\(\int \frac {(1+\sqrt {3}) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {-a-b x^3}} \, dx\) [70]

Optimal result

Integrand size = 36, antiderivative size = 266 \[ \int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {-a-b x^3}} \, dx=-\frac {2 \sqrt {-a-b x^3}}{\sqrt [3]{b} \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7+4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {-a-b x^3}} \] Output:

-2*(-b*x^3-a)^(1/2)/b^(1/3)/((1-3^(1/2))*a^(1/3)+b^(1/3)*x)+3^(1/4)*(1/2*6 
^(1/2)+1/2*2^(1/2))*a^(1/3)*(a^(1/3)+b^(1/3)*x)*((a^(2/3)-a^(1/3)*b^(1/3)* 
x+b^(2/3)*x^2)/((1-3^(1/2))*a^(1/3)+b^(1/3)*x)^2)^(1/2)*EllipticE(((1+3^(1 
/2))*a^(1/3)+b^(1/3)*x)/((1-3^(1/2))*a^(1/3)+b^(1/3)*x),2*I-I*3^(1/2))/b^( 
1/3)/(-a^(1/3)*(a^(1/3)+b^(1/3)*x)/((1-3^(1/2))*a^(1/3)+b^(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 = 10.05 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.35 \[ \int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {-a-b x^3}} \, dx=\frac {x \sqrt {1+\frac {b x^3}{a}} \left (2 \left (1+\sqrt {3}\right ) \sqrt [3]{a} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a}\right )+\sqrt [3]{b} x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )\right )}{2 \sqrt {-a-b x^3}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {2418}

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.

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

\(\Big \downarrow \) 2418

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2418

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 - S 
qrt[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] && NegQ[a] && 
 EqQ[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. 1011 vs. \(2 (201 ) = 402\).

Time = 0.66 (sec) , antiderivative size = 1012, normalized size of antiderivative = 3.80

method	result	size
default	\(\text {Expression too large to display}\)	\(1012\)

Input:

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

Output:

-2/3*I*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)*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*I*a^(1/3)/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^(2/3)*3^(1/2)*(-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)-...
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.21 \[ \int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {-a-b x^3}} \, dx=-\frac {2 \, {\left (a^{\frac {1}{3}} \sqrt {-b} {\left (\sqrt {3} + 1\right )} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - \sqrt {-b} b^{\frac {1}{3}} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right )\right )}}{b} \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 2.26 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.48 \[ \int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {-a-b x^3}} \, dx=- \frac {i \sqrt [3]{b} x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {5}{3}\right )} - \frac {\sqrt {3} i x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [6]{a} \Gamma \left (\frac {4}{3}\right )} - \frac {i x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [6]{a} \Gamma \left (\frac {4}{3}\right )} \] Input:

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

Output:

-I*b**(1/3)*x**2*gamma(2/3)*hyper((1/2, 2/3), (5/3,), b*x**3*exp_polar(I*p 
i)/a)/(3*sqrt(a)*gamma(5/3)) - sqrt(3)*I*x*gamma(1/3)*hyper((1/3, 1/2), (4 
/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(1/6)*gamma(4/3)) - I*x*gamma(1/3)*h 
yper((1/3, 1/2), (4/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(1/6)*gamma(4/3))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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