\(\int \frac {1}{(-a+x^2) (\sqrt {2} a+x^2) (\sqrt {3} a+x^2)^2} \, dx\) [27]

Optimal result

Integrand size = 36, antiderivative size = 251 \[ \int \frac {1}{\left (-a+x^2\right ) \left (\sqrt {2} a+x^2\right ) \left (\sqrt {3} a+x^2\right )^2} \, dx=\frac {x}{2 \sqrt {3} \left (3-\sqrt {2}+\sqrt {3}-\sqrt {6}\right ) a^3 \left (\sqrt {3} a+x^2\right )}+\frac {\left (1-\sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt [4]{2} \sqrt {a}}\right )}{\sqrt [4]{2} \left (1+\sqrt {3}\right )^2 a^{7/2}}-\frac {\left (1-\sqrt {2}+2 \sqrt {3}\right ) \left (5+2 \sqrt {6}\right ) \arctan \left (\frac {x}{\sqrt [4]{2} \sqrt {a}}\right )}{\sqrt [4]{2} \left (1+\sqrt {3}\right )^2 a^{7/2}}+\frac {\left (15-\sqrt {2}+3 \sqrt {3}-3 \sqrt {6}\right ) \left (5+2 \sqrt {6}\right ) \arctan \left (\frac {x}{\sqrt [4]{3} \sqrt {a}}\right )}{2\ 3^{3/4} \left (1+\sqrt {3}\right )^2 a^{7/2}}+\frac {\left (1-\sqrt {2}\right ) \text {arctanh}\left (\frac {x}{\sqrt {a}}\right )}{\left (1+\sqrt {3}\right )^2 a^{7/2}} \] Output:

1/6*x*3^(1/2)/(3-2^(1/2)+3^(1/2)-6^(1/2))/a^3/(3^(1/2)*a+x^2)+1/2*(1-2^(1/ 
2))*arctan(1/2*x*2^(3/4)/a^(1/2))*2^(3/4)/(1+3^(1/2))^2/a^(7/2)-1/2*(1-2^( 
1/2)+2*3^(1/2))*(5+2*6^(1/2))*arctan(1/2*x*2^(3/4)/a^(1/2))*2^(3/4)/(1+3^( 
1/2))^2/a^(7/2)+1/6*(15-2^(1/2)+3*3^(1/2)-3*6^(1/2))*(5+2*6^(1/2))*arctan( 
1/3*x*3^(3/4)/a^(1/2))*3^(1/4)/(1+3^(1/2))^2/a^(7/2)+(1-2^(1/2))*arctanh(x 
/a^(1/2))/(1+3^(1/2))^2/a^(7/2)
 

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (-a+x^2\right ) \left (\sqrt {2} a+x^2\right ) \left (\sqrt {3} a+x^2\right )^2} \, dx=\frac {\frac {\sqrt {3} \left (-3+\sqrt {2}-\sqrt {3}+\sqrt {6}\right ) \sqrt {a} x}{\left (-10+6 \sqrt {2}-5 \sqrt {3}+4 \sqrt {6}\right ) \left (\sqrt {3} a+x^2\right )}+\frac {2\ 2^{3/4} \sqrt {3} \left (-2729524173+1929898200 \sqrt {2}-1575755281 \sqrt {3}[phone] \sqrt {6}\right ) \arctan \left (\frac {x}{\sqrt [4]{2} \sqrt {a}}\right )}{-6462963267+4567702969 \sqrt {2}-3729513857 \sqrt {3}[phone] \sqrt {6}}+\frac {3^{3/4} \left (-93230995+65540631 \sqrt {2}-53513703 \sqrt {3}[phone] \sqrt {6}\right ) \arctan \left (\frac {x}{\sqrt [4]{3} \sqrt {a}}\right )}{-205326435+145299702 \sqrt {2}-118636710 \sqrt {3}[phone] \sqrt {6}}+\frac {6 \left (-4177234609+2957309786 \sqrt {2}-2414633329 \sqrt {3}[phone] \sqrt {6}\right ) \text {arctanh}\left (\frac {x}{\sqrt {a}}\right )}{-6462963267+4567702969 \sqrt {2}-3729513857 \sqrt {3}[phone] \sqrt {6}}}{12 a^{7/2}} \] Input:

