\(\int \frac {(1+\sqrt {3}) \sqrt [3]{a}+\sqrt [3]{b} x}{((1-\sqrt {3}) \sqrt [3]{a}+\sqrt [3]{b} x) \sqrt {-a-b x^3}} \, dx\) [154]

Optimal result

Integrand size = 61, antiderivative size = 72 \[ \int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {-a-b x^3}}\right )}{\sqrt {-3+2 \sqrt {3}} \sqrt [6]{a} \sqrt [3]{b}} \] Output:

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

Mathematica [A] (verified)

Time = 7.45 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.21 \[ \int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {1+\frac {2}{\sqrt {3}}} \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [6]{a} \sqrt {-a-b x^3}}\right )}{\sqrt {-3+2 \sqrt {3}} \sqrt [6]{a} \sqrt [3]{b}} \] Input:

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

Output:

(-2*ArcTan[(Sqrt[1 + 2/Sqrt[3]]*(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2 
))/(a^(1/6)*Sqrt[-a - b*x^3])])/(Sqrt[-3 + 2*Sqrt[3]]*a^(1/6)*b^(1/3))
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {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.

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

\(\Big \downarrow \) 2565

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

\(\Big \downarrow \) 216

\(\displaystyle -\frac {2 \arctan \left (\frac {\sqrt {2 \sqrt {3}-3} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {-a-b x^3}}\right )}{\sqrt {2 \sqrt {3}-3} \sqrt [6]{a} \sqrt [3]{b}}\)

Input:

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

Output:

(-2*ArcTan[(Sqrt[-3 + 2*Sqrt[3]]*a^(1/6)*(a^(1/3) + b^(1/3)*x))/Sqrt[-a - 
b*x^3]])/(Sqrt[-3 + 2*Sqrt[3]]*a^(1/6)*b^(1/3))
 

Defintions of rubi rules used

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 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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (52) = 104\).

