\(\int \frac {1}{\sqrt {a+b x^2} (c+d x^2) (e+f x^2)^3} \, dx\) [333]

Optimal result

Integrand size = 30, antiderivative size = 345 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=\frac {f^2 x \sqrt {a+b x^2}}{4 e (b e-a f) (d e-c f) \left (e+f x^2\right )^2}+\frac {f^2 (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {a+b x^2}}{8 e^2 (b e-a f)^2 (d e-c f)^2 \left (e+f x^2\right )}+\frac {d^3 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^3}-\frac {f \left (8 b^2 e^2 \left (3 d^2 e^2-3 c d e f+c^2 f^2\right )-4 a b e f \left (9 d^2 e^2-7 c d e f+2 c^2 f^2\right )+a^2 f^2 \left (15 d^2 e^2-10 c d e f+3 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{8 e^{5/2} (b e-a f)^{5/2} (d e-c f)^3} \] Output:

1/4*f^2*x*(b*x^2+a)^(1/2)/e/(-a*f+b*e)/(-c*f+d*e)/(f*x^2+e)^2+1/8*f^2*(2*b 
*e*(-3*c*f+5*d*e)-a*f*(-3*c*f+7*d*e))*x*(b*x^2+a)^(1/2)/e^2/(-a*f+b*e)^2/( 
-c*f+d*e)^2/(f*x^2+e)+d^3*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)^3-1/8*f*(8*b^2*e^2*(c^2*f^2-3*c*d* 
e*f+3*d^2*e^2)-4*a*b*e*f*(2*c^2*f^2-7*c*d*e*f+9*d^2*e^2)+a^2*f^2*(3*c^2*f^ 
2-10*c*d*e*f+15*d^2*e^2))*arctanh((-a*f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/ 
2))/e^(5/2)/(-a*f+b*e)^(5/2)/(-c*f+d*e)^3
 

Mathematica [A] (verified)

Time = 11.49 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=\frac {\frac {4 d f (d e-c f) x \left (f \left (a+b x^2\right )-\frac {(2 b e-a f) \left (e+f x^2\right ) \text {arctanh}\left (\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}\right )}{e \sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}}\right )}{e (b e-a f) \sqrt {a+b x^2} \left (e+f x^2\right )}+\frac {f (d e-c f)^2 x \left (e f \left (a+b x^2\right ) \left (2 b e \left (4 e+3 f x^2\right )-a f \left (5 e+3 f x^2\right )\right )-\frac {\left (8 b^2 e^2-8 a b e f+3 a^2 f^2\right ) \left (e+f x^2\right )^2 \text {arctanh}\left (\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}\right )}{\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}}\right )}{e^3 (b e-a f)^2 \sqrt {a+b x^2} \left (e+f x^2\right )^2}+\frac {8 d^3 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} \sqrt {b c-a d}}-\frac {8 d^2 f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} \sqrt {b e-a f}}}{8 (d e-c f)^3} \] Input:

Integrate[1/(Sqrt[a + b*x^2]*(c + d*x^2)*(e + f*x^2)^3),x]
 

Output:

((4*d*f*(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)) + (f*(d*e - c*f) 
^2*x*(e*f*(a + b*x^2)*(2*b*e*(4*e + 3*f*x^2) - a*f*(5*e + 3*f*x^2)) - ((8* 
b^2*e^2 - 8*a*b*e*f + 3*a^2*f^2)*(e + f*x^2)^2*ArcTanh[Sqrt[((b*e - a*f)*x 
^2)/(e*(a + b*x^2))]])/Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]))/(e^3*(b*e 
 - a*f)^2*Sqrt[a + b*x^2]*(e + f*x^2)^2) + (8*d^3*ArcTanh[(Sqrt[b*c - a*d] 
*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sqrt[c]*Sqrt[b*c - a*d]) - (8*d^2*f*ArcTa 
nh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(Sqrt[e]*Sqrt[b*e - a*f 
]))/(8*(d*e - c*f)^3)
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.11, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {421, 402, 25, 402, 25, 27, 291, 221, 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.

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {\int -\frac {2 b f (d e-c f) x^2+a f (7 d e-3 c f)-4 b e (2 d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\int \frac {2 b f (d e-c f) x^2+a f (7 d e-3 c f)-4 b e (2 d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {\int -\frac {8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {\int \frac {8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {\left (a^2 f^2 (7 d e-3 c f)-4 a b e f (5 d e-2 c f)+8 b^2 e^2 (2 d e-c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {\left (a^2 f^2 (7 d e-3 c f)-4 a b e f (5 d e-2 c f)+8 b^2 e^2 (2 d e-c f)\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (a^2 f^2 (7 d e-3 c f)-4 a b e f (5 d e-2 c f)+8 b^2 e^2 (2 d e-c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 407