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

Output:

((Sqrt[3]*(-3 + Sqrt[2] - Sqrt[3] + Sqrt[6])*Sqrt[a]*x)/((-10 + 6*Sqrt[2] 
- 5*Sqrt[3] + 4*Sqrt[6])*(Sqrt[3]*a + x^2)) + (2*2^(3/4)*Sqrt[3]*(-2729524 
173 + 1929898200*Sqrt[2] - 1575755281*Sqrt[3] + 1114323578*Sqrt[6])*ArcTan 
[x/(2^(1/4)*Sqrt[a])])/(-6462963267 + 4567702969*Sqrt[2] - 3729513857*Sqrt 
[3] + 2638493705*Sqrt[6]) + (3^(3/4)*(-93230995 + 65540631*Sqrt[2] - 53513 
703*Sqrt[3] + 38061393*Sqrt[6])*ArcTan[x/(3^(1/4)*Sqrt[a])])/(-205326435 + 
 145299702*Sqrt[2] - 118636710*Sqrt[3] + 83824166*Sqrt[6]) + (6*(-41772346 
09 + 2957309786*Sqrt[2] - 2414633329*Sqrt[3] + 1705348888*Sqrt[6])*ArcTanh 
[x/Sqrt[a]])/(-6462963267 + 4567702969*Sqrt[2] - 3729513857*Sqrt[3] + 2638 
493705*Sqrt[6]))/(12*a^(7/2))
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {421, 25, 303, 216, 219, 402, 27, 397, 216}

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.

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 303

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 402

\(\displaystyle -\frac {\frac {\left (3+\sqrt {3}\right ) x}{6 \left (\sqrt {2}-\sqrt {3}\right ) a \left (\sqrt {3} a+x^2\right )}-\frac {\int \frac {a \left (\left (12-\sqrt {2}+2 \sqrt {3}-3 \sqrt {6}\right ) a-\left (1+\sqrt {3}\right ) x^2\right )}{\left (x^2+\sqrt {2} a\right ) \left (x^2+\sqrt {3} a\right )}dx}{2 \sqrt {3} \left (\sqrt {2}-\sqrt {3}\right ) a^2}}{\left (1+\sqrt {3}\right )^2 a^2}-\frac {-\frac {\left (1-\sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt [4]{2} \sqrt {a}}\right )}{\sqrt [4]{2} a^{3/2}}-\frac {\left (1-\sqrt {2}\right ) \text {arctanh}\left (\frac {x}{\sqrt {a}}\right )}{a^{3/2}}}{\left (1+\sqrt {3}\right )^2 a^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\left (3+\sqrt {3}\right ) x}{6 \left (\sqrt {2}-\sqrt {3}\right ) a \left (\sqrt {3} a+x^2\right )}-\frac {\int \frac {\left (12-\sqrt {2}+2 \sqrt {3}-3 \sqrt {6}\right ) a-\left (1+\sqrt {3}\right ) x^2}{\left (x^2+\sqrt {2} a\right ) \left (x^2+\sqrt {3} a\right )}dx}{2 \sqrt {3} \left (\sqrt {2}-\sqrt {3}\right ) a}}{\left (1+\sqrt {3}\right )^2 a^2}-\frac {-\frac {\left (1-\sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt [4]{2} \sqrt {a}}\right )}{\sqrt [4]{2} a^{3/2}}-\frac {\left (1-\sqrt {2}\right ) \text {arctanh}\left (\frac {x}{\sqrt {a}}\right )}{a^{3/2}}}{\left (1+\sqrt {3}\right )^2 a^2}\)

