\(\int \frac {1-\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\sqrt {a-b x^3}} \, dx\) [88]

Optimal result

Integrand size = 34, antiderivative size = 248 \[ \int \frac {1-\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\sqrt {a-b x^3}} \, dx=-\frac {2 \left (\frac {b}{a}\right )^{2/3} \sqrt {a-b x^3}}{b \left (1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x\right )}+\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1-\sqrt [3]{\frac {b}{a}} x\right ) \sqrt {\frac {1+\sqrt [3]{\frac {b}{a}} x+\left (\frac {b}{a}\right )^{2/3} x^2}{\left (1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x\right )^2}} E\left (\arcsin \left (\frac {1-\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{\frac {b}{a}} \sqrt {\frac {1-\sqrt [3]{\frac {b}{a}} x}{\left (1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x\right )^2}} \sqrt {a-b x^3}} \] Output:

-2*(b/a)^(2/3)*(-b*x^3+a)^(1/2)/b/(1+3^(1/2)-(b/a)^(1/3)*x)+3^(1/4)*(1/2*6 
^(1/2)-1/2*2^(1/2))*(1-(b/a)^(1/3)*x)*((1+(b/a)^(1/3)*x+(b/a)^(2/3)*x^2)/( 
1+3^(1/2)-(b/a)^(1/3)*x)^2)^(1/2)*EllipticE((1-3^(1/2)-(b/a)^(1/3)*x)/(1+3 
^(1/2)-(b/a)^(1/3)*x),I*3^(1/2)+2*I)/(b/a)^(1/3)/((1-(b/a)^(1/3)*x)/(1+3^( 
1/2)-(b/a)^(1/3)*x)^2)^(1/2)/(-b*x^3+a)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.04 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.36 \[ \int \frac {1-\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\sqrt {a-b x^3}} \, dx=-\frac {x \sqrt {1-\frac {b x^3}{a}} \left (2 \left (-1+\sqrt {3}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\frac {b x^3}{a}\right )+\sqrt [3]{\frac {b}{a}} x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\frac {b x^3}{a}\right )\right )}{2 \sqrt {a-b x^3}} \] Input:

Integrate[(1 - Sqrt[3] - (b/a)^(1/3)*x)/Sqrt[a - b*x^3],x]
 

Output:

-1/2*(x*Sqrt[1 - (b*x^3)/a]*(2*(-1 + Sqrt[3])*Hypergeometric2F1[1/3, 1/2, 
4/3, (b*x^3)/a] + (b/a)^(1/3)*x*Hypergeometric2F1[1/2, 2/3, 5/3, (b*x^3)/a 
]))/Sqrt[a - b*x^3]
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {2416}

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.

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

\(\Big \downarrow \) 2416

\(\displaystyle \frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1-x \sqrt [3]{\frac {b}{a}}\right ) \sqrt {\frac {x^2 \left (\frac {b}{a}\right )^{2/3}+x \sqrt [3]{\frac {b}{a}}+1}{\left (x \left (-\sqrt [3]{\frac {b}{a}}\right )+\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{\frac {b}{a}} x-\sqrt {3}+1}{-\sqrt [3]{\frac {b}{a}} x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{\frac {b}{a}} \sqrt {\frac {1-x \sqrt [3]{\frac {b}{a}}}{\left (x \left (-\sqrt [3]{\frac {b}{a}}\right )+\sqrt {3}+1\right )^2}} \sqrt {a-b x^3}}-\frac {2 \left (\frac {b}{a}\right )^{2/3} \sqrt {a-b x^3}}{b \left (x \left (-\sqrt [3]{\frac {b}{a}}\right )+\sqrt {3}+1\right )}\)

Input:

Int[(1 - Sqrt[3] - (b/a)^(1/3)*x)/Sqrt[a - b*x^3],x]
 

Output:

(-2*(b/a)^(2/3)*Sqrt[a - b*x^3])/(b*(1 + Sqrt[3] - (b/a)^(1/3)*x)) + (3^(1 
/4)*Sqrt[2 - Sqrt[3]]*(1 - (b/a)^(1/3)*x)*Sqrt[(1 + (b/a)^(1/3)*x + (b/a)^ 
(2/3)*x^2)/(1 + Sqrt[3] - (b/a)^(1/3)*x)^2]*EllipticE[ArcSin[(1 - Sqrt[3] 
- (b/a)^(1/3)*x)/(1 + Sqrt[3] - (b/a)^(1/3)*x)], -7 - 4*Sqrt[3]])/((b/a)^( 
1/3)*Sqrt[(1 - (b/a)^(1/3)*x)/(1 + Sqrt[3] - (b/a)^(1/3)*x)^2]*Sqrt[a - b* 
x^3])
 

Defintions of rubi rules used

rule 2416

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 - Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 - Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x] - S 
imp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 + Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt 
[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3]) 
*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && Eq 
Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 949 vs. \(2 (203 ) = 406\).

