\(\int \frac {1}{\sqrt [4]{a+3 x^2} (2 a+3 x^2)} \, dx\) [391]

Optimal result

Integrand size = 23, antiderivative size = 120 \[ \int \frac {1}{\sqrt [4]{a+3 x^2} \left (2 a+3 x^2\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {3} x \sqrt [4]{a+3 x^2}}{a^{3/4} \left (1+\frac {\sqrt {a+3 x^2}}{\sqrt {a}}\right )}\right )}{2 \sqrt {3} a^{3/4}}-\frac {\text {arctanh}\left (\frac {a^{3/4} \left (1-\frac {\sqrt {a+3 x^2}}{\sqrt {a}}\right )}{\sqrt {3} x \sqrt [4]{a+3 x^2}}\right )}{2 \sqrt {3} a^{3/4}} \] Output:

1/6*arctan(3^(1/2)*x*(3*x^2+a)^(1/4)/a^(3/4)/(1+(3*x^2+a)^(1/2)/a^(1/2)))* 
3^(1/2)/a^(3/4)-1/6*arctanh(1/3*a^(3/4)*(1-(3*x^2+a)^(1/2)/a^(1/2))*3^(1/2 
)/x/(3*x^2+a)^(1/4))*3^(1/2)/a^(3/4)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\sqrt [4]{a+3 x^2} \left (2 a+3 x^2\right )} \, dx=\frac {-\arctan \left (\frac {-3 x^2+2 \sqrt {a} \sqrt {a+3 x^2}}{2 \sqrt {3} \sqrt [4]{a} x \sqrt [4]{a+3 x^2}}\right )+\text {arctanh}\left (\frac {2 \sqrt {3} \sqrt [4]{a} x \sqrt [4]{a+3 x^2}}{3 x^2+2 \sqrt {a} \sqrt {a+3 x^2}}\right )}{4 \sqrt {3} a^{3/4}} \] Input:

Integrate[1/((a + 3*x^2)^(1/4)*(2*a + 3*x^2)),x]
 

Output:

(-ArcTan[(-3*x^2 + 2*Sqrt[a]*Sqrt[a + 3*x^2])/(2*Sqrt[3]*a^(1/4)*x*(a + 3* 
x^2)^(1/4))] + ArcTanh[(2*Sqrt[3]*a^(1/4)*x*(a + 3*x^2)^(1/4))/(3*x^2 + 2* 
Sqrt[a]*Sqrt[a + 3*x^2])])/(4*Sqrt[3]*a^(3/4))
 

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {308}

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.

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

\(\Big \downarrow \) 308

\(\displaystyle -\frac {\arctan \left (\frac {a^{3/4} \left (\frac {\sqrt {a+3 x^2}}{\sqrt {a}}+1\right )}{\sqrt {3} x \sqrt [4]{a+3 x^2}}\right )}{2 \sqrt {3} a^{3/4}}-\frac {\text {arctanh}\left (\frac {a^{3/4} \left (1-\frac {\sqrt {a+3 x^2}}{\sqrt {a}}\right )}{\sqrt {3} x \sqrt [4]{a+3 x^2}}\right )}{2 \sqrt {3} a^{3/4}}\)

Input:

Int[1/((a + 3*x^2)^(1/4)*(2*a + 3*x^2)),x]
 

Output:

-1/2*ArcTan[(a^(3/4)*(1 + Sqrt[a + 3*x^2]/Sqrt[a]))/(Sqrt[3]*x*(a + 3*x^2) 
^(1/4))]/(Sqrt[3]*a^(3/4)) - ArcTanh[(a^(3/4)*(1 - Sqrt[a + 3*x^2]/Sqrt[a] 
))/(Sqrt[3]*x*(a + 3*x^2)^(1/4))]/(2*Sqrt[3]*a^(3/4))
 

Defintions of rubi rules used

rule 308

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Wit 
h[{q = Rt[b^2/a, 4]}, Simp[(-b/(2*a*d*q))*ArcTan[(b + q^2*Sqrt[a + b*x^2])/ 
(q^3*x*(a + b*x^2)^(1/4))], x] - Simp[(b/(2*a*d*q))*ArcTanh[(b - q^2*Sqrt[a 
 + b*x^2])/(q^3*x*(a + b*x^2)^(1/4))], x]] /; FreeQ[{a, b, c, d}, x] && EqQ 
[b*c - 2*a*d, 0] && PosQ[b^2/a]
 

Maple [F]

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

Input:

int(1/(3*x^2+a)^(1/4)/(3*x^2+2*a),x)
 

Output:

