\(\int \frac {1}{(a+b x^2)^{3/2} (c+d x^2)^3 (e+f x^2)^3} \, dx\) [357]

Optimal result

Integrand size = 30, antiderivative size = 943 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\frac {b \left (8 b^5 c^2 e^2 (d e-c f)^4+3 a^5 d^3 f^3 \left (d^3 e^3-5 c d^2 e^2 f-5 c^2 d e f^2+c^3 f^3\right )-a^4 b d^2 f^2 \left (9 d^4 e^4-35 c d^3 e^3 f-44 c^2 d^2 e^2 f^2-35 c^3 d e f^3+9 c^4 f^4\right )+3 a^3 b^2 d f \left (3 d^5 e^5-5 c d^4 e^4 f-22 c^2 d^3 e^3 f^2-22 c^3 d^2 e^2 f^3-5 c^4 d e f^4+3 c^5 f^5\right )+2 a b^4 c e \left (5 d^5 e^5-11 c d^4 e^4 f-11 c^4 d e f^4+5 c^5 f^5\right )-3 a^2 b^3 \left (d^6 e^6+5 c d^5 e^5 f-22 c^2 d^4 e^4 f^2-22 c^4 d^2 e^2 f^4+5 c^5 d e f^5+c^6 f^6\right )\right ) x}{8 a c^2 (b c-a d)^3 e^2 (b e-a f)^3 (d e-c f)^4 \sqrt {a+b x^2}}-\frac {d^4 x}{4 c (b c-a d) (d e-c f)^3 \sqrt {a+b x^2} \left (c+d x^2\right )^2}+\frac {d^4 (3 a d (d e-5 c f)-4 b c (2 d e-5 c f)) x}{8 c^2 (b c-a d)^2 (d e-c f)^4 \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {f^4 x}{4 e (b e-a f) (d e-c f)^3 \sqrt {a+b x^2} \left (e+f x^2\right )^2}+\frac {f^4 (4 b e (5 d e-2 c f)-3 a f (5 d e-c f)) x}{8 e^2 (b e-a f)^2 (d e-c f)^4 \sqrt {a+b x^2} \left (e+f x^2\right )}-\frac {3 d^4 \left (8 b^2 c^2 \left (d^2 e^2-4 c d e f+5 c^2 f^2\right )-4 a b c d \left (d^2 e^2-7 c d e f+14 c^2 f^2\right )+a^2 d^2 \left (d^2 e^2-6 c d e f+21 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{7/2} (d e-c f)^5}-\frac {3 f^4 \left (4 a b e f \left (14 d^2 e^2-7 c d e f+c^2 f^2\right )-a^2 f^2 \left (21 d^2 e^2-6 c d e f+c^2 f^2\right )-8 b^2 e^2 \left (5 d^2 e^2-4 c d e f+c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{8 e^{5/2} (b e-a f)^{7/2} (d e-c f)^5} \] Output:

1/8*b*(8*b^5*c^2*e^2*(-c*f+d*e)^4+3*a^5*d^3*f^3*(c^3*f^3-5*c^2*d*e*f^2-5*c 
*d^2*e^2*f+d^3*e^3)-a^4*b*d^2*f^2*(9*c^4*f^4-35*c^3*d*e*f^3-44*c^2*d^2*e^2 
*f^2-35*c*d^3*e^3*f+9*d^4*e^4)+3*a^3*b^2*d*f*(3*c^5*f^5-5*c^4*d*e*f^4-22*c 
^3*d^2*e^2*f^3-22*c^2*d^3*e^3*f^2-5*c*d^4*e^4*f+3*d^5*e^5)+2*a*b^4*c*e*(5* 
c^5*f^5-11*c^4*d*e*f^4-11*c*d^4*e^4*f+5*d^5*e^5)-3*a^2*b^3*(c^6*f^6+5*c^5* 
d*e*f^5-22*c^4*d^2*e^2*f^4-22*c^2*d^4*e^4*f^2+5*c*d^5*e^5*f+d^6*e^6))*x/a/ 
c^2/(-a*d+b*c)^3/e^2/(-a*f+b*e)^3/(-c*f+d*e)^4/(b*x^2+a)^(1/2)-1/4*d^4*x/c 
/(-a*d+b*c)/(-c*f+d*e)^3/(b*x^2+a)^(1/2)/(d*x^2+c)^2+1/8*d^4*(3*a*d*(-5*c* 
f+d*e)-4*b*c*(-5*c*f+2*d*e))*x/c^2/(-a*d+b*c)^2/(-c*f+d*e)^4/(b*x^2+a)^(1/ 
2)/(d*x^2+c)+1/4*f^4*x/e/(-a*f+b*e)/(-c*f+d*e)^3/(b*x^2+a)^(1/2)/(f*x^2+e) 
^2+1/8*f^4*(4*b*e*(-2*c*f+5*d*e)-3*a*f*(-c*f+5*d*e))*x/e^2/(-a*f+b*e)^2/(- 
c*f+d*e)^4/(b*x^2+a)^(1/2)/(f*x^2+e)-3/8*d^4*(8*b^2*c^2*(5*c^2*f^2-4*c*d*e 
*f+d^2*e^2)-4*a*b*c*d*(14*c^2*f^2-7*c*d*e*f+d^2*e^2)+a^2*d^2*(21*c^2*f^2-6 
*c*d*e*f+d^2*e^2))*arctanh((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2))/c^( 
5/2)/(-a*d+b*c)^(7/2)/(-c*f+d*e)^5-3/8*f^4*(4*a*b*e*f*(c^2*f^2-7*c*d*e*f+1 
4*d^2*e^2)-a^2*f^2*(c^2*f^2-6*c*d*e*f+21*d^2*e^2)-8*b^2*e^2*(c^2*f^2-4*c*d 
*e*f+5*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)^(7/2)/(-c*f+d*e)^5
 

