\(\int \frac {1}{(a+b x^2) (c+d x^2) (e+f x^2)^3} \, dx\) [255]

Optimal result

Integrand size = 28, antiderivative size = 333 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=\frac {f^2 x}{4 e (b e-a f) (d e-c f) \left (e+f x^2\right )^2}+\frac {f^2 (b e (11 d e-7 c f)-a f (7 d e-3 c f)) x}{8 e^2 (b e-a f)^2 (d e-c f)^2 \left (e+f x^2\right )}+\frac {b^{7/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (b c-a d) (b e-a f)^3}-\frac {d^{7/2} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d) (d e-c f)^3}+\frac {f^{3/2} \left (a^2 f^2 \left (15 d^2 e^2-10 c d e f+3 c^2 f^2\right )-2 a b e f \left (21 d^2 e^2-18 c d e f+5 c^2 f^2\right )+b^2 e^2 \left (35 d^2 e^2-42 c d e f+15 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{8 e^{5/2} (b e-a f)^3 (d e-c f)^3} \] Output:

1/4*f^2*x/e/(-a*f+b*e)/(-c*f+d*e)/(f*x^2+e)^2+1/8*f^2*(b*e*(-7*c*f+11*d*e) 
-a*f*(-3*c*f+7*d*e))*x/e^2/(-a*f+b*e)^2/(-c*f+d*e)^2/(f*x^2+e)+b^(7/2)*arc 
tan(b^(1/2)*x/a^(1/2))/a^(1/2)/(-a*d+b*c)/(-a*f+b*e)^3-d^(7/2)*arctan(d^(1 
/2)*x/c^(1/2))/c^(1/2)/(-a*d+b*c)/(-c*f+d*e)^3+1/8*f^(3/2)*(a^2*f^2*(3*c^2 
*f^2-10*c*d*e*f+15*d^2*e^2)-2*a*b*e*f*(5*c^2*f^2-18*c*d*e*f+21*d^2*e^2)+b^ 
2*e^2*(15*c^2*f^2-42*c*d*e*f+35*d^2*e^2))*arctan(f^(1/2)*x/e^(1/2))/e^(5/2 
)/(-a*f+b*e)^3/(-c*f+d*e)^3
 

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 329, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=\frac {1}{8} \left (\frac {2 f^2 x}{e (b e-a f) (d e-c f) \left (e+f x^2\right )^2}+\frac {f^2 (b e (11 d e-7 c f)+a f (-7 d e+3 c f)) x}{e^2 (b e-a f)^2 (d e-c f)^2 \left (e+f x^2\right )}+\frac {8 b^{7/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (-b c+a d) (-b e+a f)^3}+\frac {8 d^{7/2} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d) (-d e+c f)^3}+\frac {f^{3/2} \left (a^2 f^2 \left (15 d^2 e^2-10 c d e f+3 c^2 f^2\right )-2 a b e f \left (21 d^2 e^2-18 c d e f+5 c^2 f^2\right )+b^2 e^2 \left (35 d^2 e^2-42 c d e f+15 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{e^{5/2} (b e-a f)^3 (d e-c f)^3}\right ) \] Input:

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

Output:

((2*f^2*x)/(e*(b*e - a*f)*(d*e - c*f)*(e + f*x^2)^2) + (f^2*(b*e*(11*d*e - 
 7*c*f) + a*f*(-7*d*e + 3*c*f))*x)/(e^2*(b*e - a*f)^2*(d*e - c*f)^2*(e + f 
*x^2)) + (8*b^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*(-(b*c) + a*d)*( 
-(b*e) + a*f)^3) + (8*d^(7/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*(b*c - 
 a*d)*(-(d*e) + c*f)^3) + (f^(3/2)*(a^2*f^2*(15*d^2*e^2 - 10*c*d*e*f + 3*c 
^2*f^2) - 2*a*b*e*f*(21*d^2*e^2 - 18*c*d*e*f + 5*c^2*f^2) + b^2*e^2*(35*d^ 
2*e^2 - 42*c*d*e*f + 15*c^2*f^2))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(e^(5/2)*(b 
*e - a*f)^3*(d*e - c*f)^3))/8
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.39, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {421, 402, 402, 25, 397, 218, 422, 303, 218}

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.