Time = 0.94 (sec) , antiderivative size = 1303, normalized size of antiderivative = 18.10 \[ \int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/2*sqrt(1/3)*a^(1/3)*sqrt(-(2*sqrt(3) + 3)/(a*b^(2/3)))*log((b^8*x^24 - 
1840*a*b^7*x^21 + 67264*a^2*b^6*x^18 - 58624*a^3*b^5*x^15 + 504064*a^4*b^4 
*x^12 + 2140160*a^5*b^3*x^9 + 3100672*a^6*b^2*x^6 + 1089536*a^7*b*x^3 + 28 
672*a^8 + 32*(9*b^7*x^22 - 846*a*b^6*x^19 + 4617*a^2*b^5*x^16 + 5472*a^3*b 
^4*x^13 + 43776*a^4*b^3*x^10 + 98496*a^5*b^2*x^7 + 59328*a^6*b*x^4 + 4608* 
a^7*x - sqrt(3)*(5*b^7*x^22 - 505*a*b^6*x^19 + 2130*a^2*b^5*x^16 - 4928*a^ 
3*b^4*x^13 - 28688*a^4*b^3*x^10 - 53760*a^5*b^2*x^7 - 35200*a^6*b*x^4 - 25 
60*a^7*x))*a^(2/3)*b^(1/3) - 8*(3*b^7*x^23 - 1077*a*b^6*x^20 + 13320*a^2*b 
^5*x^17 - 19200*a^3*b^4*x^14 - 111360*a^4*b^3*x^11 - 345024*a^5*b^2*x^8 - 
328704*a^6*b*x^5 - 61440*a^7*x^2 - 2*sqrt(3)*(b^7*x^23 - 299*a*b^6*x^20 + 
4260*a^2*b^5*x^17 + 1520*a^3*b^4*x^14 + 26720*a^4*b^3*x^11 + 105024*a^5*b^ 
2*x^8 + 93184*a^6*b*x^5 + 17920*a^7*x^2))*a^(1/3)*b^(2/3) + 4*sqrt(1/3)*(( 
3*b^7*x^22 - 2688*a*b^6*x^19 + 56952*a^2*b^5*x^16 - 93504*a^3*b^4*x^13 - 6 
3552*a^4*b^3*x^10 - 377856*a^5*b^2*x^7 - 314880*a^6*b*x^4 - 24576*a^7*x - 
2*sqrt(3)*(b^7*x^22 - 764*a*b^6*x^19 + 16860*a^2*b^5*x^16 - 19792*a^3*b^4* 
x^13 + 42368*a^4*b^3*x^10 + 104448*a^5*b^2*x^7 + 90880*a^6*b*x^4 + 7168*a^ 
7*x))*sqrt(-b*x^3 - a)*a^(2/3)*b^(2/3) + 6*(81*a*b^7*x^20 - 4752*a^2*b^6*x 
^17 + 14472*a^3*b^5*x^14 - 24192*a^4*b^4*x^11 - 39744*a^5*b^3*x^8 - 69120* 
a^6*b^2*x^5 - 13824*a^7*b*x^2 - sqrt(3)*(47*a*b^7*x^20 - 2724*a^2*b^6*x^17 
 + 8976*a^3*b^5*x^14 - 4928*a^4*b^4*x^11 + 32448*a^5*b^3*x^8 + 37632*a^...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx=i \left (-6 b^{\frac {2}{3}} a^{\frac {2}{3}} \sqrt {3}\, \left (\int \frac {\sqrt {b \,x^{3}+a}\, x^{2}}{4 a^{\frac {7}{3}}+8 a^{\frac {4}{3}} b \,x^{3}+4 a^{\frac {1}{3}} b^{2} x^{6}-8 b^{\frac {1}{3}} a^{2} x -7 b^{\frac {4}{3}} a \,x^{4}+b^{\frac {7}{3}} x^{7}}d x \right )-6 b^{\frac {2}{3}} a^{\frac {2}{3}} \left (\int \frac {\sqrt {b \,x^{3}+a}\, x^{2}}{4 a^{\frac {7}{3}}+8 a^{\frac {4}{3}} b \,x^{3}+4 a^{\frac {1}{3}} b^{2} x^{6}-8 b^{\frac {1}{3}} a^{2} x -7 b^{\frac {4}{3}} a \,x^{4}+b^{\frac {7}{3}} x^{7}}d x \right )+4 a^{\frac {4}{3}} \sqrt {3}\, \left (\int \frac {\sqrt {b \,x^{3}+a}}{4 a^{\frac {7}{3}}+8 a^{\frac {4}{3}} b \,x^{3}+4 a^{\frac {1}{3}} b^{2} x^{6}-8 b^{\frac {1}{3}} a^{2} x -7 b^{\frac {4}{3}} a \,x^{4}+b^{\frac {7}{3}} x^{7}}d x \right )-2 a^{\frac {1}{3}} \sqrt {3}\, \left (\int \frac {\sqrt {b \,x^{3}+a}\, x^{3}}{4 a^{\frac {7}{3}}+8 a^{\frac {4}{3}} b \,x^{3}+4 a^{\frac {1}{3}} b^{2} x^{6}-8 b^{\frac {1}{3}} a^{2} x -7 b^{\frac {4}{3}} a \,x^{4}+b^{\frac {7}{3}} x^{7}}d x \right ) b +8 a^{\frac {4}{3}} \left (\int \frac {\sqrt {b \,x^{3}+a}}{4 a^{\frac {7}{3}}+8 a^{\frac {4}{3}} b \,x^{3}+4 a^{\frac {1}{3}} b^{2} x^{6}-8 b^{\frac {1}{3}} a^{2} x -7 b^{\frac {4}{3}} a \,x^{4}+b^{\frac {7}{3}} x^{7}}d x \right )-4 a^{\frac {1}{3}} \left (\int \frac {\sqrt {b \,x^{3}+a}\, x^{3}}{4 a^{\frac {7}{3}}+8 a^{\frac {4}{3}} b \,x^{3}+4 a^{\frac {1}{3}} b^{2} x^{6}-8 b^{\frac {1}{3}} a^{2} x -7 b^{\frac {4}{3}} a \,x^{4}+b^{\frac {7}{3}} x^{7}}d x \right ) b -b^{\frac {4}{3}} \left (\int \frac {\sqrt {b \,x^{3}+a}\, x^{4}}{4 a^{\frac {7}{3}}+8 a^{\frac {4}{3}} b \,x^{3}+4 a^{\frac {1}{3}} b^{2} x^{6}-8 b^{\frac {1}{3}} a^{2} x -7 b^{\frac {4}{3}} a \,x^{4}+b^{\frac {7}{3}} x^{7}}d x \right )-4 b^{\frac {1}{3}} \left (\int \frac {\sqrt {b \,x^{3}+a}\, x}{4 a^{\frac {7}{3}}+8 a^{\frac {4}{3}} b \,x^{3}+4 a^{\frac {1}{3}} b^{2} x^{6}-8 b^{\frac {1}{3}} a^{2} x -7 b^{\frac {4}{3}} a \,x^{4}+b^{\frac {7}{3}} x^{7}}d x \right ) a \right ) \] Input:

