\(\int \frac {2^{2/3} \sqrt [3]{a}+2 \sqrt [3]{b} x}{(2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x) \sqrt {a-b x^3}} \, dx\) [94]

Optimal result

Integrand size = 55, antiderivative size = 65 \[ \int \frac {2^{2/3} \sqrt [3]{a}+2 \sqrt [3]{b} x}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=-\frac {2\ 2^{2/3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{b} x\right )}{\sqrt {a-b x^3}}\right )}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{b}} \] Output:

-2/3*2^(2/3)*arctan(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 = 5.99 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.03 \[ \int \frac {2^{2/3} \sqrt [3]{a}+2 \sqrt [3]{b} x}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=\frac {2\ 2^{2/3} \arctan \left (\frac {\sqrt {a-b x^3}}{\sqrt {3} \left (\sqrt {a}-\sqrt [3]{2} \sqrt [6]{a} \sqrt [3]{b} x\right )}\right )}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{b}} \] 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)*ArcTan[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 = 0.55 (sec) , antiderivative size = 65, 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, 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.

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

\(\Big \downarrow \) 2562

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

\(\Big \downarrow \) 216

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

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)*ArcTan[(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 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 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)*sqrt 
(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 = 25.23 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.65 \[ \int \frac {2^{2/3} \sqrt [3]{a}+2 \sqrt [3]{b} x}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=\frac {2^{2/3}\,\sqrt {3}\,\ln \left (\frac {\left (\sqrt {a-b\,x^3}-\sqrt {3}\,\sqrt {a}\,1{}\mathrm {i}+2^{1/3}\,\sqrt {3}\,a^{1/6}\,b^{1/3}\,x\,1{}\mathrm {i}\right )\,{\left (\sqrt {3}\,\sqrt {a}\,1{}\mathrm {i}+\sqrt {a-b\,x^3}-2^{1/3}\,\sqrt {3}\,a^{1/6}\,b^{1/3}\,x\,1{}\mathrm {i}\right )}^3}{{\left (2^{2/3}\,a^{1/3}-b^{1/3}\,x\right )}^6}\right )\,1{}\mathrm {i}}{3\,a^{1/6}\,b^{1/3}} \] Input:

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

Output:

(2^(2/3)*3^(1/2)*log((((a - b*x^3)^(1/2) - 3^(1/2)*a^(1/2)*1i + 2^(1/3)*3^ 
(1/2)*a^(1/6)*b^(1/3)*x*1i)*(3^(1/2)*a^(1/2)*1i + (a - b*x^3)^(1/2) - 2^(1 
/3)*3^(1/2)*a^(1/6)*b^(1/3)*x*1i)^3)/(2^(2/3)*a^(1/3) - b^(1/3)*x)^6)*1i)/ 
(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)*sqr 
t(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)