\(\Big \downarrow \) 397

\(\displaystyle -\frac {\frac {\left (3+\sqrt {3}\right ) x}{6 \left (\sqrt {2}-\sqrt {3}\right ) a \left (\sqrt {3} a+x^2\right )}-\frac {\frac {\left (15-\sqrt {2}+3 \sqrt {3}-3 \sqrt {6}\right ) \int \frac {1}{x^2+\sqrt {3} a}dx}{\sqrt {2}-\sqrt {3}}-\frac {2 \left (6+\sqrt {3} \left (1-\sqrt {2}\right )\right ) \int \frac {1}{x^2+\sqrt {2} a}dx}{\sqrt {2}-\sqrt {3}}}{2 \sqrt {3} \left (\sqrt {2}-\sqrt {3}\right ) a}}{\left (1+\sqrt {3}\right )^2 a^2}-\frac {-\frac {\left (1-\sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt [4]{2} \sqrt {a}}\right )}{\sqrt [4]{2} a^{3/2}}-\frac {\left (1-\sqrt {2}\right ) \text {arctanh}\left (\frac {x}{\sqrt {a}}\right )}{a^{3/2}}}{\left (1+\sqrt {3}\right )^2 a^2}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {\frac {\left (3+\sqrt {3}\right ) x}{6 \left (\sqrt {2}-\sqrt {3}\right ) a \left (\sqrt {3} a+x^2\right )}-\frac {\frac {\left (15-\sqrt {2}+3 \sqrt {3}-3 \sqrt {6}\right ) \arctan \left (\frac {x}{\sqrt [4]{3} \sqrt {a}}\right )}{\sqrt [4]{3} \left (\sqrt {2}-\sqrt {3}\right ) \sqrt {a}}-\frac {2^{3/4} \left (6+\sqrt {3} \left (1-\sqrt {2}\right )\right ) \arctan \left (\frac {x}{\sqrt [4]{2} \sqrt {a}}\right )}{\left (\sqrt {2}-\sqrt {3}\right ) \sqrt {a}}}{2 \sqrt {3} \left (\sqrt {2}-\sqrt {3}\right ) a}}{\left (1+\sqrt {3}\right )^2 a^2}-\frac {-\frac {\left (1-\sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt [4]{2} \sqrt {a}}\right )}{\sqrt [4]{2} a^{3/2}}-\frac {\left (1-\sqrt {2}\right ) \text {arctanh}\left (\frac {x}{\sqrt {a}}\right )}{a^{3/2}}}{\left (1+\sqrt {3}\right )^2 a^2}\)

Input:

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

Output:

-((((3 + Sqrt[3])*x)/(6*(Sqrt[2] - Sqrt[3])*a*(Sqrt[3]*a + x^2)) - (-((2^( 
3/4)*(6 + Sqrt[3]*(1 - Sqrt[2]))*ArcTan[x/(2^(1/4)*Sqrt[a])])/((Sqrt[2] - 
Sqrt[3])*Sqrt[a])) + ((15 - Sqrt[2] + 3*Sqrt[3] - 3*Sqrt[6])*ArcTan[x/(3^( 
1/4)*Sqrt[a])])/(3^(1/4)*(Sqrt[2] - Sqrt[3])*Sqrt[a]))/(2*Sqrt[3]*(Sqrt[2] 
 - Sqrt[3])*a))/((1 + Sqrt[3])^2*a^2)) - (-(((1 - Sqrt[2])*ArcTan[x/(2^(1/ 
4)*Sqrt[a])])/(2^(1/4)*a^(3/2))) - ((1 - Sqrt[2])*ArcTanh[x/Sqrt[a]])/a^(3 
/2))/((1 + Sqrt[3])^2*a^2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 303

Int[1/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Simp[b/(b 
*c - a*d)   Int[1/(a + b*x^2), x], x] - Simp[d/(b*c - a*d)   Int[1/(c + d*x 
^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
 

rule 397

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ 
Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^2), x], x] - Simp[ 
(d*e - c*f)/(b*c - a*d)   Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e 
, f}, x]
 

rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x 
_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ 
(q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) 
*(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, q}, x] && LtQ[p, -1]
 

rule 421

Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( 
x_)^2), x_Symbol] :> Simp[b^2/(b*c - a*d)^2   Int[(c + d*x^2)^(q + 2)*((e + 
 f*x^2)^r/(a + b*x^2)), x], x] - Simp[d/(b*c - a*d)^2   Int[(c + d*x^2)^q*( 
e + f*x^2)^r*(2*b*c - a*d + b*d*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, r} 
, x] && LtQ[q, -1]
 

Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.66

method	result	size
default	\(-\frac {6 \left (\frac {\left (\frac {\sqrt {2}}{6}-\frac {1}{6}-\frac {\sqrt {3}}{6}+\frac {\sqrt {3}\, \sqrt {2}}{18}\right ) x}{\sqrt {3}\, a +x^{2}}+\frac {73 \left (\frac {3 \sqrt {2}}{73}-\frac {3}{73}-\frac {5 \sqrt {3}}{73}+\frac {\sqrt {3}\, \sqrt {2}}{219}\right ) \arctan \left (\frac {x}{\sqrt {\sqrt {3}\, a}}\right )}{6 \sqrt {\sqrt {3}\, a}}\right )}{\left (\sqrt {3}\, \sqrt {2}-2\right )^{2} \left (1+\sqrt {3}\right )^{2} a^{3}}-\frac {3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {x}{\sqrt {a}}\right )}{a^{\frac {7}{2}} \left (2+\sqrt {2}\right ) \left (3+\sqrt {3}\right )^{2}}-\frac {3 \arctan \left (\frac {x}{\sqrt {\sqrt {2}\, a}}\right )}{a^{3} \left (1+\sqrt {2}\right ) \left (\sqrt {3}\, \sqrt {2}-3\right )^{2} \sqrt {\sqrt {2}\, a}}\)	\(166\)

Input:

int(1/(x^2-a)/(2^(1/2)*a+x^2)/(3^(1/2)*a+x^2)^2,x,method=_RETURNVERBOSE)
 

Output:

-6/(3^(1/2)*2^(1/2)-2)^2/(1+3^(1/2))^2/a^3*((1/6*2^(1/2)-1/6-1/6*3^(1/2)+1 
/18*3^(1/2)*2^(1/2))*x/(3^(1/2)*a+x^2)+73/6*(3/73*2^(1/2)-3/73-5/73*3^(1/2 
)+1/219*3^(1/2)*2^(1/2))/(3^(1/2)*a)^(1/2)*arctan(x/(3^(1/2)*a)^(1/2)))-3* 
2^(1/2)/a^(7/2)/(2+2^(1/2))/(3+3^(1/2))^2*arctanh(x/a^(1/2))-3/a^3/(1+2^(1 
/2))/(3^(1/2)*2^(1/2)-3)^2/(2^(1/2)*a)^(1/2)*arctan(x/(2^(1/2)*a)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 3389, normalized size of antiderivative = 13.50 \[ \int \frac {1}{\left (-a+x^2\right ) \left (\sqrt {2} a+x^2\right ) \left (\sqrt {3} a+x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (-a+x^2\right ) \left (\sqrt {2} a+x^2\right ) \left (\sqrt {3} a+x^2\right )^2} \, dx=-\frac {x}{2 \, {\left ({\left (\sqrt {3} \sqrt {2} - 3 \, \sqrt {3} + 3 \, \sqrt {2} - 3\right )} a^{3} x^{2} + 3 \, {\left (\sqrt {3} {\left (\sqrt {2} - 1\right )} + \sqrt {2} - 3\right )} a^{4}\right )}} + \frac {3^{\frac {3}{4}} {\left (3 \, \sqrt {3} {\left (\sqrt {2} - 1\right )} + \sqrt {2} - 15\right )} \arctan \left (\frac {3^{\frac {3}{4}} x}{3 \, \sqrt {a}}\right )}{6 \, {\left (3 \, \sqrt {3} {\left (4 \, \sqrt {2} - 3\right )} - 11 \, \sqrt {3} + 24 \, \sqrt {2} - 30\right )} a^{\frac {7}{2}}} + \frac {2^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {3}{4}} x}{2 \, \sqrt {a}}\right )}{2 \, {\left (2 \, \sqrt {3} {\left (\sqrt {2} + 2\right )} - 5 \, \sqrt {2} - 5\right )} a^{\frac {7}{2}}} + \frac {\log \left (\frac {x - \sqrt {a}}{x + \sqrt {a}}\right )}{4 \, {\left (\sqrt {3} {\left (\sqrt {2} + 1\right )} + 2 \, \sqrt {2} + 2\right )} a^{\frac {7}{2}}} \] Input:

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

Output:

-1/2*x/((sqrt(3)*sqrt(2) - 3*sqrt(3) + 3*sqrt(2) - 3)*a^3*x^2 + 3*(sqrt(3) 
*(sqrt(2) - 1) + sqrt(2) - 3)*a^4) + 1/6*3^(3/4)*(3*sqrt(3)*(sqrt(2) - 1) 
+ sqrt(2) - 15)*arctan(1/3*3^(3/4)*x/sqrt(a))/((3*sqrt(3)*(4*sqrt(2) - 3) 
- 11*sqrt(3) + 24*sqrt(2) - 30)*a^(7/2)) + 1/2*2^(3/4)*arctan(1/2*2^(3/4)* 
x/sqrt(a))/((2*sqrt(3)*(sqrt(2) + 2) - 5*sqrt(2) - 5)*a^(7/2)) + 1/4*log(( 
x - sqrt(a))/(x + sqrt(a)))/((sqrt(3)*(sqrt(2) + 1) + 2*sqrt(2) + 2)*a^(7/ 
2))
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\left (-a+x^2\right ) \left (\sqrt {2} a+x^2\right ) \left (\sqrt {3} a+x^2\right )^2} \, dx=\frac {2^{\frac {3}{4}} {\left (202 \, \sqrt {3} \sqrt {2} - 284 \, \sqrt {3} + 359 \, \sqrt {2} - 479\right )} \arctan \left (\frac {2^{\frac {3}{4}} x}{2 \, \sqrt {a}}\right )}{2 \, {\left (888 \, \sqrt {3} \sqrt {2} - 1080 \, \sqrt {3} + 1320 \, \sqrt {2} - 2179\right )} a^{\frac {7}{2}}} - \frac {{\left (437 \, \sqrt {3} \sqrt {2} - 893 \, \sqrt {3} + 1094 \, \sqrt {2} - 1070\right )} \arctan \left (\frac {x}{\sqrt {-a}}\right )}{2 \, {\left (888 \, \sqrt {3} \sqrt {2} - 1080 \, \sqrt {3} + 1320 \, \sqrt {2} - 2179\right )} \sqrt {-a} a^{3}} + \frac {{\left (4 \, \sqrt {3} \sqrt {2} - 9 \, \sqrt {3} + 10 \, \sqrt {2} - 11\right )} x}{4 \, {\left (x^{2} + \sqrt {3} a\right )} {\left (40 \, \sqrt {3} \sqrt {2} - 52 \, \sqrt {3} + 63 \, \sqrt {2} - 99\right )} a^{3}} - \frac {{\left (6417 \, \sqrt {3} \sqrt {2} - 9409 \, \sqrt {3} + 11499 \, \sqrt {2} - 15753\right )} \arctan \left (\frac {40 \, \sqrt {3} \sqrt {2} x - 52 \, \sqrt {3} x + 63 \, \sqrt {2} x - 99 \, x}{\sqrt {-24954 \, \sqrt {3} \sqrt {2} + 35451 \, \sqrt {3} - 43416 \, \sqrt {2} + 61128} \sqrt {a}}\right )}{4 \, {\left (40 \, \sqrt {3} \sqrt {2} - 52 \, \sqrt {3} + 63 \, \sqrt {2} - 99\right )} \sqrt {-24954 \, \sqrt {3} \sqrt {2} + 35451 \, \sqrt {3} - 43416 \, \sqrt {2} + 61128} a^{\frac {7}{2}}} \] Input:

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

Output:

1/2*2^(3/4)*(202*sqrt(3)*sqrt(2) - 284*sqrt(3) + 359*sqrt(2) - 479)*arctan 
(1/2*2^(3/4)*x/sqrt(a))/((888*sqrt(3)*sqrt(2) - 1080*sqrt(3) + 1320*sqrt(2 
) - 2179)*a^(7/2)) - 1/2*(437*sqrt(3)*sqrt(2) - 893*sqrt(3) + 1094*sqrt(2) 
 - 1070)*arctan(x/sqrt(-a))/((888*sqrt(3)*sqrt(2) - 1080*sqrt(3) + 1320*sq 
rt(2) - 2179)*sqrt(-a)*a^3) + 1/4*(4*sqrt(3)*sqrt(2) - 9*sqrt(3) + 10*sqrt 
(2) - 11)*x/((x^2 + sqrt(3)*a)*(40*sqrt(3)*sqrt(2) - 52*sqrt(3) + 63*sqrt( 
2) - 99)*a^3) - 1/4*(6417*sqrt(3)*sqrt(2) - 9409*sqrt(3) + 11499*sqrt(2) - 
 15753)*arctan((40*sqrt(3)*sqrt(2)*x - 52*sqrt(3)*x + 63*sqrt(2)*x - 99*x) 
/(sqrt(-24954*sqrt(3)*sqrt(2) + 35451*sqrt(3) - 43416*sqrt(2) + 61128)*sqr 
t(a)))/((40*sqrt(3)*sqrt(2) - 52*sqrt(3) + 63*sqrt(2) - 99)*sqrt(-24954*sq 
rt(3)*sqrt(2) + 35451*sqrt(3) - 43416*sqrt(2) + 61128)*a^(7/2))
 

Mupad [B] (verification not implemented)

Time = 68.70 (sec) , antiderivative size = 3037, normalized size of antiderivative = 12.10 \[ \int \frac {1}{\left (-a+x^2\right ) \left (\sqrt {2} a+x^2\right ) \left (\sqrt {3} a+x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

symsum(log(-root(3039952896*2^(1/2)*3^(1/2)*6^(1/2)*a^21*z^6 + 15904014336 
*2^(1/2)*3^(1/2)*a^21*z^6 - 2452488192*2^(1/2)*6^(1/2)*a^21*z^6 + 27014529 
024*3^(1/2)*6^(1/2)*a^21*z^6 + 4904976384*3^(1/2)*a^21*z^6 - 81043587072*2 
^(1/2)*a^21*z^6 - 15904014336*6^(1/2)*a^21*z^6 - 18239717376*a^21*z^6 + 68 
281344*2^(1/2)*3^(1/2)*6^(1/2)*a^14*z^4 + 150165504*2^(1/2)*3^(1/2)*a^14*z 
^4 + 102274560*2^(1/2)*6^(1/2)*a^14*z^4 - 568320*3^(1/2)*6^(1/2)*a^14*z^4 
- 204549120*3^(1/2)*a^14*z^4 - 150165504*6^(1/2)*a^14*z^4 + 1704960*2^(1/2 
)*a^14*z^4 - 409688064*a^14*z^4 + 10880*2^(1/2)*3^(1/2)*6^(1/2)*a^7*z^2 + 
46120*2^(1/2)*6^(1/2)*a^7*z^2 + 41952*3^(1/2)*6^(1/2)*a^7*z^2 - 22848*2^(1 
/2)*3^(1/2)*a^7*z^2 - 126444*2^(1/2)*a^7*z^2 - 91760*3^(1/2)*a^7*z^2 + 231 
68*6^(1/2)*a^7*z^2 - 66064*a^7*z^2 - 1, z, k)*(8*x + root(3039952896*2^(1/ 
2)*3^(1/2)*6^(1/2)*a^21*z^6 + 15904014336*2^(1/2)*3^(1/2)*a^21*z^6 - 24524 
88192*2^(1/2)*6^(1/2)*a^21*z^6 + 27014529024*3^(1/2)*6^(1/2)*a^21*z^6 + 49 
04976384*3^(1/2)*a^21*z^6 - 81043587072*2^(1/2)*a^21*z^6 - 15904014336*6^( 
1/2)*a^21*z^6 - 18239717376*a^21*z^6 + 68281344*2^(1/2)*3^(1/2)*6^(1/2)*a^ 
14*z^4 + 150165504*2^(1/2)*3^(1/2)*a^14*z^4 + 102274560*2^(1/2)*6^(1/2)*a^ 
14*z^4 - 568320*3^(1/2)*6^(1/2)*a^14*z^4 - 204549120*3^(1/2)*a^14*z^4 - 15 
0165504*6^(1/2)*a^14*z^4 + 1704960*2^(1/2)*a^14*z^4 - 409688064*a^14*z^4 + 
 10880*2^(1/2)*3^(1/2)*6^(1/2)*a^7*z^2 + 46120*2^(1/2)*6^(1/2)*a^7*z^2 + 4 
1952*3^(1/2)*6^(1/2)*a^7*z^2 - 22848*2^(1/2)*3^(1/2)*a^7*z^2 - 126444*2...
 