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

Output:

i*( - 6*b**(2/3)*a**(2/3)*sqrt(3)*int((sqrt(a + b*x**3)*x**2)/(4*a**(1/3)* 
a**2 + 8*a**(1/3)*a*b*x**3 + 4*a**(1/3)*b**2*x**6 - 8*b**(1/3)*a**2*x - 7* 
b**(1/3)*a*b*x**4 + b**(1/3)*b**2*x**7),x) - 6*b**(2/3)*a**(2/3)*int((sqrt 
(a + b*x**3)*x**2)/(4*a**(1/3)*a**2 + 8*a**(1/3)*a*b*x**3 + 4*a**(1/3)*b** 
2*x**6 - 8*b**(1/3)*a**2*x - 7*b**(1/3)*a*b*x**4 + b**(1/3)*b**2*x**7),x) 
+ 4*a**(1/3)*sqrt(3)*int(sqrt(a + b*x**3)/(4*a**(1/3)*a**2 + 8*a**(1/3)*a* 
b*x**3 + 4*a**(1/3)*b**2*x**6 - 8*b**(1/3)*a**2*x - 7*b**(1/3)*a*b*x**4 + 
b**(1/3)*b**2*x**7),x)*a - 2*a**(1/3)*sqrt(3)*int((sqrt(a + b*x**3)*x**3)/ 
(4*a**(1/3)*a**2 + 8*a**(1/3)*a*b*x**3 + 4*a**(1/3)*b**2*x**6 - 8*b**(1/3) 
*a**2*x - 7*b**(1/3)*a*b*x**4 + b**(1/3)*b**2*x**7),x)*b + 8*a**(1/3)*int( 
sqrt(a + b*x**3)/(4*a**(1/3)*a**2 + 8*a**(1/3)*a*b*x**3 + 4*a**(1/3)*b**2* 
x**6 - 8*b**(1/3)*a**2*x - 7*b**(1/3)*a*b*x**4 + b**(1/3)*b**2*x**7),x)*a 
- 4*a**(1/3)*int((sqrt(a + b*x**3)*x**3)/(4*a**(1/3)*a**2 + 8*a**(1/3)*a*b 
*x**3 + 4*a**(1/3)*b**2*x**6 - 8*b**(1/3)*a**2*x - 7*b**(1/3)*a*b*x**4 + b 
**(1/3)*b**2*x**7),x)*b - b**(1/3)*int((sqrt(a + b*x**3)*x**4)/(4*a**(1/3) 
*a**2 + 8*a**(1/3)*a*b*x**3 + 4*a**(1/3)*b**2*x**6 - 8*b**(1/3)*a**2*x - 7 
*b**(1/3)*a*b*x**4 + b**(1/3)*b**2*x**7),x)*b - 4*b**(1/3)*int((sqrt(a + b 
*x**3)*x)/(4*a**(1/3)*a**2 + 8*a**(1/3)*a*b*x**3 + 4*a**(1/3)*b**2*x**6 - 
8*b**(1/3)*a**2*x - 7*b**(1/3)*a*b*x**4 + b**(1/3)*b**2*x**7),x)*a)