\(\int \frac {1}{(a-b x)^{4/3} (a+b x)^{4/3}} \, dx\) [220]

Optimal result

Integrand size = 20, antiderivative size = 652 \[ \int \frac {1}{(a-b x)^{4/3} (a+b x)^{4/3}} \, dx=\frac {3 x}{2 a^2 \sqrt [3]{a-b x} \sqrt [3]{a+b x}}+\frac {3 x \sqrt [3]{1-\frac {b^2 x^2}{a^2}}}{2 a^2 \sqrt [3]{a-b x} \sqrt [3]{a+b x} \left (1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right )}+\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{1-\frac {b^2 x^2}{a^2}} \left (1-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right ) \sqrt {\frac {1+\sqrt [3]{1-\frac {b^2 x^2}{a^2}}+\left (1-\frac {b^2 x^2}{a^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right )^2}} E\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}{1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}\right )|-7+4 \sqrt {3}\right )}{4 b^2 x \sqrt [3]{a-b x} \sqrt [3]{a+b x} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}{\left (1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right )^2}}}-\frac {3^{3/4} \sqrt [3]{1-\frac {b^2 x^2}{a^2}} \left (1-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right ) \sqrt {\frac {1+\sqrt [3]{1-\frac {b^2 x^2}{a^2}}+\left (1-\frac {b^2 x^2}{a^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}{1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}\right ),-7+4 \sqrt {3}\right )}{\sqrt {2} b^2 x \sqrt [3]{a-b x} \sqrt [3]{a+b x} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}{\left (1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right )^2}}} \] Output:

3/2*x/a^2/(-b*x+a)^(1/3)/(b*x+a)^(1/3)+3/2*x*(1-b^2*x^2/a^2)^(1/3)/a^2/(-b 
*x+a)^(1/3)/(b*x+a)^(1/3)/(1-3^(1/2)-(1-b^2*x^2/a^2)^(1/3))+3/4*3^(1/4)*(1 
/2*6^(1/2)+1/2*2^(1/2))*(1-b^2*x^2/a^2)^(1/3)*(1-(1-b^2*x^2/a^2)^(1/3))*(( 
1+(1-b^2*x^2/a^2)^(1/3)+(1-b^2*x^2/a^2)^(2/3))/(1-3^(1/2)-(1-b^2*x^2/a^2)^ 
(1/3))^2)^(1/2)*EllipticE((1+3^(1/2)-(1-b^2*x^2/a^2)^(1/3))/(1-3^(1/2)-(1- 
b^2*x^2/a^2)^(1/3)),2*I-I*3^(1/2))/b^2/x/(-b*x+a)^(1/3)/(b*x+a)^(1/3)/(-(1 
-(1-b^2*x^2/a^2)^(1/3))/(1-3^(1/2)-(1-b^2*x^2/a^2)^(1/3))^2)^(1/2)-1/2*3^( 
3/4)*(1-b^2*x^2/a^2)^(1/3)*(1-(1-b^2*x^2/a^2)^(1/3))*((1+(1-b^2*x^2/a^2)^( 
1/3)+(1-b^2*x^2/a^2)^(2/3))/(1-3^(1/2)-(1-b^2*x^2/a^2)^(1/3))^2)^(1/2)*Ell 
ipticF((1+3^(1/2)-(1-b^2*x^2/a^2)^(1/3))/(1-3^(1/2)-(1-b^2*x^2/a^2)^(1/3)) 
,2*I-I*3^(1/2))*2^(1/2)/b^2/x/(-b*x+a)^(1/3)/(b*x+a)^(1/3)/(-(1-(1-b^2*x^2 
/a^2)^(1/3))/(1-3^(1/2)-(1-b^2*x^2/a^2)^(1/3))^2)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.11 \[ \int \frac {1}{(a-b x)^{4/3} (a+b x)^{4/3}} \, dx=\frac {3 \sqrt [3]{\frac {a+b x}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {4}{3},\frac {2}{3},\frac {a-b x}{2 a}\right )}{2 \sqrt [3]{2} a b \sqrt [3]{a-b x} \sqrt [3]{a+b x}} \] Input:

Integrate[1/((a - b*x)^(4/3)*(a + b*x)^(4/3)),x]
 

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.53 (sec) , antiderivative size = 735, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {46, 215, 233, 833, 760, 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.

