\(\int \frac {2^{2/3} \sqrt [3]{a}+2 \sqrt [3]{b} x}{(2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x) \sqrt {-a+b x^3}} \, dx\) [95]

Optimal result

Integrand size = 56, antiderivative size = 66 \[ \int \frac {2^{2/3} \sqrt [3]{a}+2 \sqrt [3]{b} x}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx=-\frac {2\ 2^{2/3} \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{b} x\right )}{\sqrt {-a+b x^3}}\right )}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{b}} \] Output:

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

Mathematica [A] (verified)

Time = 6.04 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.03 \[ \int \frac {2^{2/3} \sqrt [3]{a}+2 \sqrt [3]{b} x}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx=-\frac {2\ 2^{2/3} \text {arctanh}\left (\frac {\sqrt {-a+b x^3}}{\sqrt {3} \left (\sqrt {a}-\sqrt [3]{2} \sqrt [6]{a} \sqrt [3]{b} x\right )}\right )}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{b}} \] Input:

Integrate[(2^(2/3)*a^(1/3) + 2*b^(1/3)*x)/((2^(2/3)*a^(1/3) - b^(1/3)*x)*S 
qrt[-a + b*x^3]),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {2562, 219}

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.

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

\(\Big \downarrow \) 2562

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

-Integral(2**(2/3)*a**(1/3)/(-2**(2/3)*a**(1/3)*sqrt(-a + b*x**3) + b**(1/ 
3)*x*sqrt(-a + b*x**3)), x) - Integral(2*b**(1/3)*x/(-2**(2/3)*a**(1/3)*sq 
rt(-a + b*x**3) + b**(1/3)*x*sqrt(-a + b*x**3)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 23.61 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.55 \[ \int \frac {2^{2/3} \sqrt [3]{a}+2 \sqrt [3]{b} x}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx=\frac {\sqrt {3}\,4^{1/3}\,\ln \left (\frac {\left (\sqrt {b\,x^3-a}+\sqrt {3}\,\sqrt {a}-2^{1/3}\,\sqrt {3}\,a^{1/6}\,b^{1/3}\,x\right )\,{\left (\sqrt {b\,x^3-a}-\sqrt {3}\,\sqrt {a}+2^{1/3}\,\sqrt {3}\,a^{1/6}\,b^{1/3}\,x\right )}^3}{{\left (2^{2/3}\,a^{1/3}-b^{1/3}\,x\right )}^6}\right )}{3\,a^{1/6}\,b^{1/3}} \] Input:

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

Output:

(3^(1/2)*4^(1/3)*log((((b*x^3 - a)^(1/2) + 3^(1/2)*a^(1/2) - 2^(1/3)*3^(1/ 
2)*a^(1/6)*b^(1/3)*x)*((b*x^3 - a)^(1/2) - 3^(1/2)*a^(1/2) + 2^(1/3)*3^(1/ 
2)*a^(1/6)*b^(1/3)*x)^3)/(2^(2/3)*a^(1/3) - b^(1/3)*x)^6))/(3*a^(1/6)*b^(1 
/3))
 

Reduce [F]

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

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

Output:

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