\(\int \frac {1}{\sqrt {a+b x^2} (c+d x^2)^2 (e+f x^2)^2} \, dx\) [334]

Optimal result

Integrand size = 30, antiderivative size = 284 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=-\frac {d^3 x \sqrt {a+b x^2}}{2 c (b c-a d) (d e-c f)^2 \left (c+d x^2\right )}-\frac {f^3 x \sqrt {a+b x^2}}{2 e (b e-a f) (d e-c f)^2 \left (e+f x^2\right )}-\frac {d^2 (a d (d e-5 c f)-2 b c (d e-3 c f)) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{3/2} (d e-c f)^3}+\frac {f^2 (2 b e (3 d e-c f)-a f (5 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{2 e^{3/2} (b e-a f)^{3/2} (d e-c f)^3} \] Output:

-1/2*d^3*x*(b*x^2+a)^(1/2)/c/(-a*d+b*c)/(-c*f+d*e)^2/(d*x^2+c)-1/2*f^3*x*( 
b*x^2+a)^(1/2)/e/(-a*f+b*e)/(-c*f+d*e)^2/(f*x^2+e)-1/2*d^2*(a*d*(-5*c*f+d* 
e)-2*b*c*(-3*c*f+d*e))*arctanh((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2)) 
/c^(3/2)/(-a*d+b*c)^(3/2)/(-c*f+d*e)^3+1/2*f^2*(2*b*e*(-c*f+3*d*e)-a*f*(-c 
*f+5*d*e))*arctanh((-a*f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/2))/e^(3/2)/(-a 
*f+b*e)^(3/2)/(-c*f+d*e)^3
 

Mathematica [A] (verified)

Time = 13.04 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.38 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=-\frac {\frac {d^2 (d e-c f) x \left (d \left (a+b x^2\right )-\frac {(2 b c-a d) \left (c+d x^2\right ) \text {arctanh}\left (\sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}\right )}{c \sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}}\right )}{c (b c-a d) \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {f^2 (d e-c f) x \left (f \left (a+b x^2\right )-\frac {(2 b e-a f) \left (e+f x^2\right ) \text {arctanh}\left (\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}\right )}{e \sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}}\right )}{e (b e-a f) \sqrt {a+b x^2} \left (e+f x^2\right )}+\frac {4 d^2 f \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} \sqrt {b c-a d}}-\frac {4 d f^2 \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} \sqrt {b e-a f}}}{2 (d e-c f)^3} \] Input:

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

Output:

-1/2*((d^2*(d*e - c*f)*x*(d*(a + b*x^2) - ((2*b*c - a*d)*(c + d*x^2)*ArcTa 
nh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/(c*Sqrt[((b*c - a*d)*x^2)/(c* 
(a + b*x^2))])))/(c*(b*c - a*d)*Sqrt[a + b*x^2]*(c + d*x^2)) + (f^2*(d*e - 
 c*f)*x*(f*(a + b*x^2) - ((2*b*e - a*f)*(e + f*x^2)*ArcTanh[Sqrt[((b*e - a 
*f)*x^2)/(e*(a + b*x^2))]])/(e*Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))])))/ 
(e*(b*e - a*f)*Sqrt[a + b*x^2]*(e + f*x^2)) + (4*d^2*f*ArcTanh[(Sqrt[b*c - 
 a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sqrt[c]*Sqrt[b*c - a*d]) - (4*d*f^2* 
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)^3
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.51, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {426, 421, 25, 291, 221, 402, 25, 27, 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.