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {b^2 \int \frac {1}{\left (b x^2+a\right ) \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(b e-a f)^2}-\frac {f \left (\frac {\frac {\int -\frac {d f (b e (11 d e-7 c f)-a f (7 d e-3 c f)) x^2+a f \left (8 d^2 e^2-7 c d f e+3 c^2 f^2\right )-b e \left (16 d^2 e^2-19 c d f e+7 c^2 f^2\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{2 e (d e-c f)}-\frac {f x (b e (11 d e-7 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (d e-c f)}}{4 e (d e-c f)}-\frac {f x (b e-a f)}{4 e \left (e+f x^2\right )^2 (d e-c f)}\right )}{(b e-a f)^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b^2 \int \frac {1}{\left (b x^2+a\right ) \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(b e-a f)^2}-\frac {f \left (\frac {-\frac {\int \frac {d f (b e (11 d e-7 c f)-a f (7 d e-3 c f)) x^2+a f \left (8 d^2 e^2-7 c d f e+3 c^2 f^2\right )-b e \left (16 d^2 e^2-19 c d f e+7 c^2 f^2\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{2 e (d e-c f)}-\frac {f x (b e (11 d e-7 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (d e-c f)}}{4 e (d e-c f)}-\frac {f x (b e-a f)}{4 e \left (e+f x^2\right )^2 (d e-c f)}\right )}{(b e-a f)^2}\)

\(\Big \downarrow \) 397

\(\displaystyle \frac {b^2 \int \frac {1}{\left (b x^2+a\right ) \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(b e-a f)^2}-\frac {f \left (\frac {-\frac {-\frac {f \left (a f \left (3 c^2 f^2-10 c d e f+15 d^2 e^2\right )-b e \left (7 c^2 f^2-26 c d e f+27 d^2 e^2\right )\right ) \int \frac {1}{f x^2+e}dx}{d e-c f}-\frac {8 d^2 e^2 (-a d f-b c f+2 b d e) \int \frac {1}{d x^2+c}dx}{d e-c f}}{2 e (d e-c f)}-\frac {f x (b e (11 d e-7 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (d e-c f)}}{4 e (d e-c f)}-\frac {f x (b e-a f)}{4 e \left (e+f x^2\right )^2 (d e-c f)}\right )}{(b e-a f)^2}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {b^2 \int \frac {1}{\left (b x^2+a\right ) \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(b e-a f)^2}-\frac {f \left (\frac {-\frac {-\frac {\sqrt {f} \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a f \left (3 c^2 f^2-10 c d e f+15 d^2 e^2\right )-b e \left (7 c^2 f^2-26 c d e f+27 d^2 e^2\right )\right )}{\sqrt {e} (d e-c f)}-\frac {8 d^{3/2} e^2 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) (-a d f-b c f+2 b d e)}{\sqrt {c} (d e-c f)}}{2 e (d e-c f)}-\frac {f x (b e (11 d e-7 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (d e-c f)}}{4 e (d e-c f)}-\frac {f x (b e-a f)}{4 e \left (e+f x^2\right )^2 (d e-c f)}\right )}{(b e-a f)^2}\)

