}-2 x^{3}+x^{2}+2 x -2}d x \right ) e +\sqrt {3}\, \left (\int \frac {\sqrt {x^{3}+1}\, x}{x^{5}+2 x^{4}-2 x^{3}+x^{2}+2 x -2}d x \right ) f -3 \left (\int \frac {\sqrt {x^{3}+1}}{x^{5}+2 x^{4}-2 x^{3}+x^{2}+2 x -2}d x \right ) e -3 \left (\int \frac {\sqrt {x^{3}+1}\, x}{x^{5}+2 x^{4}-2 x^{3}+x^{2}+2 x -2}d x \right ) f}{\sqrt {3}} \] Input:

int((f*x+e)/(1+3^(1/2)+x)/(x^3+1)^(1/2),x)
 

Output:

(sqrt(3)*int(sqrt(x**3 + 1)/(x**5 + 2*x**4 - 2*x**3 + x**2 + 2*x - 2),x)*e 
 + sqrt(3)*int((sqrt(x**3 + 1)*x**2)/(x**5 + 2*x**4 - 2*x**3 + x**2 + 2*x 
- 2),x)*f + sqrt(3)*int((sqrt(x**3 + 1)*x)/(x**5 + 2*x**4 - 2*x**3 + x**2 
+ 2*x - 2),x)*e + sqrt(3)*int((sqrt(x**3 + 1)*x)/(x**5 + 2*x**4 - 2*x**3 + 
 x**2 + 2*x - 2),x)*f - 3*int(sqrt(x**3 + 1)/(x**5 + 2*x**4 - 2*x**3 + x** 
2 + 2*x - 2),x)*e - 3*int((sqrt(x**3 + 1)*x)/(x**5 + 2*x**4 - 2*x**3 + x** 
2 + 2*x - 2),x)*f)/sqrt(3)