Mathematica [A] (verified)

Time = 24.14 (sec) , antiderivative size = 635, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\sqrt {a+b x^2} \left (\frac {b^6 x}{a (-b c+a d)^3 (-b e+a f)^3 \left (a+b x^2\right )}-\frac {d^5 x}{4 c (b c-a d)^2 (-d e+c f)^3 \left (c+d x^2\right )^2}-\frac {d^5 \left (-10 b c d e+3 a d^2 e+22 b c^2 f-15 a c d f\right ) x}{8 c^2 (b c-a d)^3 (-d e+c f)^4 \left (c+d x^2\right )}-\frac {f^5 x}{4 e (b e-a f)^2 (d e-c f)^3 \left (e+f x^2\right )^2}-\frac {f^5 \left (22 b d e^2-10 b c e f-15 a d e f+3 a c f^2\right ) x}{8 e^2 (b e-a f)^3 (d e-c f)^4 \left (e+f x^2\right )}\right )+\frac {3 \left (8 b^2 c^2 d^6 e^2-4 a b c d^7 e^2+a^2 d^8 e^2-32 b^2 c^3 d^5 e f+28 a b c^2 d^6 e f-6 a^2 c d^7 e f+40 b^2 c^4 d^4 f^2-56 a b c^3 d^5 f^2+21 a^2 c^2 d^6 f^2\right ) \arctan \left (\frac {\sqrt {-b c+a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^3 \sqrt {-b c+a d} (-d e+c f)^5}+\frac {3 \left (40 b^2 d^2 e^4 f^4-32 b^2 c d e^3 f^5-56 a b d^2 e^3 f^5+8 b^2 c^2 e^2 f^6+28 a b c d e^2 f^6+21 a^2 d^2 e^2 f^6-4 a b c^2 e f^7-6 a^2 c d e f^7+a^2 c^2 f^8\right ) \arctan \left (\frac {\sqrt {-b e+a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{8 e^{5/2} (b e-a f)^3 \sqrt {-b e+a f} (d e-c f)^5} \] Input:

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

Output:

Sqrt[a + b*x^2]*((b^6*x)/(a*(-(b*c) + a*d)^3*(-(b*e) + a*f)^3*(a + b*x^2)) 
 - (d^5*x)/(4*c*(b*c - a*d)^2*(-(d*e) + c*f)^3*(c + d*x^2)^2) - (d^5*(-10* 
b*c*d*e + 3*a*d^2*e + 22*b*c^2*f - 15*a*c*d*f)*x)/(8*c^2*(b*c - a*d)^3*(-( 
d*e) + c*f)^4*(c + d*x^2)) - (f^5*x)/(4*e*(b*e - a*f)^2*(d*e - c*f)^3*(e + 
 f*x^2)^2) - (f^5*(22*b*d*e^2 - 10*b*c*e*f - 15*a*d*e*f + 3*a*c*f^2)*x)/(8 
*e^2*(b*e - a*f)^3*(d*e - c*f)^4*(e + f*x^2))) + (3*(8*b^2*c^2*d^6*e^2 - 4 
*a*b*c*d^7*e^2 + a^2*d^8*e^2 - 32*b^2*c^3*d^5*e*f + 28*a*b*c^2*d^6*e*f - 6 
*a^2*c*d^7*e*f + 40*b^2*c^4*d^4*f^2 - 56*a*b*c^3*d^5*f^2 + 21*a^2*c^2*d^6* 
f^2)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(8*c^(5/2)* 
(b*c - a*d)^3*Sqrt[-(b*c) + a*d]*(-(d*e) + c*f)^5) + (3*(40*b^2*d^2*e^4*f^ 
4 - 32*b^2*c*d*e^3*f^5 - 56*a*b*d^2*e^3*f^5 + 8*b^2*c^2*e^2*f^6 + 28*a*b*c 
*d*e^2*f^6 + 21*a^2*d^2*e^2*f^6 - 4*a*b*c^2*e*f^7 - 6*a^2*c*d*e*f^7 + a^2* 
c^2*f^8)*ArcTan[(Sqrt[-(b*e) + a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(8*e^(5 
/2)*(b*e - a*f)^3*Sqrt[-(b*e) + a*f]*(d*e - c*f)^5)
 

Rubi [F]

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.

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

\(\Big \downarrow \) 426

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

\(\Big \downarrow \) 426

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {b \left (\frac {b \left (\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {\int \frac {a \left (4 e (d e+2 c f) b^2-3 a f (5 d e+c f) b+6 a^2 d f^2\right )-2 b f \left (-2 d f a^2-b (3 d e-c f) a+4 b^2 c e\right ) x^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{b c-a d}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {b \left (\frac {b \left (\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {\frac {\int -\frac {a \left (-8 e^2 (d e+3 c f) b^3+4 a e f (10 d e+3 c f) b^2-a^2 f^2 (23 d e+3 c f) b+6 a^3 d f^3\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{b c-a d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \left (\frac {b \left (\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {\int \frac {a \left (-8 e^2 (d e+3 c f) b^3+4 a e f (10 d e+3 c f) b^2-a^2 f^2 (23 d e+3 c f) b+6 a^3 d f^3\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{b c-a d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \left (\frac {b \left (\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{b c-a d}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {b \left (\frac {b \left (\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{b c-a d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {b \left (\frac {b \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{b c-a d}\)

\(\Big \downarrow \) 421

\(\displaystyle \frac {b \left (\frac {b \left (\frac {d^2 \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \int \frac {\sqrt {b x^2+a} \left (d f x^2+2 d e-c f\right )}{\left (f x^2+e\right )^3}dx}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{b c-a d}\)

\(\Big \downarrow \) 401

\(\displaystyle \frac {b \left (\frac {b \left (\frac {d^2 \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}-\frac {\int -\frac {f \left (2 b (3 d e-c f) x^2+a (7 d e-3 c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e f}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{b c-a d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \left (\frac {b \left (\frac {d^2 \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {\int \frac {f \left (2 b (3 d e-c f) x^2+a (7 d e-3 c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e f}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{b c-a d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \left (\frac {b \left (\frac {d^2 \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {\int \frac {2 b (3 d e-c f) x^2+a (7 d e-3 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{b c-a d}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {b \left (\frac {b \left (\frac {d^2 \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {\frac {\int -\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{b c-a d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \left (\frac {b \left (\frac {d^2 \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {\int \frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{b c-a d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \left (\frac {b \left (\frac {d^2 \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{b c-a d}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {b \left (\frac {b \left (\frac {\left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e \left (f x^2+e\right )^2}+\frac {-\frac {(a f (7 d e-3 c f)-2 b e (3 d e-c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}}{4 e}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}-\frac {\frac {-\frac {f \left (6 d f^2 a^3-b f (19 d e+3 c f) a^2-2 b^2 e (d e-5 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a \left (-8 e^2 (d e+3 c f) b^3+4 a e f (10 d e+3 c f) b^2-a^2 f^2 (23 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}-\frac {f \left (-2 d f a^2-b (3 d e-c f) a+4 b^2 c e\right ) x \sqrt {b x^2+a}}{4 e (b e-a f) \left (f x^2+e\right )^2}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{b c-a d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {b \left (\frac {b \left (\frac {\left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e \left (f x^2+e\right )^2}+\frac {-\frac {(a f (7 d e-3 c f)-2 b e (3 d e-c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}-\frac {\frac {-\frac {f \left (6 d f^2 a^3-b f (19 d e+3 c f) a^2-2 b^2 e (d e-5 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a \left (-8 e^2 (d e+3 c f) b^3+4 a e f (10 d e+3 c f) b^2-a^2 f^2 (23 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}-\frac {f \left (-2 d f a^2-b (3 d e-c f) a+4 b^2 c e\right ) x \sqrt {b x^2+a}}{4 e (b e-a f) \left (f x^2+e\right )^2}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{b c-a d}\)