\(\displaystyle \frac {d^2 \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (a^2 f^2 (7 d e-3 c f)-4 a b e f (5 d e-2 c f)+8 b^2 e^2 (2 d e-c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {d^2 \left (\frac {d \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d e-c f}-\frac {f \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (a^2 f^2 (7 d e-3 c f)-4 a b e f (5 d e-2 c f)+8 b^2 e^2 (2 d e-c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (a^2 f^2 (7 d e-3 c f)-4 a b e f (5 d e-2 c f)+8 b^2 e^2 (2 d e-c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\)

Input:

Int[1/(Sqrt[a + b*x^2]*(c + d*x^2)*(e + f*x^2)^3),x]
 

Output:

(d^2*((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]*Sqr 
t[a + b*x^2])])/(Sqrt[e]*Sqrt[b*e - a*f]*(d*e - c*f))))/(d*e - c*f)^2 - (f 
*(-1/4*(f*(d*e - c*f)*x*Sqrt[a + b*x^2])/(e*(b*e - a*f)*(e + f*x^2)^2) - ( 
(f*(2*b*e*(5*d*e - 3*c*f) - a*f*(7*d*e - 3*c*f))*x*Sqrt[a + b*x^2])/(2*e*( 
b*e - a*f)*(e + f*x^2)) - ((a^2*f^2*(7*d*e - 3*c*f) - 4*a*b*e*f*(5*d*e - 2 
*c*f) + 8*b^2*e^2*(2*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)))/(4*e*(b*e - a*f))))/(d*e - c 
*f)^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 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 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]
 

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 = 1.74 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.04

method	result	size
pseudoelliptic	\(\frac {\frac {5 \left (\left (\frac {12 b d \,e^{3}}{5}-\frac {9 \left (a d +\frac {8 b \left (-\frac {5 x^{2} d}{4}+c \right )}{9}\right ) f \,e^{2}}{5}+\left (\left (-\frac {7 x^{2} d}{5}+c \right ) a -\frac {6 x^{2} b c}{5}\right ) f^{2} e +\frac {3 a c \,f^{3} x^{2}}{5}\right ) \sqrt {\left (a f -b e \right ) e}\, \left (c f -d e \right ) x f \sqrt {b \,x^{2}+a}-\frac {3 \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right ) \left (f \,x^{2}+e \right )^{2} \left (8 b^{2} d^{2} e^{4}+\left (-12 a b \,d^{2}-8 b^{2} c d \right ) f \,e^{3}+5 \left (a^{2} d^{2}+\frac {28}{15} a b c d +\frac {8}{15} b^{2} c^{2}\right ) f^{2} e^{2}-\frac {10 \left (a d +\frac {4 b c}{5}\right ) a c \,f^{3} e}{3}+a^{2} c^{2} f^{4}\right )}{5}\right ) f \sqrt {\left (a d -b c \right ) c}}{8}+\sqrt {\left (a f -b e \right ) e}\, d^{3} e^{2} \left (f \,x^{2}+e \right )^{2} \left (a f -b e \right )^{2} \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )}{\sqrt {\left (a d -b c \right ) c}\, \sqrt {\left (a f -b e \right ) e}\, \left (a f -b e \right )^{2} \left (c f -d e \right )^{3} \left (f \,x^{2}+e \right )^{2} e^{2}}\)	\(360\)
default	\(\text {Expression too large to display}\)	\(2545\)

Input:

int(1/(b*x^2+a)^(1/2)/(d*x^2+c)/(f*x^2+e)^3,x,method=_RETURNVERBOSE)
 

Output:

(5/8*((12/5*b*d*e^3-9/5*(a*d+8/9*b*(-5/4*x^2*d+c))*f*e^2+((-7/5*x^2*d+c)*a 
-6/5*x^2*b*c)*f^2*e+3/5*a*c*f^3*x^2)*((a*f-b*e)*e)^(1/2)*(c*f-d*e)*x*f*(b* 
x^2+a)^(1/2)-3/5*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2))*(f*x^2+e) 
^2*(8*b^2*d^2*e^4+(-12*a*b*d^2-8*b^2*c*d)*f*e^3+5*(a^2*d^2+28/15*a*b*c*d+8 
/15*b^2*c^2)*f^2*e^2-10/3*(a*d+4/5*b*c)*a*c*f^3*e+a^2*c^2*f^4))*f*((a*d-b* 
c)*c)^(1/2)+((a*f-b*e)*e)^(1/2)*d^3*e^2*(f*x^2+e)^2*(a*f-b*e)^2*arctan(c*( 
b*x^2+a)^(1/2)/x/((a*d-b*c)*c)^(1/2)))/((a*d-b*c)*c)^(1/2)/((a*f-b*e)*e)^( 
1/2)/(a*f-b*e)^2/(c*f-d*e)^3/(f*x^2+e)^2/e^2
 

Fricas [F(-1)]

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

integrate(1/(b*x^2+a)^(1/2)/(d*x^2+c)/(f*x^2+e)^3,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

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

integrate(1/(b*x**2+a)**(1/2)/(d*x**2+c)/(f*x**2+e)**3,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(1/(b*x^2+a)^(1/2)/(d*x^2+c)/(f*x^2+e)^3,x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

integrate(1/(sqrt(b*x^2 + a)*(d*x^2 + c)*(f*x^2 + e)^3), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1344 vs. \(2 (315) = 630\).

