\(\int \frac {1}{\sqrt {a+b x^2} (c+d x^2) (e+f x^2)} \, dx\) [331]

Optimal result

Integrand size = 30, antiderivative size = 122 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)} \] Output:

d*arctanh((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2))/c^(1/2)/(-a*d+b*c)^( 
1/2)/(-c*f+d*e)-f*arctanh((-a*f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/2))/e^(1 
/2)/(-a*f+b*e)^(1/2)/(-c*f+d*e)
 

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\frac {\frac {d \arctan \left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {-b c+a d}}\right )}{\sqrt {c} \sqrt {-b c+a d}}-\frac {f \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{\sqrt {e} \sqrt {-b e+a f}}}{-d e+c f} \] Input:

Integrate[1/(Sqrt[a + b*x^2]*(c + d*x^2)*(e + f*x^2)),x]
 

Output:

((d*ArcTan[(-(d*x*Sqrt[a + b*x^2]) + Sqrt[b]*(c + d*x^2))/(Sqrt[c]*Sqrt[-( 
b*c) + a*d])])/(Sqrt[c]*Sqrt[-(b*c) + a*d]) - (f*ArcTan[(-(f*x*Sqrt[a + b* 
x^2]) + Sqrt[b]*(e + f*x^2))/(Sqrt[e]*Sqrt[-(b*e) + a*f])])/(Sqrt[e]*Sqrt[ 
-(b*e) + a*f]))/(-(d*e) + c*f)
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {407, 291, 221}

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.

\(\displaystyle \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )} \, dx\)

\(\Big \downarrow \) 407

\(\displaystyle \frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{d e-c f}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {d \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d e-c f}-\frac {f \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d e-c f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {d \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\)

Input:

Int[1/(Sqrt[a + b*x^2]*(c + d*x^2)*(e + f*x^2)),x]
 

Output:

(d*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sqrt[c]*Sqrt[b 
*c - a*d]*(d*e - c*f)) - (f*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + 
b*x^2])])/(Sqrt[e]*Sqrt[b*e - a*f]*(d*e - c*f))
 

Defintions of rubi rules used

rule 221

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

rule 291

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

rule 407

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

Maple [A] (verified)

Time = 1.34 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.79

method	result	size
pseudoelliptic	\(\frac {\frac {d \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )}{\sqrt {\left (a d -b c \right ) c}}-\frac {f \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )}{\sqrt {\left (a f -b e \right ) e}}}{c f -d e}\)	\(96\)
default	\(-\frac {d \,f^{2} \ln \left (\frac {\frac {2 a f -2 b e}{f}+\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x -\frac {\sqrt {-e f}}{f}\right )^{2} b +\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x -\frac {\sqrt {-e f}}{f}}\right )}{2 \left (\sqrt {-c d}\, f +\sqrt {-e f}\, d \right ) \left (\sqrt {-e f}\, d -\sqrt {-c d}\, f \right ) \sqrt {-e f}\, \sqrt {\frac {a f -b e}{f}}}+\frac {d \,f^{2} \ln \left (\frac {\frac {2 a f -2 b e}{f}-\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x +\frac {\sqrt {-e f}}{f}\right )^{2} b -\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x +\frac {\sqrt {-e f}}{f}}\right )}{2 \left (\sqrt {-c d}\, f +\sqrt {-e f}\, d \right ) \left (\sqrt {-e f}\, d -\sqrt {-c d}\, f \right ) \sqrt {-e f}\, \sqrt {\frac {a f -b e}{f}}}+\frac {d^{2} f \ln \left (\frac {\frac {2 a d -2 b c}{d}+\frac {2 b \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}+2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} b +\frac {2 b \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}+\frac {a d -b c}{d}}}{x -\frac {\sqrt {-c d}}{d}}\right )}{2 \sqrt {-c d}\, \left (\sqrt {-c d}\, f +\sqrt {-e f}\, d \right ) \left (\sqrt {-e f}\, d -\sqrt {-c d}\, f \right ) \sqrt {\frac {a d -b c}{d}}}-\frac {d^{2} f \ln \left (\frac {\frac {2 a d -2 b c}{d}-\frac {2 b \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}+2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} b -\frac {2 b \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}+\frac {a d -b c}{d}}}{x +\frac {\sqrt {-c d}}{d}}\right )}{2 \sqrt {-c d}\, \left (\sqrt {-c d}\, f +\sqrt {-e f}\, d \right ) \left (\sqrt {-e f}\, d -\sqrt {-c d}\, f \right ) \sqrt {\frac {a d -b c}{d}}}\)	\(770\)

Input:

int(1/(b*x^2+a)^(1/2)/(d*x^2+c)/(f*x^2+e),x,method=_RETURNVERBOSE)
 

Output:

1/(c*f-d*e)*(d/((a*d-b*c)*c)^(1/2)*arctan(c*(b*x^2+a)^(1/2)/x/((a*d-b*c)*c 
)^(1/2))-f/((a*f-b*e)*e)^(1/2)*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1 
/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (102) = 204\).