Time = 0.62 (sec) , antiderivative size = 950, normalized size of antiderivative = 3.83

method	result	size
default	\(\text {Expression too large to display}\)	\(950\)

Input:

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

Output:

-2*I/b*(a*b^2)^(1/3)*(-I*(x+1/2/b*(a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(a*b^2)^(1 
/3))*3^(1/2)*b/(a*b^2)^(1/3))^(1/2)*((x-1/b*(a*b^2)^(1/3))/(-3/2/b*(a*b^2) 
^(1/3)-1/2*I*3^(1/2)/b*(a*b^2)^(1/3)))^(1/2)*(I*(x+1/2/b*(a*b^2)^(1/3)-1/2 
*I*3^(1/2)/b*(a*b^2)^(1/3))*3^(1/2)*b/(a*b^2)^(1/3))^(1/2)/(-b*x^3+a)^(1/2 
)*EllipticF(1/3*3^(1/2)*(-I*(x+1/2/b*(a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(a*b^2) 
^(1/3))*3^(1/2)*b/(a*b^2)^(1/3))^(1/2),(-I*3^(1/2)/b*(a*b^2)^(1/3)/(-3/2/b 
*(a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(a*b^2)^(1/3)))^(1/2))-2/3*I*(b/a)^(1/3)*3^ 
(1/2)/b*(a*b^2)^(1/3)*(-I*(x+1/2/b*(a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(a*b^2)^( 
1/3))*3^(1/2)*b/(a*b^2)^(1/3))^(1/2)*((x-1/b*(a*b^2)^(1/3))/(-3/2/b*(a*b^2 
)^(1/3)-1/2*I*3^(1/2)/b*(a*b^2)^(1/3)))^(1/2)*(I*(x+1/2/b*(a*b^2)^(1/3)-1/ 
2*I*3^(1/2)/b*(a*b^2)^(1/3))*3^(1/2)*b/(a*b^2)^(1/3))^(1/2)/(-b*x^3+a)^(1/ 
2)*((-3/2/b*(a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(a*b^2)^(1/3))*EllipticE(1/3*3^( 
1/2)*(-I*(x+1/2/b*(a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(a*b^2)^(1/3))*3^(1/2)*b/( 
a*b^2)^(1/3))^(1/2),(-I*3^(1/2)/b*(a*b^2)^(1/3)/(-3/2/b*(a*b^2)^(1/3)-1/2* 
I*3^(1/2)/b*(a*b^2)^(1/3)))^(1/2))+1/b*(a*b^2)^(1/3)*EllipticF(1/3*3^(1/2) 
*(-I*(x+1/2/b*(a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(a*b^2)^(1/3))*3^(1/2)*b/(a*b^ 
2)^(1/3))^(1/2),(-I*3^(1/2)/b*(a*b^2)^(1/3)/(-3/2/b*(a*b^2)^(1/3)-1/2*I*3^ 
(1/2)/b*(a*b^2)^(1/3)))^(1/2)))+2/3*I*3^(1/2)/b*(a*b^2)^(1/3)*(-I*(x+1/2/b 
*(a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(a*b^2)^(1/3))*3^(1/2)*b/(a*b^2)^(1/3))^(1/ 
2)*((x-1/b*(a*b^2)^(1/3))/(-3/2/b*(a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(a*b^2)...
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.23 \[ \int \frac {1-\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\sqrt {a-b x^3}} \, dx=\frac {2 \, {\left (\sqrt {-b} {\left (\sqrt {3} - 1\right )} {\rm weierstrassPInverse}\left (0, \frac {4 \, a}{b}, x\right ) - \sqrt {-b} \left (\frac {b}{a}\right )^{\frac {1}{3}} {\rm weierstrassZeta}\left (0, \frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, \frac {4 \, a}{b}, x\right )\right )\right )}}{b} \] Input:

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

Output:

2*(sqrt(-b)*(sqrt(3) - 1)*weierstrassPInverse(0, 4*a/b, x) - sqrt(-b)*(b/a 
)^(1/3)*weierstrassZeta(0, 4*a/b, weierstrassPInverse(0, 4*a/b, x)))/b
 

Sympy [A] (verification not implemented)

Time = 1.62 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.52 \[ \int \frac {1-\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\sqrt {a-b x^3}} \, dx=- \frac {x^{2} \sqrt [3]{\frac {b}{a}} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{2 i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {5}{3}\right )} - \frac {\sqrt {3} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{2 i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {4}{3}\right )} + \frac {x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{2 i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {4}{3}\right )} \] Input:

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

Output:

-x**2*(b/a)**(1/3)*gamma(2/3)*hyper((1/2, 2/3), (5/3,), b*x**3*exp_polar(2 
*I*pi)/a)/(3*sqrt(a)*gamma(5/3)) - sqrt(3)*x*gamma(1/3)*hyper((1/3, 1/2), 
(4/3,), b*x**3*exp_polar(2*I*pi)/a)/(3*sqrt(a)*gamma(4/3)) + x*gamma(1/3)* 
hyper((1/3, 1/2), (4/3,), b*x**3*exp_polar(2*I*pi)/a)/(3*sqrt(a)*gamma(4/3 
))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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