\(\int \frac {1}{\sqrt [3]{a-b x} \sqrt [3]{a+b x}} \, dx\) [194]

Optimal result

Integrand size = 20, antiderivative size = 625 \[ \int \frac {1}{\sqrt [3]{a-b x} \sqrt [3]{a+b x}} \, dx=-\frac {3 x \sqrt [3]{1-\frac {b^2 x^2}{a^2}}}{\sqrt [3]{a-b x} \sqrt [3]{a+b x} \left (1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right )}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^2 \sqrt [3]{1-\frac {b^2 x^2}{a^2}} \left (1-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right ) \sqrt {\frac {1+\sqrt [3]{1-\frac {b^2 x^2}{a^2}}+\left (1-\frac {b^2 x^2}{a^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right )^2}} E\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}{1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}\right )|-7+4 \sqrt {3}\right )}{2 b^2 x \sqrt [3]{a-b x} \sqrt [3]{a+b x} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}{\left (1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right )^2}}}+\frac {\sqrt {2} 3^{3/4} a^2 \sqrt [3]{1-\frac {b^2 x^2}{a^2}} \left (1-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right ) \sqrt {\frac {1+\sqrt [3]{1-\frac {b^2 x^2}{a^2}}+\left (1-\frac {b^2 x^2}{a^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}{1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}\right ),-7+4 \sqrt {3}\right )}{b^2 x \sqrt [3]{a-b x} \sqrt [3]{a+b x} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}{\left (1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right )^2}}} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.01 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.11 \[ \int \frac {1}{\sqrt [3]{a-b x} \sqrt [3]{a+b x}} \, dx=-\frac {3 (a-b x)^{2/3} \sqrt [3]{\frac {a+b x}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},\frac {a-b x}{2 a}\right )}{2 \sqrt [3]{2} b \sqrt [3]{a+b x}} \] Input:

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

Output:

(-3*(a - b*x)^(2/3)*((a + b*x)/a)^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, ( 
a - b*x)/(2*a)])/(2*2^(1/3)*b*(a + b*x)^(1/3))
 

Rubi [A] (warning: unable to verify)

Time = 0.49 (sec) , antiderivative size = 706, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {46, 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.

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

\(\Big \downarrow \) 46

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

\(\Big \downarrow \) 233

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

\(\Big \downarrow \) 833

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

\(\Big \downarrow \) 760

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

\(\Big \downarrow \) 2418

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

Input:

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

Output:

(-3*Sqrt[-(b^2*x^2)]*(a^2 - b^2*x^2)^(1/3)*((-2*Sqrt[-(b^2*x^2)])/((1 - Sq 
rt[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 + 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]])/(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 + Sqr 
t[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[ArcSin[((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)]*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*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 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [A] (verification not implemented)

Time = 1.59 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.17 \[ \int \frac {1}{\sqrt [3]{a-b x} \sqrt [3]{a+b x}} \, dx=\frac {\sqrt [3]{a} {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{6}, \frac {2}{3}, 1 & \frac {1}{3}, \frac {1}{2}, \frac {5}{6} \\- \frac {1}{6}, \frac {1}{6}, \frac {1}{3}, \frac {2}{3}, \frac {5}{6} & 0 \end {matrix} \middle | {\frac {a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )} e^{\frac {i \pi }{3}}}{4 \pi b \Gamma \left (\frac {1}{3}\right )} - \frac {\sqrt [3]{a} {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{3}, 0, \frac {1}{6}, \frac {1}{2}, 1 & \\- \frac {1}{3}, \frac {1}{6} & - \frac {1}{2}, - \frac {1}{6}, 0, 0 \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi b \Gamma \left (\frac {1}{3}\right )} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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