Reduce [F]

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

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

Output:

(6*sqrt(a)*sqrt(3)*3**(1/4)*atan(x/(sqrt(a)*3**(1/4)))*a**2 - 2*sqrt(a)*sq 
rt(3)*3**(1/4)*atan(x/(sqrt(a)*3**(1/4)))*x**4 - 3*sqrt(a)*sqrt(3)*3**(1/4 
)*log(sqrt(a)*3**(1/4) - x)*a**2 + sqrt(a)*sqrt(3)*3**(1/4)*log(sqrt(a)*3* 
*(1/4) - x)*x**4 + 3*sqrt(a)*sqrt(3)*3**(1/4)*log(sqrt(a)*3**(1/4) + x)*a* 
*2 - sqrt(a)*sqrt(3)*3**(1/4)*log(sqrt(a)*3**(1/4) + x)*x**4 - 144*sqrt(6) 
*int(x**4/(18*a**7 - 18*a**6*x**2 - 21*a**5*x**4 + 21*a**4*x**6 + 8*a**3*x 
**8 - 8*a**2*x**10 - a*x**12 + x**14),x)*a**7 + 48*sqrt(6)*int(x**4/(18*a* 
*7 - 18*a**6*x**2 - 21*a**5*x**4 + 21*a**4*x**6 + 8*a**3*x**8 - 8*a**2*x** 
10 - a*x**12 + x**14),x)*a**5*x**4 - 432*sqrt(6)*int(1/(18*a**7 - 18*a**6* 
x**2 - 21*a**5*x**4 + 21*a**4*x**6 + 8*a**3*x**8 - 8*a**2*x**10 - a*x**12 
+ x**14),x)*a**9 + 144*sqrt(6)*int(1/(18*a**7 - 18*a**6*x**2 - 21*a**5*x** 
4 + 21*a**4*x**6 + 8*a**3*x**8 - 8*a**2*x**10 - a*x**12 + x**14),x)*a**7*x 
**4 + 144*sqrt(3)*int(x**4/(18*a**7 - 18*a**6*x**2 - 21*a**5*x**4 + 21*a** 
4*x**6 + 8*a**3*x**8 - 8*a**2*x**10 - a*x**12 + x**14),x)*a**7 - 48*sqrt(3 
)*int(x**4/(18*a**7 - 18*a**6*x**2 - 21*a**5*x**4 + 21*a**4*x**6 + 8*a**3* 
x**8 - 8*a**2*x**10 - a*x**12 + x**14),x)*a**5*x**4 + 720*sqrt(3)*int(x**2 
/(18*a**7 - 18*a**6*x**2 - 21*a**5*x**4 + 21*a**4*x**6 + 8*a**3*x**8 - 8*a 
**2*x**10 - a*x**12 + x**14),x)*a**8 - 240*sqrt(3)*int(x**2/(18*a**7 - 18* 
a**6*x**2 - 21*a**5*x**4 + 21*a**4*x**6 + 8*a**3*x**8 - 8*a**2*x**10 - a*x 
**12 + x**14),x)*a**6*x**4 - 288*sqrt(3)*int(1/(18*a**7 - 18*a**6*x**2 ...