Time = 52.72 (sec) , antiderivative size = 1289, normalized size of antiderivative = 10.57 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\text {Too large to display} \] Input:

integrate(1/(b*x^2+a)^(1/2)/(d*x^2+c)/(f*x^2+e),x, algorithm="fricas")
 

Output:

[-1/4*((b*c^2 - a*c*d)*sqrt(b*e^2 - a*e*f)*f*log(((8*b^2*e^2 - 8*a*b*e*f + 
 a^2*f^2)*x^4 + a^2*e^2 + 2*(4*a*b*e^2 - 3*a^2*e*f)*x^2 + 4*((2*b*e - a*f) 
*x^3 + a*e*x)*sqrt(b*e^2 - a*e*f)*sqrt(b*x^2 + a))/(f^2*x^4 + 2*e*f*x^2 + 
e^2)) + (b*d*e^2 - a*d*e*f)*sqrt(b*c^2 - a*c*d)*log(((8*b^2*c^2 - 8*a*b*c* 
d + a^2*d^2)*x^4 + a^2*c^2 + 2*(4*a*b*c^2 - 3*a^2*c*d)*x^2 - 4*((2*b*c - a 
*d)*x^3 + a*c*x)*sqrt(b*c^2 - a*c*d)*sqrt(b*x^2 + a))/(d^2*x^4 + 2*c*d*x^2 
 + c^2)))/((b^2*c^2*d - a*b*c*d^2)*e^3 - (b^2*c^3 - a^2*c*d^2)*e^2*f + (a* 
b*c^3 - a^2*c^2*d)*e*f^2), 1/4*(2*(b*c^2 - a*c*d)*sqrt(-b*e^2 + a*e*f)*f*a 
rctan(1/2*sqrt(-b*e^2 + a*e*f)*((2*b*e - a*f)*x^2 + a*e)*sqrt(b*x^2 + a)/( 
(b^2*e^2 - a*b*e*f)*x^3 + (a*b*e^2 - a^2*e*f)*x)) - (b*d*e^2 - a*d*e*f)*sq 
rt(b*c^2 - a*c*d)*log(((8*b^2*c^2 - 8*a*b*c*d + a^2*d^2)*x^4 + a^2*c^2 + 2 
*(4*a*b*c^2 - 3*a^2*c*d)*x^2 - 4*((2*b*c - a*d)*x^3 + a*c*x)*sqrt(b*c^2 - 
a*c*d)*sqrt(b*x^2 + a))/(d^2*x^4 + 2*c*d*x^2 + c^2)))/((b^2*c^2*d - a*b*c* 
d^2)*e^3 - (b^2*c^3 - a^2*c*d^2)*e^2*f + (a*b*c^3 - a^2*c^2*d)*e*f^2), -1/ 
4*((b*c^2 - a*c*d)*sqrt(b*e^2 - a*e*f)*f*log(((8*b^2*e^2 - 8*a*b*e*f + a^2 
*f^2)*x^4 + a^2*e^2 + 2*(4*a*b*e^2 - 3*a^2*e*f)*x^2 + 4*((2*b*e - a*f)*x^3 
 + a*e*x)*sqrt(b*e^2 - a*e*f)*sqrt(b*x^2 + a))/(f^2*x^4 + 2*e*f*x^2 + e^2) 
) + 2*(b*d*e^2 - a*d*e*f)*sqrt(-b*c^2 + a*c*d)*arctan(1/2*sqrt(-b*c^2 + a* 
c*d)*((2*b*c - a*d)*x^2 + a*c)*sqrt(b*x^2 + a)/((b^2*c^2 - a*b*c*d)*x^3 + 
(a*b*c^2 - a^2*c*d)*x)))/((b^2*c^2*d - a*b*c*d^2)*e^3 - (b^2*c^3 - a^2*...
 