\(\Big \downarrow \) 422

\(\displaystyle \frac {b \left (\frac {b \left (\frac {\left (\frac {d^2 \left (\frac {d \int \frac {\sqrt {b x^2+a}}{d x^2+c}dx}{d e-c f}-\frac {f \int \frac {\sqrt {b x^2+a}}{f x^2+e}dx}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e \left (f x^2+e\right )^2}+\frac {-\frac {(a f (7 d e-3 c f)-2 b e (3 d e-c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}-\frac {\frac {-\frac {f \left (6 d f^2 a^3-b f (19 d e+3 c f) a^2-2 b^2 e (d e-5 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a \left (-8 e^2 (d e+3 c f) b^3+4 a e f (10 d e+3 c f) b^2-a^2 f^2 (23 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}-\frac {f \left (-2 d f a^2-b (3 d e-c f) a+4 b^2 c e\right ) x \sqrt {b x^2+a}}{4 e (b e-a f) \left (f x^2+e\right )^2}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{b c-a d}\)

\(\Big \downarrow \) 301

\(\displaystyle \frac {b \left (\frac {b \left (\frac {\left (\frac {d^2 \left (\frac {d \left (\frac {b \int \frac {1}{\sqrt {b x^2+a}}dx}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{d e-c f}-\frac {f \left (\frac {b \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e \left (f x^2+e\right )^2}+\frac {-\frac {(a f (7 d e-3 c f)-2 b e (3 d e-c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}-\frac {\frac {-\frac {f \left (6 d f^2 a^3-b f (19 d e+3 c f) a^2-2 b^2 e (d e-5 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a \left (-8 e^2 (d e+3 c f) b^3+4 a e f (10 d e+3 c f) b^2-a^2 f^2 (23 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}-\frac {f \left (-2 d f a^2-b (3 d e-c f) a+4 b^2 c e\right ) x \sqrt {b x^2+a}}{4 e (b e-a f) \left (f x^2+e\right )^2}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{b c-a d}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {b \left (\frac {b \left (\frac {\left (\frac {d^2 \left (\frac {d \left (\frac {b \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{d e-c f}-\frac {f \left (\frac {b \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e \left (f x^2+e\right )^2}+\frac {-\frac {(a f (7 d e-3 c f)-2 b e (3 d e-c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}-\frac {\frac {-\frac {f \left (6 d f^2 a^3-b f (19 d e+3 c f) a^2-2 b^2 e (d e-5 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a \left (-8 e^2 (d e+3 c f) b^3+4 a e f (10 d e+3 c f) b^2-a^2 f^2 (23 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}-\frac {f \left (-2 d f a^2-b (3 d e-c f) a+4 b^2 c e\right ) x \sqrt {b x^2+a}}{4 e (b e-a f) \left (f x^2+e\right )^2}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{b c-a d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {b \left (\frac {b \left (\frac {\left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e \left (f x^2+e\right )^2}+\frac {-\frac {(a f (7 d e-3 c f)-2 b e (3 d e-c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}-\frac {\frac {-\frac {f \left (6 d f^2 a^3-b f (19 d e+3 c f) a^2-2 b^2 e (d e-5 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a \left (-8 e^2 (d e+3 c f) b^3+4 a e f (10 d e+3 c f) b^2-a^2 f^2 (23 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}-\frac {f \left (-2 d f a^2-b (3 d e-c f) a+4 b^2 c e\right ) x \sqrt {b x^2+a}}{4 e (b e-a f) \left (f x^2+e\right )^2}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{b c-a d}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {b \left (\frac {b \left (\frac {\left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{d}-\frac {(b c-a d) \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{f}-\frac {(b e-a f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e \left (f x^2+e\right )^2}+\frac {-\frac {(a f (7 d e-3 c f)-2 b e (3 d e-c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}-\frac {\frac {-\frac {f \left (6 d f^2 a^3-b f (19 d e+3 c f) a^2-2 b^2 e (d e-5 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a \left (-8 e^2 (d e+3 c f) b^3+4 a e f (10 d e+3 c f) b^2-a^2 f^2 (23 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}-\frac {f \left (-2 d f a^2-b (3 d e-c f) a+4 b^2 c e\right ) x \sqrt {b x^2+a}}{4 e (b e-a f) \left (f x^2+e\right )^2}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{b c-a d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {b \left (\frac {b \left (\frac {\left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e \left (f x^2+e\right )^2}+\frac {-\frac {(a f (7 d e-3 c f)-2 b e (3 d e-c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}-\frac {\frac {-\frac {f \left (6 d f^2 a^3-b f (19 d e+3 c f) a^2-2 b^2 e (d e-5 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a \left (-8 e^2 (d e+3 c f) b^3+4 a e f (10 d e+3 c f) b^2-a^2 f^2 (23 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}-\frac {f \left (-2 d f a^2-b (3 d e-c f) a+4 b^2 c e\right ) x \sqrt {b x^2+a}}{4 e (b e-a f) \left (f x^2+e\right )^2}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{b c-a d}\)