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

\(\Big \downarrow \) 426

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {d \left (\frac {d \left (\frac {\int -\frac {a d (d e-3 c f)-2 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{2 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{2 c \left (c+d x^2\right ) (b c-a d)}\right )}{(d e-c f)^2}+\frac {f^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {\int \frac {2 b e (2 d e-c f)-a f (3 d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {d \left (\frac {d \left (-\frac {\int \frac {a d (d e-3 c f)-2 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{2 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{2 c \left (c+d x^2\right ) (b c-a d)}\right )}{(d e-c f)^2}+\frac {f^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {\int \frac {2 b e (2 d e-c f)-a f (3 d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d \left (\frac {d \left (-\frac {(a d (d e-3 c f)-2 b c (d e-2 c f)) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{2 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{2 c \left (c+d x^2\right ) (b c-a d)}\right )}{(d e-c f)^2}+\frac {f^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {(2 b e (2 d e-c f)-a f (3 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {d \left (\frac {d \left (-\frac {(a d (d e-3 c f)-2 b c (d e-2 c f)) \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{2 c \left (c+d x^2\right ) (b c-a d)}\right )}{(d e-c f)^2}+\frac {f^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {(2 b e (2 d e-c f)-a f (3 d e-c f)) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {d \left (\frac {d \left (-\frac {\text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right ) (a d (d e-3 c f)-2 b c (d e-2 c f))}{2 c^{3/2} (b c-a d)^{3/2}}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{2 c \left (c+d x^2\right ) (b c-a d)}\right )}{(d e-c f)^2}+\frac {f^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (2 b e (2 d e-c f)-a f (3 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\)

Input:

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

Output:

(d*((d*(-1/2*(d*(d*e - c*f)*x*Sqrt[a + b*x^2])/(c*(b*c - a*d)*(c + d*x^2)) 
 - ((a*d*(d*e - 3*c*f) - 2*b*c*(d*e - 2*c*f))*ArcTanh[(Sqrt[b*c - a*d]*x)/ 
(Sqrt[c]*Sqrt[a + b*x^2])])/(2*c^(3/2)*(b*c - a*d)^(3/2))))/(d*e - c*f)^2 
+ (f^2*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(Sqrt[e]*Sq 
rt[b*e - a*f]*(d*e - c*f)^2)))/(d*e - c*f) - (f*((d^2*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)^2 
) - (f*(-1/2*(f*(d*e - c*f)*x*Sqrt[a + b*x^2])/(e*(b*e - a*f)*(e + f*x^2)) 
 + ((2*b*e*(2*d*e - c*f) - a*f*(3*d*e - c*f))*ArcTanh[(Sqrt[b*e - a*f]*x)/ 
(Sqrt[e]*Sqrt[a + b*x^2])])/(2*e^(3/2)*(b*e - a*f)^(3/2))))/(d*e - c*f)^2) 
)/(d*e - c*f)
 

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 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 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]
 

rule 426

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ 
)^2)^(r_), x_Symbol] :> Simp[b/(b*c - a*d)   Int[(a + b*x^2)^p*(c + d*x^2)^ 
(q + 1)*(e + f*x^2)^r, x], x] - Simp[d/(b*c - a*d)   Int[(a + b*x^2)^(p + 1 
)*(c + d*x^2)^q*(e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && 
 ILtQ[p, 0] && LeQ[q, -1]
 

Maple [A] (verified)

Time = 1.68 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.27

method	result	size
pseudoelliptic	\(\frac {-5 d^{2} \left (-\frac {6 b \,c^{2} f}{5}+d \left (a f +\frac {2 b e}{5}\right ) c -\frac {a \,d^{2} e}{5}\right ) \left (x^{2} d +c \right ) \sqrt {\left (a f -b e \right ) e}\, \left (a f -b e \right ) \left (f \,x^{2}+e \right ) e \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )+\sqrt {\left (a d -b c \right ) c}\, \left (-\left (a d -b c \right ) c \left (\left (a \,f^{2}-2 b e f \right ) c -5 a d e f +6 b d \,e^{2}\right ) \left (x^{2} d +c \right ) \left (f \,x^{2}+e \right ) f^{2} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\sqrt {\left (a f -b e \right ) e}\, \left (c f -d e \right ) \sqrt {b \,x^{2}+a}\, x \left (-b \,c^{3} f^{3}+d \,f^{3} \left (-b \,x^{2}+a \right ) c^{2}+a c \,d^{2} f^{3} x^{2}+d^{3} e \left (f \,x^{2}+e \right ) \left (a f -b e \right )\right )\right )}{2 \sqrt {\left (a d -b c \right ) c}\, \sqrt {\left (a f -b e \right ) e}\, \left (a d -b c \right ) \left (x^{2} d +c \right ) \left (c f -d e \right )^{3} c \left (a f -b e \right ) e \left (f \,x^{2}+e \right )}\)	\(360\)
default	\(\text {Expression too large to display}\)	\(2000\)