\(\Big \downarrow \) 422

\(\displaystyle \frac {b^2 \left (\frac {b \int \frac {1}{\left (b x^2+a\right ) \left (f x^2+e\right )}dx}{b c-a d}-\frac {d \int \frac {1}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{b c-a d}\right )}{(b e-a f)^2}-\frac {f \left (\frac {-\frac {-\frac {\sqrt {f} \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a f \left (3 c^2 f^2-10 c d e f+15 d^2 e^2\right )-b e \left (7 c^2 f^2-26 c d e f+27 d^2 e^2\right )\right )}{\sqrt {e} (d e-c f)}-\frac {8 d^{3/2} e^2 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) (-a d f-b c f+2 b d e)}{\sqrt {c} (d e-c f)}}{2 e (d e-c f)}-\frac {f x (b e (11 d e-7 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (d e-c f)}}{4 e (d e-c f)}-\frac {f x (b e-a f)}{4 e \left (e+f x^2\right )^2 (d e-c f)}\right )}{(b e-a f)^2}\)

\(\Big \downarrow \) 303

\(\displaystyle \frac {b^2 \left (\frac {b \left (\frac {b \int \frac {1}{b x^2+a}dx}{b e-a f}-\frac {f \int \frac {1}{f x^2+e}dx}{b e-a f}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{d x^2+c}dx}{d e-c f}-\frac {f \int \frac {1}{f x^2+e}dx}{d e-c f}\right )}{b c-a d}\right )}{(b e-a f)^2}-\frac {f \left (\frac {-\frac {-\frac {\sqrt {f} \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a f \left (3 c^2 f^2-10 c d e f+15 d^2 e^2\right )-b e \left (7 c^2 f^2-26 c d e f+27 d^2 e^2\right )\right )}{\sqrt {e} (d e-c f)}-\frac {8 d^{3/2} e^2 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) (-a d f-b c f+2 b d e)}{\sqrt {c} (d e-c f)}}{2 e (d e-c f)}-\frac {f x (b e (11 d e-7 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (d e-c f)}}{4 e (d e-c f)}-\frac {f x (b e-a f)}{4 e \left (e+f x^2\right )^2 (d e-c f)}\right )}{(b e-a f)^2}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {b^2 \left (\frac {b \left (\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (b e-a f)}-\frac {\sqrt {f} \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} (b e-a f)}\right )}{b c-a d}-\frac {d \left (\frac {\sqrt {d} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (d e-c f)}-\frac {\sqrt {f} \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} (d e-c f)}\right )}{b c-a d}\right )}{(b e-a f)^2}-\frac {f \left (\frac {-\frac {-\frac {\sqrt {f} \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a f \left (3 c^2 f^2-10 c d e f+15 d^2 e^2\right )-b e \left (7 c^2 f^2-26 c d e f+27 d^2 e^2\right )\right )}{\sqrt {e} (d e-c f)}-\frac {8 d^{3/2} e^2 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) (-a d f-b c f+2 b d e)}{\sqrt {c} (d e-c f)}}{2 e (d e-c f)}-\frac {f x (b e (11 d e-7 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (d e-c f)}}{4 e (d e-c f)}-\frac {f x (b e-a f)}{4 e \left (e+f x^2\right )^2 (d e-c f)}\right )}{(b e-a f)^2}\)

Input:

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

Output:

(b^2*((b*((Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*(b*e - a*f)) - (S 
qrt[f]*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*(b*e - a*f))))/(b*c - a*d) - 
(d*((Sqrt[d]*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*(d*e - c*f)) - (Sqrt[f] 
*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*(d*e - c*f))))/(b*c - a*d)))/(b*e - 
 a*f)^2 - (f*(-1/4*(f*(b*e - a*f)*x)/(e*(d*e - c*f)*(e + f*x^2)^2) + (-1/2 
*(f*(b*e*(11*d*e - 7*c*f) - a*f*(7*d*e - 3*c*f))*x)/(e*(d*e - c*f)*(e + f* 
x^2)) - ((-8*d^(3/2)*e^2*(2*b*d*e - b*c*f - a*d*f)*ArcTan[(Sqrt[d]*x)/Sqrt 
[c]])/(Sqrt[c]*(d*e - c*f)) - (Sqrt[f]*(a*f*(15*d^2*e^2 - 10*c*d*e*f + 3*c 
^2*f^2) - b*e*(27*d^2*e^2 - 26*c*d*e*f + 7*c^2*f^2))*ArcTan[(Sqrt[f]*x)/Sq 
rt[e]])/(Sqrt[e]*(d*e - c*f)))/(2*e*(d*e - c*f)))/(4*e*(d*e - c*f))))/(b*e 
 - a*f)^2
 