\(\Big \downarrow \) 426

\(\displaystyle \frac {b \left (\frac {b \left (\frac {\left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e \left (f x^2+e\right )^2}+\frac {-\frac {(a f (7 d e-3 c f)-2 b e (3 d e-c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}-\frac {\frac {-\frac {f \left (6 d f^2 a^3-b f (19 d e+3 c f) a^2-2 b^2 e (d e-5 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a \left (-8 e^2 (d e+3 c f) b^3+4 a e f (10 d e+3 c f) b^2-a^2 f^2 (23 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}-\frac {f \left (-2 d f a^2-b (3 d e-c f) a+4 b^2 c e\right ) x \sqrt {b x^2+a}}{4 e (b e-a f) \left (f x^2+e\right )^2}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}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 )^3}dx}{d e-c f}\right )}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \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 )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}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 )^3}dx}{d e-c f}\right )}{d e-c f}\right )}{b c-a d}\)

\(\Big \downarrow \) 421

\(\displaystyle \frac {b \left (\frac {b \left (\frac {\left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e \left (f x^2+e\right )^2}+\frac {-\frac {(a f (7 d e-3 c f)-2 b e (3 d e-c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}-\frac {\frac {-\frac {f \left (6 d f^2 a^3-b f (19 d e+3 c f) a^2-2 b^2 e (d e-5 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a \left (-8 e^2 (d e+3 c f) b^3+4 a e f (10 d e+3 c f) b^2-a^2 f^2 (23 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}-\frac {f \left (-2 d f a^2-b (3 d e-c f) a+4 b^2 c e\right ) x \sqrt {b x^2+a}}{4 e (b e-a f) \left (f x^2+e\right )^2}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\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}\right )}{d e-c f}\right )}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \left (\frac {d \left (\frac {f^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 {d \int -\frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^3}dx}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\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}\right )}{d e-c f}\right )}{d e-c f}\right )}{b c-a d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \left (\frac {b \left (\frac {\left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e \left (f x^2+e\right )^2}+\frac {-\frac {(a f (7 d e-3 c f)-2 b e (3 d e-c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}-\frac {\frac {-\frac {f \left (6 d f^2 a^3-b f (19 d e+3 c f) a^2-2 b^2 e (d e-5 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a \left (-8 e^2 (d e+3 c f) b^3+4 a e f (10 d e+3 c f) b^2-a^2 f^2 (23 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}-\frac {f \left (-2 d f a^2-b (3 d e-c f) a+4 b^2 c e\right ) x \sqrt {b x^2+a}}{4 e (b e-a f) \left (f x^2+e\right )^2}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\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}\right )}{d e-c f}\right )}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx f^2}{(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 )^3}dx}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\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}\right )}{d e-c f}\right )}{d e-c f}\right )}{b c-a d}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {b \left (\frac {b \left (\frac {\left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e \left (f x^2+e\right )^2}+\frac {-\frac {(a f (7 d e-3 c f)-2 b e (3 d e-c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}-\frac {\frac {-\frac {f \left (6 d f^2 a^3-b f (19 d e+3 c f) a^2-2 b^2 e (d e-5 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a \left (-8 e^2 (d e+3 c f) b^3+4 a e f (10 d e+3 c f) b^2-a^2 f^2 (23 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}-\frac {f \left (-2 d f a^2-b (3 d e-c f) a+4 b^2 c e\right ) x \sqrt {b x^2+a}}{4 e (b e-a f) \left (f x^2+e\right )^2}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\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 (d e-c f) x \sqrt {b x^2+a}}{4 e (b e-a f) \left (f x^2+e\right )^2}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (\frac {\int -\frac {2 b d (d e-c f) x^2+a d (3 d e-7 c f)-4 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^2}dx}{4 c (b c-a d)}-\frac {d (d e-c f) x \sqrt {b x^2+a}}{4 c (b c-a d) \left (d x^2+c\right )^2}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\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 (d e-c f) x \sqrt {b x^2+a}}{4 e (b e-a f) \left (f x^2+e\right )^2}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\right )}{b c-a d}\)

