\(\int \frac {(1+\sqrt {3}) \sqrt [3]{a}-\sqrt [3]{b} x}{((1-\sqrt {3}) \sqrt [3]{a}-\sqrt [3]{b} x) \sqrt {-a+b x^3}} \, dx\) [153]

Optimal result

Integrand size = 62, 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 = 7.51 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.18 \[ \int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {1+\frac {2}{\sqrt {3}}} \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [6]{a} \sqrt {-a+b x^3}}\right )}{\sqrt {-3+2 \sqrt {3}} \sqrt [6]{a} \sqrt [3]{b}} \] 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 = 0.55 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, 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.

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

\(\Big \downarrow \) 2565

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

\(\Big \downarrow \) 216

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

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 [A] (verification not implemented)

Time = 0.91 (sec) , antiderivative size = 1245, normalized size of antiderivative = 17.29 \[ \int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx=\text {Too large to display} \] 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 - 4*sqrt(1/3)*sqrt(b*x^3 - a)*((3*b^7*x^22 + 2688*a*b^6*x^19 + 569 
52*a^2*b^5*x^16 + 93504*a^3*b^4*x^13 - 63552*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))*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 - 397 
44*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^6*b^2*x^5 + 8192*a^7*b*x^2))*a^(1/3) + 2*(30*a*b^7*x^2 
1 + 5010*a^2*b^6*x^18 + 44640*a^3*b^5*x^15 + 21360*a^4*b^4*x^12 + 79872*a^ 
5*b^3*x^9 - 233856*a^6*b^2*x^6 + 86016*a^7*b*x^3 - 3072*a^8 - sqrt(3)*(17* 
a*b^7*x^21 + 2920*a^2*b^6*x^18 + 24864*a^3*b^5*x^15 + 26576*a^4*b^4*x^12 - 
 56000*a^5*b^3*x^9 + 115968*a^6*b^2*x^6 - 56320*a^7*b*x^3 + 1024*a^8))*b^( 
1/3))*sqrt(-(2*sqrt(3) + 3)/(a*b^(2/3))) + 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 + 5376...
 

Sympy [F]

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

 - 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