Defintions of rubi rules used

rule 25

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

rule 218

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

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]
 

rule 422

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

Maple [A] (verified)

Time = 1.72 (sec) , antiderivative size = 470, normalized size of antiderivative = 1.41

method	result	size
default	\(\frac {b^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\left (a d -b c \right ) \left (a f -b e \right )^{3} \sqrt {a b}}+\frac {f^{2} \left (\frac {\frac {f \left (3 a^{2} c^{2} f^{4}-10 a^{2} c d e \,f^{3}+7 a^{2} d^{2} e^{2} f^{2}-10 a b \,c^{2} e \,f^{3}+28 a b c d \,e^{2} f^{2}-18 a b \,d^{2} e^{3} f +7 b^{2} c^{2} e^{2} f^{2}-18 b^{2} c d \,e^{3} f +11 b^{2} d^{2} e^{4}\right ) x^{3}}{8 e^{2}}+\frac {\left (5 a^{2} c^{2} f^{4}-14 a^{2} c d e \,f^{3}+9 a^{2} d^{2} e^{2} f^{2}-14 a b \,c^{2} e \,f^{3}+36 a b c d \,e^{2} f^{2}-22 a b \,d^{2} e^{3} f +9 b^{2} c^{2} e^{2} f^{2}-22 b^{2} c d \,e^{3} f +13 b^{2} d^{2} e^{4}\right ) x}{8 e}}{\left (f \,x^{2}+e \right )^{2}}+\frac {\left (3 a^{2} c^{2} f^{4}-10 a^{2} c d e \,f^{3}+15 a^{2} d^{2} e^{2} f^{2}-10 a b \,c^{2} e \,f^{3}+36 a b c d \,e^{2} f^{2}-42 a b \,d^{2} e^{3} f +15 b^{2} c^{2} e^{2} f^{2}-42 b^{2} c d \,e^{3} f +35 b^{2} d^{2} e^{4}\right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 e^{2} \sqrt {e f}}\right )}{\left (c f -d e \right )^{3} \left (a f -b e \right )^{3}}-\frac {d^{4} \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{\left (a d -b c \right ) \left (c f -d e \right )^{3} \sqrt {c d}}\)	\(470\)
risch	\(\text {Expression too large to display}\)	\(312011\)

Input:

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

Output:

b^4/(a*d-b*c)/(a*f-b*e)^3/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))+f^2/(c*f-d*e 
)^3/(a*f-b*e)^3*((1/8*f*(3*a^2*c^2*f^4-10*a^2*c*d*e*f^3+7*a^2*d^2*e^2*f^2- 
10*a*b*c^2*e*f^3+28*a*b*c*d*e^2*f^2-18*a*b*d^2*e^3*f+7*b^2*c^2*e^2*f^2-18* 
b^2*c*d*e^3*f+11*b^2*d^2*e^4)/e^2*x^3+1/8*(5*a^2*c^2*f^4-14*a^2*c*d*e*f^3+ 
9*a^2*d^2*e^2*f^2-14*a*b*c^2*e*f^3+36*a*b*c*d*e^2*f^2-22*a*b*d^2*e^3*f+9*b 
^2*c^2*e^2*f^2-22*b^2*c*d*e^3*f+13*b^2*d^2*e^4)/e*x)/(f*x^2+e)^2+1/8*(3*a^ 
2*c^2*f^4-10*a^2*c*d*e*f^3+15*a^2*d^2*e^2*f^2-10*a*b*c^2*e*f^3+36*a*b*c*d* 
e^2*f^2-42*a*b*d^2*e^3*f+15*b^2*c^2*e^2*f^2-42*b^2*c*d*e^3*f+35*b^2*d^2*e^ 
4)/e^2/(e*f)^(1/2)*arctan(f*x/(e*f)^(1/2)))-d^4/(a*d-b*c)/(c*f-d*e)^3/(c*d 
)^(1/2)*arctan(x*d/(c*d)^(1/2))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 787 vs. \(2 (303) = 606\).