Input:

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

Output:

1/2/((a*d-b*c)*c)^(1/2)*(-5*d^2*(-6/5*b*c^2*f+d*(a*f+2/5*b*e)*c-1/5*a*d^2* 
e)*(d*x^2+c)*((a*f-b*e)*e)^(1/2)*(a*f-b*e)*(f*x^2+e)*e*arctan(c*(b*x^2+a)^ 
(1/2)/x/((a*d-b*c)*c)^(1/2))+((a*d-b*c)*c)^(1/2)*(-(a*d-b*c)*c*((a*f^2-2*b 
*e*f)*c-5*a*d*e*f+6*b*d*e^2)*(d*x^2+c)*(f*x^2+e)*f^2*arctan(e*(b*x^2+a)^(1 
/2)/x/((a*f-b*e)*e)^(1/2))+((a*f-b*e)*e)^(1/2)*(c*f-d*e)*(b*x^2+a)^(1/2)*x 
*(-b*c^3*f^3+d*f^3*(-b*x^2+a)*c^2+a*c*d^2*f^3*x^2+d^3*e*(f*x^2+e)*(a*f-b*e 
))))/((a*f-b*e)*e)^(1/2)/(a*d-b*c)/(d*x^2+c)/(c*f-d*e)^3/c/(a*f-b*e)/e/(f* 
x^2+e)
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1772 vs. \(2 (253) = 506\).

Time = 1.76 (sec) , antiderivative size = 1772, normalized size of antiderivative = 6.24 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*b^(7/2)*((2*b*c*d^3*e - a*d^4*e - 6*b*c^2*d^2*f + 5*a*c*d^3*f)*arctan 
(1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2*d + 2*b*c - a*d)/sqrt(-b^2*c^2 + a*b 
*c*d))/((b^4*c^2*d^3*e^3 - a*b^3*c*d^4*e^3 - 3*b^4*c^3*d^2*e^2*f + 3*a*b^3 
*c^2*d^3*e^2*f + 3*b^4*c^4*d*e*f^2 - 3*a*b^3*c^3*d^2*e*f^2 - b^4*c^5*f^3 + 
 a*b^3*c^4*d*f^3)*sqrt(-b^2*c^2 + a*b*c*d)) + (6*b*d*e^2*f^2 - 2*b*c*e*f^3 
 - 5*a*d*e*f^3 + a*c*f^4)*arctan(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))/((b^4*d^3*e^5 - 3*b^4*c*d^2*e^4*f - 
 a*b^3*d^3*e^4*f + 3*b^4*c^2*d*e^3*f^2 + 3*a*b^3*c*d^2*e^3*f^2 - b^4*c^3*e 
^2*f^3 - 3*a*b^3*c^2*d*e^2*f^3 + a*b^3*c^3*e*f^4)*sqrt(-b^2*e^2 + a*b*e*f) 
) + 2*(2*(sqrt(b)*x - sqrt(b*x^2 + a))^6*b^2*c*d^2*e^2*f - (sqrt(b)*x - sq 
rt(b*x^2 + a))^6*a*b*d^3*e^2*f + 2*(sqrt(b)*x - sqrt(b*x^2 + a))^6*b^2*c^2 
*d*e*f^2 - 4*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b*c*d^2*e*f^2 + (sqrt(b)*x 
- sqrt(b*x^2 + a))^6*a^2*d^3*e*f^2 - (sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b*c 
^2*d*f^3 + (sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*c*d^2*f^3 + 8*(sqrt(b)*x - 
sqrt(b*x^2 + a))^4*b^3*c*d^2*e^3 - 4*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b^2 
*d^3*e^3 - 12*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b^2*c*d^2*e^2*f + 7*(sqrt( 
b)*x - sqrt(b*x^2 + a))^4*a^2*b*d^3*e^2*f + 8*(sqrt(b)*x - sqrt(b*x^2 + a) 
)^4*b^3*c^3*e*f^2 - 12*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b^2*c^2*d*e*f^2 + 
 8*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b*c*d^2*e*f^2 - 3*(sqrt(b)*x - sqrt 
(b*x^2 + a))^4*a^3*d^3*e*f^2 - 4*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b^2*...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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