Time = 1.92 (sec) , antiderivative size = 1344, normalized size of antiderivative = 3.90 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(1/(b*x^2+a)^(1/2)/(d*x^2+c)/(f*x^2+e)^3,x, algorithm="giac")
 

Output:

-1/8*(8*d^3*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))/((b^3*d^3*e^3 - 3*b^3*c*d^2*e^2*f + 3*b^3*c^2*d*e 
*f^2 - b^3*c^3*f^3)*sqrt(-b^2*c^2 + a*b*c*d)) - (24*b^2*d^2*e^4*f - 24*b^2 
*c*d*e^3*f^2 - 36*a*b*d^2*e^3*f^2 + 8*b^2*c^2*e^2*f^3 + 28*a*b*c*d*e^2*f^3 
 + 15*a^2*d^2*e^2*f^3 - 8*a*b*c^2*e*f^4 - 10*a^2*c*d*e*f^4 + 3*a^2*c^2*f^5 
)*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^5*d^3*e^7 - 3*b^5*c*d^2*e^6*f - 2*a*b^4*d^3*e^6*f + 3*b 
^5*c^2*d*e^5*f^2 + 6*a*b^4*c*d^2*e^5*f^2 + a^2*b^3*d^3*e^5*f^2 - b^5*c^3*e 
^4*f^3 - 6*a*b^4*c^2*d*e^4*f^3 - 3*a^2*b^3*c*d^2*e^4*f^3 + 2*a*b^4*c^3*e^3 
*f^4 + 3*a^2*b^3*c^2*d*e^3*f^4 - a^2*b^3*c^3*e^2*f^5)*sqrt(-b^2*e^2 + a*b* 
e*f)) - 2*(16*(sqrt(b)*x - sqrt(b*x^2 + a))^6*b^2*d*e^3*f^2 - 8*(sqrt(b)*x 
 - sqrt(b*x^2 + a))^6*b^2*c*e^2*f^3 - 20*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a 
*b*d*e^2*f^3 + 8*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b*c*e*f^4 + 7*(sqrt(b)* 
x - sqrt(b*x^2 + a))^6*a^2*d*e*f^4 - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2 
*c*f^5 + 80*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^3*d*e^4*f - 48*(sqrt(b)*x - 
sqrt(b*x^2 + a))^4*b^3*c*e^3*f^2 - 136*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b 
^2*d*e^3*f^2 + 72*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b^2*c*e^2*f^3 + 86*(sq 
rt(b)*x - sqrt(b*x^2 + a))^4*a^2*b*d*e^2*f^3 - 42*(sqrt(b)*x - sqrt(b*x^2 
+ a))^4*a^2*b*c*e*f^4 - 21*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^3*d*e*f^4 + 9 
*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^3*c*f^5 + 64*(sqrt(b)*x - sqrt(b*x^2...
 

Mupad [F(-1)]

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

int(1/((a + b*x^2)^(1/2)*(c + d*x^2)*(e + f*x^2)^3),x)
 

Output:

int(1/((a + b*x^2)^(1/2)*(c + d*x^2)*(e + f*x^2)^3), x)
 

Reduce [B] (verification not implemented)

Time = 6.30 (sec) , antiderivative size = 12786, normalized size of antiderivative = 37.06 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int(1/(b*x^2+a)^(1/2)/(d*x^2+c)/(f*x^2+e)^3,x)
 

Output:

(16*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**4*d**3*e**5*f**4 + 32*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**4*d**3*e**4*f**5*x**2 + 16*sqrt(c)*sqr 
t(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sq 
rt(b)*x)/(sqrt(c)*sqrt(b)))*a**4*d**3*e**3*f**6*x**4 - 80*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**3*b*d**3*e**6*f**3 - 160*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)/(sq 
rt(c)*sqrt(b)))*a**3*b*d**3*e**5*f**4*x**2 - 80*sqrt(c)*sqrt(a*d - b*c)*at 
an((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt( 
c)*sqrt(b)))*a**3*b*d**3*e**4*f**5*x**4 + 144*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**2*b**2*d**3*e**7*f**2 + 288*sqrt(c)*sqrt(a*d - b*c)*atan((sq 
rt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqr 
t(b)))*a**2*b**2*d**3*e**6*f**3*x**2 + 144*sqrt(c)*sqrt(a*d - b*c)*atan((s 
qrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*sq 
rt(b)))*a**2*b**2*d**3*e**5*f**4*x**4 - 112*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)*s 
qrt(b)))*a*b**3*d**3*e**8*f - 224*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*...