Time = 0.13 (sec) , antiderivative size = 787, normalized size of antiderivative = 2.36 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=\frac {b^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (b^{4} c e^{3} - a b^{3} d e^{3} - 3 \, a b^{3} c e^{2} f + 3 \, a^{2} b^{2} d e^{2} f + 3 \, a^{2} b^{2} c e f^{2} - 3 \, a^{3} b d e f^{2} - a^{3} b c f^{3} + a^{4} d f^{3}\right )} \sqrt {a b}} - \frac {d^{4} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b c d^{3} e^{3} - a d^{4} e^{3} - 3 \, b c^{2} d^{2} e^{2} f + 3 \, a c d^{3} e^{2} f + 3 \, b c^{3} d e f^{2} - 3 \, a c^{2} d^{2} e f^{2} - b c^{4} f^{3} + a c^{3} d f^{3}\right )} \sqrt {c d}} + \frac {{\left (35 \, b^{2} d^{2} e^{4} f^{2} - 42 \, b^{2} c d e^{3} f^{3} - 42 \, a b d^{2} e^{3} f^{3} + 15 \, b^{2} c^{2} e^{2} f^{4} + 36 \, a b c d e^{2} f^{4} + 15 \, a^{2} d^{2} e^{2} f^{4} - 10 \, a b c^{2} e f^{5} - 10 \, a^{2} c d e f^{5} + 3 \, a^{2} c^{2} f^{6}\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \, {\left (b^{3} d^{3} e^{8} - 3 \, b^{3} c d^{2} e^{7} f - 3 \, a b^{2} d^{3} e^{7} f + 3 \, b^{3} c^{2} d e^{6} f^{2} + 9 \, a b^{2} c d^{2} e^{6} f^{2} + 3 \, a^{2} b d^{3} e^{6} f^{2} - b^{3} c^{3} e^{5} f^{3} - 9 \, a b^{2} c^{2} d e^{5} f^{3} - 9 \, a^{2} b c d^{2} e^{5} f^{3} - a^{3} d^{3} e^{5} f^{3} + 3 \, a b^{2} c^{3} e^{4} f^{4} + 9 \, a^{2} b c^{2} d e^{4} f^{4} + 3 \, a^{3} c d^{2} e^{4} f^{4} - 3 \, a^{2} b c^{3} e^{3} f^{5} - 3 \, a^{3} c^{2} d e^{3} f^{5} + a^{3} c^{3} e^{2} f^{6}\right )} \sqrt {e f}} + \frac {11 \, b d e^{2} f^{3} x^{3} - 7 \, b c e f^{4} x^{3} - 7 \, a d e f^{4} x^{3} + 3 \, a c f^{5} x^{3} + 13 \, b d e^{3} f^{2} x - 9 \, b c e^{2} f^{3} x - 9 \, a d e^{2} f^{3} x + 5 \, a c e f^{4} x}{8 \, {\left (b^{2} d^{2} e^{6} - 2 \, b^{2} c d e^{5} f - 2 \, a b d^{2} e^{5} f + b^{2} c^{2} e^{4} f^{2} + 4 \, a b c d e^{4} f^{2} + a^{2} d^{2} e^{4} f^{2} - 2 \, a b c^{2} e^{3} f^{3} - 2 \, a^{2} c d e^{3} f^{3} + a^{2} c^{2} e^{2} f^{4}\right )} {\left (f x^{2} + e\right )}^{2}} \] Input:

