\(\int \frac {1}{(-3 a-b x^2) \sqrt [3]{-a+b x^2}} \, dx\) [351]

Optimal result

Integrand size = 26, antiderivative size = 252 \[ \int \frac {1}{\left (-3 a-b x^2\right ) \sqrt [3]{-a+b x^2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} \sqrt [3]{-a} \sqrt {a} \sqrt {b}}-\frac {\arctan \left (\frac {\sqrt {3} \sqrt {a} \left (\sqrt [3]{-a}-\sqrt [3]{2} \sqrt [3]{-a+b x^2}\right )}{\sqrt [3]{-a} \sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} \sqrt [3]{-a} \sqrt {a} \sqrt {b}}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{6\ 2^{2/3} \sqrt [3]{-a} \sqrt {a} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt [3]{-a} \sqrt {b} x}{\sqrt {a} \left (\sqrt [3]{-a}+\sqrt [3]{2} \sqrt [3]{-a+b x^2}\right )}\right )}{2\ 2^{2/3} \sqrt [3]{-a} \sqrt {a} \sqrt {b}} \] Output:

-1/12*arctan(3^(1/2)*a^(1/2)/b^(1/2)/x)*2^(1/3)*3^(1/2)/(-a)^(1/3)/a^(1/2) 
/b^(1/2)-1/12*arctan(3^(1/2)*a^(1/2)*((-a)^(1/3)-2^(1/3)*(b*x^2-a)^(1/3))/ 
(-a)^(1/3)/b^(1/2)/x)*2^(1/3)*3^(1/2)/(-a)^(1/3)/a^(1/2)/b^(1/2)+1/12*arct 
anh(b^(1/2)*x/a^(1/2))*2^(1/3)/(-a)^(1/3)/a^(1/2)/b^(1/2)-1/4*arctanh((-a) 
^(1/3)*b^(1/2)*x/a^(1/2)/((-a)^(1/3)+2^(1/3)*(b*x^2-a)^(1/3)))*2^(1/3)/(-a 
)^(1/3)/a^(1/2)/b^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 5.50 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.65 \[ \int \frac {1}{\left (-3 a-b x^2\right ) \sqrt [3]{-a+b x^2}} \, dx=-\frac {9 a x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )}{\sqrt [3]{-a+b x^2} \left (3 a+b x^2\right ) \left (9 a \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )+2 b x^2 \left (-\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )+\operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )\right )\right )} \] Input:

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

Output:

(-9*a*x*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a])/((-a + b*x^ 
2)^(1/3)*(3*a + b*x^2)*(9*a*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b* 
x^2)/a] + 2*b*x^2*(-AppellF1[3/2, 1/3, 2, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a] 
+ AppellF1[3/2, 4/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a])))
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {305}

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.

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

\(\Big \downarrow \) 305

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 305

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Wit 
h[{q = Rt[-b/a, 2]}, Simp[q*(ArcTan[Sqrt[3]/(q*x)]/(2*2^(2/3)*Sqrt[3]*a^(1/ 
3)*d)), x] + (Simp[q*(ArcTanh[(a^(1/3)*q*x)/(a^(1/3) + 2^(1/3)*(a + b*x^2)^ 
(1/3))]/(2*2^(2/3)*a^(1/3)*d)), x] - Simp[q*(ArcTanh[q*x]/(6*2^(2/3)*a^(1/3 
)*d)), x] + Simp[q*(ArcTan[Sqrt[3]*((a^(1/3) - 2^(1/3)*(a + b*x^2)^(1/3))/( 
a^(1/3)*q*x))]/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d)), x])] /; FreeQ[{a, b, c, d}, 
x] && NeQ[b*c - a*d, 0] && EqQ[b*c + 3*a*d, 0] && NegQ[b/a]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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