\(\displaystyle \int \frac {1}{(a-b x)^{4/3} (a+b x)^{4/3}} \, dx\)

\(\Big \downarrow \) 46

\(\displaystyle \frac {\sqrt [3]{a^2-b^2 x^2} \int \frac {1}{\left (a^2-b^2 x^2\right )^{4/3}}dx}{\sqrt [3]{a-b x} \sqrt [3]{a+b x}}\)

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 233

\(\displaystyle \frac {\sqrt [3]{a^2-b^2 x^2} \left (\frac {3 \sqrt {-b^2 x^2} \int \frac {\sqrt [3]{a^2-b^2 x^2}}{\sqrt {-b^2 x^2}}d\sqrt [3]{a^2-b^2 x^2}}{4 a^2 b^2 x}+\frac {3 x}{2 a^2 \sqrt [3]{a^2-b^2 x^2}}\right )}{\sqrt [3]{a-b x} \sqrt [3]{a+b x}}\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {\sqrt [3]{a^2-b^2 x^2} \left (\frac {3 \sqrt {-b^2 x^2} \left (\left (1+\sqrt {3}\right ) a^{2/3} \int \frac {1}{\sqrt {-b^2 x^2}}d\sqrt [3]{a^2-b^2 x^2}-\int \frac {\left (1+\sqrt {3}\right ) a^{2/3}-\sqrt [3]{a^2-b^2 x^2}}{\sqrt {-b^2 x^2}}d\sqrt [3]{a^2-b^2 x^2}\right )}{4 a^2 b^2 x}+\frac {3 x}{2 a^2 \sqrt [3]{a^2-b^2 x^2}}\right )}{\sqrt [3]{a-b x} \sqrt [3]{a+b x}}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {\sqrt [3]{a^2-b^2 x^2} \left (\frac {3 \sqrt {-b^2 x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) a^{2/3}-\sqrt [3]{a^2-b^2 x^2}}{\sqrt {-b^2 x^2}}d\sqrt [3]{a^2-b^2 x^2}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) a^{2/3} \left (a^{2/3}-\sqrt [3]{a^2-b^2 x^2}\right ) \sqrt {\frac {a^{4/3}+\left (a^2-b^2 x^2\right )^{2/3}+a^{2/3} \sqrt [3]{a^2-b^2 x^2}}{\left (\left (1-\sqrt {3}\right ) a^{2/3}-\sqrt [3]{a^2-b^2 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) a^{2/3}-\sqrt [3]{a^2-b^2 x^2}}{\left (1-\sqrt {3}\right ) a^{2/3}-\sqrt [3]{a^2-b^2 x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-b^2 x^2} \sqrt {-\frac {a^{2/3} \left (a^{2/3}-\sqrt [3]{a^2-b^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) a^{2/3}-\sqrt [3]{a^2-b^2 x^2}\right )^2}}}\right )}{4 a^2 b^2 x}+\frac {3 x}{2 a^2 \sqrt [3]{a^2-b^2 x^2}}\right )}{\sqrt [3]{a-b x} \sqrt [3]{a+b x}}\)

\(\Big \downarrow \) 2418

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

Input:

Int[1/((a - b*x)^(4/3)*(a + b*x)^(4/3)),x]
 

Output:

((a^2 - b^2*x^2)^(1/3)*((3*x)/(2*a^2*(a^2 - b^2*x^2)^(1/3)) + (3*Sqrt[-(b^ 
2*x^2)]*((-2*Sqrt[-(b^2*x^2)])/((1 - Sqrt[3])*a^(2/3) - (a^2 - b^2*x^2)^(1 
/3)) + (3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(2/3)*(a^(2/3) - (a^2 - b^2*x^2)^(1/3) 
)*Sqrt[(a^(4/3) + a^(2/3)*(a^2 - b^2*x^2)^(1/3) + (a^2 - b^2*x^2)^(2/3))/( 
(1 - Sqrt[3])*a^(2/3) - (a^2 - b^2*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + S 
qrt[3])*a^(2/3) - (a^2 - b^2*x^2)^(1/3))/((1 - Sqrt[3])*a^(2/3) - (a^2 - b 
^2*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(Sqrt[-(b^2*x^2)]*Sqrt[-((a^(2/3)*(a^(2/ 
3) - (a^2 - b^2*x^2)^(1/3)))/((1 - Sqrt[3])*a^(2/3) - (a^2 - b^2*x^2)^(1/3 
))^2)]) - (2*Sqrt[2 - Sqrt[3]]*(1 + Sqrt[3])*a^(2/3)*(a^(2/3) - (a^2 - b^2 
*x^2)^(1/3))*Sqrt[(a^(4/3) + a^(2/3)*(a^2 - b^2*x^2)^(1/3) + (a^2 - b^2*x^ 
2)^(2/3))/((1 - Sqrt[3])*a^(2/3) - (a^2 - b^2*x^2)^(1/3))^2]*EllipticF[Arc 
Sin[((1 + Sqrt[3])*a^(2/3) - (a^2 - b^2*x^2)^(1/3))/((1 - Sqrt[3])*a^(2/3) 
 - (a^2 - b^2*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(3^(1/4)*Sqrt[-(b^2*x^2)]*Sqr 
t[-((a^(2/3)*(a^(2/3) - (a^2 - b^2*x^2)^(1/3)))/((1 - Sqrt[3])*a^(2/3) - ( 
a^2 - b^2*x^2)^(1/3))^2)])))/(4*a^2*b^2*x)))/((a - b*x)^(1/3)*(a + b*x)^(1 
/3))
 