Sympy [F]

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

integrate(1/(b*x**2+a)**(1/2)/(d*x**2+c)/(f*x**2+e),x)
 

Output:

Integral(1/(sqrt(a + b*x**2)*(c + d*x**2)*(e + f*x**2)), x)
 

Maxima [F]

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

integrate(1/(b*x^2+a)^(1/2)/(d*x^2+c)/(f*x^2+e),x, algorithm="maxima")
 

Output:

integrate(1/(sqrt(b*x^2 + a)*(d*x^2 + c)*(f*x^2 + e)), x)
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.35 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=-b^{\frac {3}{2}} {\left (\frac {d \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{\sqrt {-b^{2} c^{2} + a b c d} {\left (b d e - b c f\right )}} - \frac {f \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} f + 2 \, b e - a f}{2 \, \sqrt {-b^{2} e^{2} + a b e f}}\right )}{\sqrt {-b^{2} e^{2} + a b e f} {\left (b d e - b c f\right )}}\right )} \] Input:

integrate(1/(b*x^2+a)^(1/2)/(d*x^2+c)/(f*x^2+e),x, algorithm="giac")
 

Output:

-b^(3/2)*(d*arctan(1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2*d + 2*b*c - a*d)/s 
qrt(-b^2*c^2 + a*b*c*d))/(sqrt(-b^2*c^2 + a*b*c*d)*(b*d*e - b*c*f)) - f*ar 
ctan(1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2*f + 2*b*e - a*f)/sqrt(-b^2*e^2 + 
 a*b*e*f))/(sqrt(-b^2*e^2 + a*b*e*f)*(b*d*e - b*c*f)))
 

Mupad [F(-1)]

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

int(1/((a + b*x^2)^(1/2)*(c + d*x^2)*(e + f*x^2)),x)
 

Output:

int(1/((a + b*x^2)^(1/2)*(c + d*x^2)*(e + f*x^2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.59 (sec) , antiderivative size = 508, normalized size of antiderivative = 4.16 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\frac {\sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b \,x^{2}+a}-\sqrt {d}\, \sqrt {b}\, x}{\sqrt {c}\, \sqrt {b}}\right ) a d e f -\sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b \,x^{2}+a}-\sqrt {d}\, \sqrt {b}\, x}{\sqrt {c}\, \sqrt {b}}\right ) b d \,e^{2}+\sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b \,x^{2}+a}+\sqrt {d}\, \sqrt {b}\, x}{\sqrt {c}\, \sqrt {b}}\right ) a d e f -\sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b \,x^{2}+a}+\sqrt {d}\, \sqrt {b}\, x}{\sqrt {c}\, \sqrt {b}}\right ) b d \,e^{2}-\sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}-\sqrt {f}\, \sqrt {b \,x^{2}+a}-\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) a c d f +\sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}-\sqrt {f}\, \sqrt {b \,x^{2}+a}-\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) b \,c^{2} f -\sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}+\sqrt {f}\, \sqrt {b \,x^{2}+a}+\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) a c d f +\sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}+\sqrt {f}\, \sqrt {b \,x^{2}+a}+\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) b \,c^{2} f}{c e \left (a^{2} c d \,f^{2}-a^{2} d^{2} e f -a b \,c^{2} f^{2}+a b \,d^{2} e^{2}+b^{2} c^{2} e f -b^{2} c d \,e^{2}\right )} \] Input:

int(1/(b*x^2+a)^(1/2)/(d*x^2+c)/(f*x^2+e),x)
 

Output:

(sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) 
- sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a*d*e*f - sqrt(c)*sqrt(a*d - b*c)* 
atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqr 
t(c)*sqrt(b)))*b*d*e**2 + sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + 
sqrt(d)*sqrt(a + b*x**2) + sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a*d*e*f - 
 sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x**2) 
+ sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*b*d*e**2 - sqrt(e)*sqrt(a*f - b*e) 
*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sq 
rt(e)*sqrt(b)))*a*c*d*f + sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - 
sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*b*c**2*f 
- sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) + sqrt(f)*sqrt(a + b*x**2) 
 + sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a*c*d*f + sqrt(e)*sqrt(a*f - b*e) 
*atan((sqrt(a*f - b*e) + sqrt(f)*sqrt(a + b*x**2) + sqrt(f)*sqrt(b)*x)/(sq 
rt(e)*sqrt(b)))*b*c**2*f)/(c*e*(a**2*c*d*f**2 - a**2*d**2*e*f - a*b*c**2*f 
**2 + a*b*d**2*e**2 + b**2*c**2*e*f - b**2*c*d*e**2))