\(\int \frac {2+3 x}{(2^{2/3}-x) \sqrt {1-x^3}} \, dx\) [99]

Optimal result

Integrand size = 28, antiderivative size = 173 \[ \int \frac {2+3 x}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}} \, dx=-\frac {2 \left (2+3\ 2^{2/3}\right ) \arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {1-x^3}}\right )}{3 \sqrt {3}}+\frac {2 \left (3-2 \sqrt [3]{2}\right ) \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}-x}{1+\sqrt {3}-x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {1-x^3}} \] Output:

-2/9*(2+3*2^(2/3))*arctan(3^(1/2)*(1-2^(1/3)*x)/(-x^3+1)^(1/2))*3^(1/2)+2/ 
9*(3-2*2^(1/3))*(1/2*6^(1/2)+1/2*2^(1/2))*(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)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 20.66 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.94 \[ \int \frac {2+3 x}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}} \, dx=\frac {2 \sqrt [6]{2} \sqrt {-\frac {i (-1+x)}{3 i+\sqrt {3}}} \left (-3 i \sqrt {-i+\sqrt {3}-2 i x} \left (-6 i-3 i \sqrt [3]{2}+2 \sqrt {3}-\sqrt [3]{2} \sqrt {3}+\left (-3 i \sqrt [3]{2}+4 \sqrt {3}+\sqrt [3]{2} \sqrt {3}\right ) x\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}+2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )+4 \sqrt {3} \left (3+\sqrt [3]{2}\right ) \sqrt {i+\sqrt {3}+2 i x} \sqrt {1+x+x^2} \operatorname {EllipticPi}\left (\frac {2 \sqrt {3}}{i+2 i 2^{2/3}+\sqrt {3}},\arcsin \left (\frac {\sqrt {i+\sqrt {3}+2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )\right )}{\sqrt {3} \left (i+2 i 2^{2/3}+\sqrt {3}\right ) \sqrt {i+\sqrt {3}+2 i x} \sqrt {1-x^3}} \] Input:

Integrate[(2 + 3*x)/((2^(2/3) - x)*Sqrt[1 - x^3]),x]
 

Output:

(2*2^(1/6)*Sqrt[((-I)*(-1 + x))/(3*I + Sqrt[3])]*((-3*I)*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])] + 4*Sqrt[3]* 
(3 + 2^(1/3))*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])
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.02, 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.

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

\(\Big \downarrow \) 2564

\(\displaystyle \frac {1}{3} \left (3+\sqrt [3]{2}\right ) \int \frac {2^{2/3} \left (\sqrt [3]{2} x+1\right )}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}}dx-\frac {1}{3} \left (3-2 \sqrt [3]{2}\right ) \int \frac {1}{\sqrt {1-x^3}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} 2^{2/3} \left (3+\sqrt [3]{2}\right ) \int \frac {\sqrt [3]{2} x+1}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}}dx-\frac {1}{3} \left (3-2 \sqrt [3]{2}\right ) \int \frac {1}{\sqrt {1-x^3}}dx\)

\(\Big \downarrow \) 759

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

\(\Big \downarrow \) 2562

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

\(\Big \downarrow \) 216

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

Input:

Int[(2 + 3*x)/((2^(2/3) - x)*Sqrt[1 - x^3]),x]
 

Output:

(-2*2^(2/3)*(3 + 2^(1/3))*ArcTan[(Sqrt[3]*(1 - 2^(1/3)*x))/Sqrt[1 - x^3]]) 
/(3*Sqrt[3]) + (2*(3 - 2*2^(1/3))*Sqrt[2 + Sqrt[3]]*(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])
 

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 2562

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]
 

rule 2564

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]
 

Maple [A] (verified)

Time = 3.83 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.49

method	result	size
default	\(\frac {2 i \left (2+3 \,2^{\frac {2}{3}}\right ) \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}-2^{\frac {2}{3}}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}\, \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}-2^{\frac {2}{3}}\right )}+\frac {2 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-x^{3}+1}}\)	\(257\)
elliptic	\(\frac {2 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-x^{3}+1}}-\frac {2 i \left (-2-3 \,2^{\frac {2}{3}}\right ) \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}-2^{\frac {2}{3}}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}\, \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}-2^{\frac {2}{3}}\right )}\)	\(257\)

Input:

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

Output:

2/3*I*(2+3*2^(2/3))*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*I*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))
 

Fricas [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.89 \[ \int \frac {2+3 x}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}} \, dx=-\frac {2}{3} \, {\left (2 i \cdot 2^{\frac {1}{3}} - 3 i\right )} {\rm weierstrassPInverse}\left (0, 4, x\right ) - \frac {1}{3} \, \sqrt {4 \cdot 2^{\frac {2}{3}} + 6 \cdot 2^{\frac {1}{3}} + \frac {4}{3}} \arctan \left (\frac {{\left (18 \, x^{5} - 42 \, x^{4} - 10 \, x^{3} - 18 \, x^{2} + 2^{\frac {2}{3}} {\left (2 \, x^{5} + 63 \, x^{4} + 15 \, x^{3} - 2 \, x^{2} - 36 \, x - 6\right )} - 2^{\frac {1}{3}} {\left (6 \, x^{5} - 14 \, x^{4} + 45 \, x^{3} - 6 \, x^{2} + 8 \, x - 18\right )} + 24 \, x + 4\right )} \sqrt {-x^{3} + 1} \sqrt {4 \cdot 2^{\frac {2}{3}} + 6 \cdot 2^{\frac {1}{3}} + \frac {4}{3}}}{116 \, {\left (2 \, x^{6} - 3 \, x^{3} + 1\right )}}\right ) \] Input:

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

Output:

-2/3*(2*I*2^(1/3) - 3*I)*weierstrassPInverse(0, 4, x) - 1/3*sqrt(4*2^(2/3) 
 + 6*2^(1/3) + 4/3)*arctan(1/116*(18*x^5 - 42*x^4 - 10*x^3 - 18*x^2 + 2^(2 
/3)*(2*x^5 + 63*x^4 + 15*x^3 - 2*x^2 - 36*x - 6) - 2^(1/3)*(6*x^5 - 14*x^4 
 + 45*x^3 - 6*x^2 + 8*x - 18) + 24*x + 4)*sqrt(-x^3 + 1)*sqrt(4*2^(2/3) + 
6*2^(1/3) + 4/3)/(2*x^6 - 3*x^3 + 1))
 

Sympy [F]

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

integrate((2+3*x)/(2**(2/3)-x)/(-x**3+1)**(1/2),x)
 

Output:

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

Maxima [F]

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

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

Output:

-integrate((3*x + 2)/(sqrt(-x^3 + 1)*(x - 2^(2/3))), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {2+3 x}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((2+3*x)/(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
 

Mupad [F(-1)]

Timed out. \[ \int \frac {2+3 x}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}} \, dx=\int -\frac {3\,x+2}{\sqrt {1-x^3}\,\left (x-2^{2/3}\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

3*int(x/(sqrt( - x**3 + 1)*2**(2/3) - sqrt( - x**3 + 1)*x),x) + 2*int(1/(s 
qrt( - x**3 + 1)*2**(2/3) - sqrt( - x**3 + 1)*x),x)