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

Output:

b^4*arctan(b*x/sqrt(a*b))/((b^4*c*e^3 - a*b^3*d*e^3 - 3*a*b^3*c*e^2*f + 3* 
a^2*b^2*d*e^2*f + 3*a^2*b^2*c*e*f^2 - 3*a^3*b*d*e*f^2 - a^3*b*c*f^3 + a^4* 
d*f^3)*sqrt(a*b)) - d^4*arctan(d*x/sqrt(c*d))/((b*c*d^3*e^3 - a*d^4*e^3 - 
3*b*c^2*d^2*e^2*f + 3*a*c*d^3*e^2*f + 3*b*c^3*d*e*f^2 - 3*a*c^2*d^2*e*f^2 
- b*c^4*f^3 + a*c^3*d*f^3)*sqrt(c*d)) + 1/8*(35*b^2*d^2*e^4*f^2 - 42*b^2*c 
*d*e^3*f^3 - 42*a*b*d^2*e^3*f^3 + 15*b^2*c^2*e^2*f^4 + 36*a*b*c*d*e^2*f^4 
+ 15*a^2*d^2*e^2*f^4 - 10*a*b*c^2*e*f^5 - 10*a^2*c*d*e*f^5 + 3*a^2*c^2*f^6 
)*arctan(f*x/sqrt(e*f))/((b^3*d^3*e^8 - 3*b^3*c*d^2*e^7*f - 3*a*b^2*d^3*e^ 
7*f + 3*b^3*c^2*d*e^6*f^2 + 9*a*b^2*c*d^2*e^6*f^2 + 3*a^2*b*d^3*e^6*f^2 - 
b^3*c^3*e^5*f^3 - 9*a*b^2*c^2*d*e^5*f^3 - 9*a^2*b*c*d^2*e^5*f^3 - a^3*d^3* 
e^5*f^3 + 3*a*b^2*c^3*e^4*f^4 + 9*a^2*b*c^2*d*e^4*f^4 + 3*a^3*c*d^2*e^4*f^ 
4 - 3*a^2*b*c^3*e^3*f^5 - 3*a^3*c^2*d*e^3*f^5 + a^3*c^3*e^2*f^6)*sqrt(e*f) 
) + 1/8*(11*b*d*e^2*f^3*x^3 - 7*b*c*e*f^4*x^3 - 7*a*d*e*f^4*x^3 + 3*a*c*f^ 
5*x^3 + 13*b*d*e^3*f^2*x - 9*b*c*e^2*f^3*x - 9*a*d*e^2*f^3*x + 5*a*c*e*f^4 
*x)/((b^2*d^2*e^6 - 2*b^2*c*d*e^5*f - 2*a*b*d^2*e^5*f + b^2*c^2*e^4*f^2 + 
4*a*b*c*d*e^4*f^2 + a^2*d^2*e^4*f^2 - 2*a*b*c^2*e^3*f^3 - 2*a^2*c*d*e^3*f^ 
3 + a^2*c^2*e^2*f^4)*(f*x^2 + e)^2)
 

Mupad [B] (verification not implemented)

Time = 17.11 (sec) , antiderivative size = 113015, normalized size of antiderivative = 339.38 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

