\(\int \frac {1}{(3 a-b x^2) \sqrt [3]{a+b x^2}} \, dx\) [352]

Optimal result

Integrand size = 24, antiderivative size = 202 \[ \int \frac {1}{\left (3 a-b x^2\right ) \sqrt [3]{a+b x^2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{6\ 2^{2/3} a^{5/6} \sqrt {b}}+\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a+b x^2}\right )}\right )}{2\ 2^{2/3} a^{5/6} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a+b x^2}\right )}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}} \] Output:

-1/12*arctan(b^(1/2)*x/a^(1/2))*2^(1/3)/a^(5/6)/b^(1/2)+1/4*arctan(b^(1/2) 
*x/a^(1/6)/(a^(1/3)+2^(1/3)*(b*x^2+a)^(1/3)))*2^(1/3)/a^(5/6)/b^(1/2)-1/12 
*arctanh(3^(1/2)*a^(1/2)/b^(1/2)/x)*2^(1/3)*3^(1/2)/a^(5/6)/b^(1/2)-1/12*a 
rctanh(3^(1/2)*a^(1/6)*(a^(1/3)-2^(1/3)*(b*x^2+a)^(1/3))/b^(1/2)/x)*2^(1/3 
)*3^(1/2)/a^(5/6)/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 = 5.46 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (3 a-b x^2\right ) \sqrt [3]{a+b x^2}} \, dx=\frac {9 a x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {b x^2}{a},\frac {b x^2}{3 a}\right )}{\left (3 a-b x^2\right ) \sqrt [3]{a+b x^2} \left (9 a \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {b x^2}{a},\frac {b x^2}{3 a}\right )+2 b x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},-\frac {b x^2}{a},\frac {b x^2}{3 a}\right )-\operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},-\frac {b x^2}{a},\frac {b x^2}{3 a}\right )\right )\right )} \] 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), (b*x^2)/(3*a)])/((3*a - b* 
x^2)*(a + b*x^2)^(1/3)*(9*a*AppellF1[1/2, 1/3, 1, 3/2, -((b*x^2)/a), (b*x^ 
2)/(3*a)] + 2*b*x^2*(AppellF1[3/2, 1/3, 2, 5/2, -((b*x^2)/a), (b*x^2)/(3*a 
)] - AppellF1[3/2, 4/3, 1, 5/2, -((b*x^2)/a), (b*x^2)/(3*a)])))
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {304}

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.

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

\(\Big \downarrow \) 304

\(\displaystyle \frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a+b x^2}+\sqrt [3]{a}\right )}\right )}{2\ 2^{2/3} a^{5/6} \sqrt {b}}-\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{6\ 2^{2/3} a^{5/6} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a+b x^2}\right )}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}\)

Input:

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

Output:

-1/6*ArcTan[(Sqrt[b]*x)/Sqrt[a]]/(2^(2/3)*a^(5/6)*Sqrt[b]) + ArcTan[(Sqrt[ 
b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a + b*x^2)^(1/3)))]/(2*2^(2/3)*a^(5/6)* 
Sqrt[b]) - ArcTanh[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)]/(2*2^(2/3)*Sqrt[3]*a^(5/ 
6)*Sqrt[b]) - ArcTanh[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a + b*x^2)^(1/3 
)))/(Sqrt[b]*x)]/(2*2^(2/3)*Sqrt[3]*a^(5/6)*Sqrt[b])
 

Defintions of rubi rules used

rule 304

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

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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