int(1/(3*x^2+a)^(1/4)/(3*x^2+2*a),x)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.81 (sec) , antiderivative size = 383, normalized size of antiderivative = 3.19 \[ \int \frac {1}{\sqrt [4]{a+3 x^2} \left (2 a+3 x^2\right )} \, dx=-\frac {1}{4} \, \left (\frac {1}{36}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3}}\right )^{\frac {1}{4}} \log \left (\frac {18 \, \left (\frac {1}{36}\right )^{\frac {3}{4}} \sqrt {3 \, x^{2} + a} a^{2} x \left (-\frac {1}{a^{3}}\right )^{\frac {3}{4}} + {\left (3 \, x^{2} + a\right )}^{\frac {1}{4}} a^{2} \sqrt {-\frac {1}{a^{3}}} - 3 \, \left (\frac {1}{36}\right )^{\frac {1}{4}} a x \left (-\frac {1}{a^{3}}\right )^{\frac {1}{4}} + {\left (3 \, x^{2} + a\right )}^{\frac {3}{4}}}{3 \, x^{2} + 2 \, a}\right ) + \frac {1}{4} \, \left (\frac {1}{36}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3}}\right )^{\frac {1}{4}} \log \left (-\frac {18 \, \left (\frac {1}{36}\right )^{\frac {3}{4}} \sqrt {3 \, x^{2} + a} a^{2} x \left (-\frac {1}{a^{3}}\right )^{\frac {3}{4}} - {\left (3 \, x^{2} + a\right )}^{\frac {1}{4}} a^{2} \sqrt {-\frac {1}{a^{3}}} - 3 \, \left (\frac {1}{36}\right )^{\frac {1}{4}} a x \left (-\frac {1}{a^{3}}\right )^{\frac {1}{4}} - {\left (3 \, x^{2} + a\right )}^{\frac {3}{4}}}{3 \, x^{2} + 2 \, a}\right ) + \frac {1}{4} i \, \left (\frac {1}{36}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3}}\right )^{\frac {1}{4}} \log \left (\frac {18 i \, \left (\frac {1}{36}\right )^{\frac {3}{4}} \sqrt {3 \, x^{2} + a} a^{2} x \left (-\frac {1}{a^{3}}\right )^{\frac {3}{4}} - {\left (3 \, x^{2} + a\right )}^{\frac {1}{4}} a^{2} \sqrt {-\frac {1}{a^{3}}} + 3 i \, \left (\frac {1}{36}\right )^{\frac {1}{4}} a x \left (-\frac {1}{a^{3}}\right )^{\frac {1}{4}} + {\left (3 \, x^{2} + a\right )}^{\frac {3}{4}}}{3 \, x^{2} + 2 \, a}\right ) - \frac {1}{4} i \, \left (\frac {1}{36}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3}}\right )^{\frac {1}{4}} \log \left (\frac {-18 i \, \left (\frac {1}{36}\right )^{\frac {3}{4}} \sqrt {3 \, x^{2} + a} a^{2} x \left (-\frac {1}{a^{3}}\right )^{\frac {3}{4}} - {\left (3 \, x^{2} + a\right )}^{\frac {1}{4}} a^{2} \sqrt {-\frac {1}{a^{3}}} - 3 i \, \left (\frac {1}{36}\right )^{\frac {1}{4}} a x \left (-\frac {1}{a^{3}}\right )^{\frac {1}{4}} + {\left (3 \, x^{2} + a\right )}^{\frac {3}{4}}}{3 \, x^{2} + 2 \, a}\right ) \] Input:

integrate(1/(3*x^2+a)^(1/4)/(3*x^2+2*a),x, algorithm="fricas")
 

Output:

-1/4*(1/36)^(1/4)*(-1/a^3)^(1/4)*log((18*(1/36)^(3/4)*sqrt(3*x^2 + a)*a^2* 
x*(-1/a^3)^(3/4) + (3*x^2 + a)^(1/4)*a^2*sqrt(-1/a^3) - 3*(1/36)^(1/4)*a*x 
*(-1/a^3)^(1/4) + (3*x^2 + a)^(3/4))/(3*x^2 + 2*a)) + 1/4*(1/36)^(1/4)*(-1 
/a^3)^(1/4)*log(-(18*(1/36)^(3/4)*sqrt(3*x^2 + a)*a^2*x*(-1/a^3)^(3/4) - ( 
3*x^2 + a)^(1/4)*a^2*sqrt(-1/a^3) - 3*(1/36)^(1/4)*a*x*(-1/a^3)^(1/4) - (3 
*x^2 + a)^(3/4))/(3*x^2 + 2*a)) + 1/4*I*(1/36)^(1/4)*(-1/a^3)^(1/4)*log((1 
8*I*(1/36)^(3/4)*sqrt(3*x^2 + a)*a^2*x*(-1/a^3)^(3/4) - (3*x^2 + a)^(1/4)* 
a^2*sqrt(-1/a^3) + 3*I*(1/36)^(1/4)*a*x*(-1/a^3)^(1/4) + (3*x^2 + a)^(3/4) 
)/(3*x^2 + 2*a)) - 1/4*I*(1/36)^(1/4)*(-1/a^3)^(1/4)*log((-18*I*(1/36)^(3/ 
4)*sqrt(3*x^2 + a)*a^2*x*(-1/a^3)^(3/4) - (3*x^2 + a)^(1/4)*a^2*sqrt(-1/a^ 
3) - 3*I*(1/36)^(1/4)*a*x*(-1/a^3)^(1/4) + (3*x^2 + a)^(3/4))/(3*x^2 + 2*a 
))
 

Sympy [F]

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

integrate(1/(3*x**2+a)**(1/4)/(3*x**2+2*a),x)
 

Output:

Integral(1/((a + 3*x**2)**(1/4)*(2*a + 3*x**2)), x)
 

Maxima [F]

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

integrate(1/(3*x^2+a)^(1/4)/(3*x^2+2*a),x, algorithm="maxima")
 

Output:

integrate(1/((3*x^2 + 2*a)*(3*x^2 + a)^(1/4)), x)
 

Giac [F]

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

integrate(1/(3*x^2+a)^(1/4)/(3*x^2+2*a),x, algorithm="giac")
 

Output:

integrate(1/((3*x^2 + 2*a)*(3*x^2 + a)^(1/4)), x)
 

Mupad [F(-1)]

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

int(1/((2*a + 3*x^2)*(a + 3*x^2)^(1/4)),x)
                                                                                    
                                                                                    
 

Output:

int(1/((2*a + 3*x^2)*(a + 3*x^2)^(1/4)), x)
 

Reduce [F]

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

int(1/(3*x^2+a)^(1/4)/(3*x^2+2*a),x)
 

Output:

int(1/(2*(a + 3*x**2)**(1/4)*a + 3*(a + 3*x**2)**(1/4)*x**2),x)