Defintions of rubi rules used

rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(a 
 + b*x)^FracPart[m]*((c + d*x)^FracPart[m]/(a*c + b*d*x^2)^FracPart[m])   I 
nt[(a*c + b*d*x^2)^m, x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 
 0] &&  !IntegerQ[2*m]
 

rule 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

rule 233

Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Simp[3*(Sqrt[b*x^2]/(2*b*x)) 
   Subst[Int[x/Sqrt[-a + x^3], x], x, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b 
}, x]
 

rule 760

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- 
s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3]) 
*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x 
] && NegQ[a]
 

rule 833

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] 
], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 + Sqrt[3]))*(s/r)   Int[1/Sqrt[a + b*x 
^3], x], x] + Simp[1/r   Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x 
]] /; FreeQ[{a, b}, x] && NegQ[a]
 

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 [F]

\[\int \frac {1}{\left (-b x +a \right )^{\frac {4}{3}} \left (b x +a \right )^{\frac {4}{3}}}d x\]

Input:

int(1/(-b*x+a)^(4/3)/(b*x+a)^(4/3),x)
 

Output:

int(1/(-b*x+a)^(4/3)/(b*x+a)^(4/3),x)
 

Fricas [F]

\[ \int \frac {1}{(a-b x)^{4/3} (a+b x)^{4/3}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {4}{3}} {\left (-b x + a\right )}^{\frac {4}{3}}} \,d x } \] Input:

integrate(1/(-b*x+a)^(4/3)/(b*x+a)^(4/3),x, algorithm="fricas")
 

Output:

integral((b*x + a)^(2/3)*(-b*x + a)^(2/3)/(b^4*x^4 - 2*a^2*b^2*x^2 + a^4), 
 x)
 

Sympy [A] (verification not implemented)

Time = 2.77 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.16 \[ \int \frac {1}{(a-b x)^{4/3} (a+b x)^{4/3}} \, dx=\frac {{G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {2}{3}, \frac {7}{6}, 1 & \frac {1}{2}, \frac {4}{3}, \frac {11}{6} \\\frac {2}{3}, \frac {5}{6}, \frac {7}{6}, \frac {4}{3}, \frac {11}{6} & 0 \end {matrix} \middle | {\frac {a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )} e^{- \frac {2 i \pi }{3}}}{4 \pi a^{\frac {5}{3}} b \Gamma \left (\frac {4}{3}\right )} - \frac {{G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, 0, \frac {1}{6}, \frac {1}{2}, \frac {2}{3}, 1 & \\\frac {1}{6}, \frac {2}{3} & - \frac {1}{2}, 0, \frac {5}{6}, 0 \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi a^{\frac {5}{3}} b \Gamma \left (\frac {4}{3}\right )} \] Input:

integrate(1/(-b*x+a)**(4/3)/(b*x+a)**(4/3),x)
 

Output:

meijerg(((2/3, 7/6, 1), (1/2, 4/3, 11/6)), ((2/3, 5/6, 7/6, 4/3, 11/6), (0 
,)), a**2*exp_polar(-2*I*pi)/(b**2*x**2))*exp(-2*I*pi/3)/(4*pi*a**(5/3)*b* 
gamma(4/3)) - meijerg(((-1/2, 0, 1/6, 1/2, 2/3, 1), ()), ((1/6, 2/3), (-1/ 
2, 0, 5/6, 0)), a**2/(b**2*x**2))/(4*pi*a**(5/3)*b*gamma(4/3))
 

Maxima [F]

\[ \int \frac {1}{(a-b x)^{4/3} (a+b x)^{4/3}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {4}{3}} {\left (-b x + a\right )}^{\frac {4}{3}}} \,d x } \] Input:

integrate(1/(-b*x+a)^(4/3)/(b*x+a)^(4/3),x, algorithm="maxima")
 

Output:

integrate(1/((b*x + a)^(4/3)*(-b*x + a)^(4/3)), x)
 

Giac [F]

\[ \int \frac {1}{(a-b x)^{4/3} (a+b x)^{4/3}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {4}{3}} {\left (-b x + a\right )}^{\frac {4}{3}}} \,d x } \] Input:

integrate(1/(-b*x+a)^(4/3)/(b*x+a)^(4/3),x, algorithm="giac")
 

Output:

integrate(1/((b*x + a)^(4/3)*(-b*x + a)^(4/3)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a-b x)^{4/3} (a+b x)^{4/3}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{4/3}\,{\left (a-b\,x\right )}^{4/3}} \,d x \] Input:

int(1/((a + b*x)^(4/3)*(a - b*x)^(4/3)),x)
 

Output:

int(1/((a + b*x)^(4/3)*(a - b*x)^(4/3)), x)
 

Reduce [F]

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

int(1/(-b*x+a)^(4/3)/(b*x+a)^(4/3),x)
                                                                                    
                                                                                    
 

Output:

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