((x^3*(3*a*c*f^5 - 7*a*d*e*f^4 - 7*b*c*e*f^4 + 11*b*d*e^2*f^3))/(8*e^2*(a^ 
2*c^2*f^4 + b^2*d^2*e^4 + a^2*d^2*e^2*f^2 + b^2*c^2*e^2*f^2 - 2*a*b*c^2*e* 
f^3 - 2*a*b*d^2*e^3*f - 2*a^2*c*d*e*f^3 - 2*b^2*c*d*e^3*f + 4*a*b*c*d*e^2* 
f^2)) + (x*(5*a*c*f^4 - 9*a*d*e*f^3 - 9*b*c*e*f^3 + 13*b*d*e^2*f^2))/(8*e* 
(a^2*c^2*f^4 + b^2*d^2*e^4 + a^2*d^2*e^2*f^2 + b^2*c^2*e^2*f^2 - 2*a*b*c^2 
*e*f^3 - 2*a*b*d^2*e^3*f - 2*a^2*c*d*e*f^3 - 2*b^2*c*d*e^3*f + 4*a*b*c*d*e 
^2*f^2)))/(e^2 + f^2*x^4 + 2*e*f*x^2) + symsum(log((1505*b^12*d^12*e^8*f^6 
 + 9*a^4*b^8*c^4*d^8*f^14 + 3270*a^2*b^10*d^12*e^6*f^8 - 1380*a^3*b^9*d^12 
*e^5*f^9 + 225*a^4*b^8*d^12*e^4*f^10 + 3270*b^12*c^2*d^10*e^6*f^8 - 1380*b 
^12*c^3*d^9*e^5*f^9 + 225*b^12*c^4*d^8*e^4*f^10 - 3556*a*b^11*d^12*e^7*f^7 
 - 3556*b^12*c*d^11*e^7*f^7 + 7288*a*b^11*c*d^11*e^6*f^8 - 5808*a*b^11*c^2 
*d^10*e^5*f^9 + 2120*a*b^11*c^3*d^9*e^4*f^10 - 300*a*b^11*c^4*d^8*e^3*f^11 
 - 5808*a^2*b^10*c*d^11*e^5*f^9 + 2120*a^3*b^9*c*d^11*e^4*f^10 - 60*a^3*b^ 
9*c^4*d^8*e*f^13 - 300*a^4*b^8*c*d^11*e^3*f^11 - 60*a^4*b^8*c^3*d^9*e*f^13 
 + 4108*a^2*b^10*c^2*d^10*e^4*f^10 - 1376*a^2*b^10*c^3*d^9*e^3*f^11 + 190* 
a^2*b^10*c^4*d^8*e^2*f^12 - 1376*a^3*b^9*c^2*d^10*e^3*f^11 + 440*a^3*b^9*c 
^3*d^9*e^2*f^12 + 190*a^4*b^8*c^2*d^10*e^2*f^12)/(64*(b^8*d^8*e^20 + a^8*c 
^8*e^4*f^16 + a^8*d^8*e^12*f^8 + b^8*c^8*e^12*f^8 + 28*a^2*b^6*c^8*e^10*f^ 
10 - 56*a^3*b^5*c^8*e^9*f^11 + 70*a^4*b^4*c^8*e^8*f^12 - 56*a^5*b^3*c^8*e^ 
7*f^13 + 28*a^6*b^2*c^8*e^6*f^14 + 28*a^2*b^6*d^8*e^18*f^2 - 56*a^3*b^5...
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 3509, normalized size of antiderivative = 10.54 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(8*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*c**4*e**5*f**3 + 16* 
sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*c**4*e**4*f**4*x**2 + 8 
*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*c**4*e**3*f**5*x**4 - 
24*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*c**3*d*e**6*f**2 - 4 
8*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*c**3*d*e**5*f**3*x**2 
 - 24*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*c**3*d*e**4*f**4* 
x**4 + 24*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*c**2*d**2*e** 
7*f + 48*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*c**2*d**2*e**6 
*f**2*x**2 + 24*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*c**2*d* 
*2*e**5*f**3*x**4 - 8*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*c 
*d**3*e**8 - 16*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*c*d**3* 
e**7*f*x**2 - 8*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*c*d**3* 
e**6*f**2*x**4 - 8*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**4*d**3 
*e**5*f**3 - 16*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**4*d**3*e* 
*4*f**4*x**2 - 8*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**4*d**3*e 
**3*f**5*x**4 + 24*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**3*b*d* 
*3*e**6*f**2 + 48*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**3*b*d** 
3*e**5*f**3*x**2 + 24*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**3*b 
*d**3*e**4*f**4*x**4 - 24*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a* 
*2*b**2*d**3*e**7*f - 48*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*...