Input:

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

Output:

$Aborted
 

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 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 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 224

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

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 301

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[b/ 
d   Int[(a + b*x^2)^(p - 1), x], x] - Simp[(b*c - a*d)/d   Int[(a + b*x^2)^ 
(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] 
&& GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4] || (EqQ[p, 2/3] && E 
qQ[b*c + 3*a*d, 0]))
 

rule 401

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/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1))   Int[(a + b*x^2)^(p + 1)*( 
c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + 
(b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L 
tQ[p, -1] && GtQ[q, 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 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]
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2017\) vs. \(2(899)=1798\).

Time = 208.01 (sec) , antiderivative size = 2018, normalized size of antiderivative = 2.14

method	result	size
pseudoelliptic	\(\text {Expression too large to display}\)	\(2018\)
default	\(\text {Expression too large to display}\)	\(8706\)

Input:

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

Output:

5/8/((a*f-b*e)*e)^(1/2)*(63/5*a*(b*x^2+a)^(1/2)*d^4*((a*f-b*e)*e)^(1/2)*(d 
*x^2+c)^2*(a*f-b*e)^3*(1/21*a^2*e^2*d^4-2/7*a*(a*f+2/3*b*e)*c*e*d^3+c^2*(a 
^2*f^2+4/3*a*b*f*e+8/21*b^2*e^2)*d^2-8/3*c^3*(a*f+4/7*b*e)*b*f*d+40/21*b^2 
*c^4*f^2)*(f*x^2+e)^2*e^2*arctan(c*(b*x^2+a)^(1/2)/x/((a*d-b*c)*c)^(1/2))+ 
(-3/5*(a*d-b*c)^3*a*(b*x^2+a)^(1/2)*c^2*(d*x^2+c)^2*((21*a^2*e^2*f^2-56*a* 
b*e^3*f+40*b^2*e^4)*d^2-6*(a*f-8/3*b*e)*c*f*(a*f-2*b*e)*e*d+c^2*f^2*(a^2*f 
^2-4*a*b*e*f+8*b^2*e^2))*(f*x^2+e)^2*f^4*arctan(e*(b*x^2+a)^(1/2)/x/((a*f- 
b*e)*e)^(1/2))+((a*f-b*e)*e)^(1/2)*(3/5*a^2*e^3*x^2*(f*x^2+e)^2*(b*x^2+a)* 
(a*f-b*e)^3*d^8+a*c*(b*x^2+a)*(a*f-b*e)^3*(-3*a*f*x^2+e*(-2*b*x^2+a))*(f*x 
^2+e)^2*e^2*d^7-17/5*c^2*(15/17*a^5*x^6*(b*x^2+a)*f^7+2*a^4*(b*x^2+a)*(-22 
/17*b*x^2+a)*x^4*e*f^6+2*a^3*(b*x^2+a)*(33/17*b^2*x^4-107/34*a*b*x^2+a^2)* 
x^2*e^2*f^5+a^2*(b*x^2+a)*(-66/17*b^3*x^6+147/17*a*b^2*x^4-100/17*a^2*b*x^ 
2+a^3)*e^3*f^4-39/17*a*(b*x^2+a)*b*e^4*(-22/39*b^3*x^6+113/39*a*b^2*x^4-32 
/13*a^2*b*x^2+a^3)*f^3+15/17*b^2*(-8/15*b^4*x^8+32/15*a*b^3*x^6+4/15*a^2*b 
^2*x^4-13/15*a^3*b*x^2+a^4)*e^5*f^2+19/17*(-16/19*b^3*x^6-2/19*a*b^2*x^4+1 
7/19*a^2*b*x^2+a^3)*b^3*e^6*f-12/17*(2/3*b^2*x^4+a*b*x^2+a^2)*b^4*e^7)*e*d 
^6-34/5*c^3*(-3/34*a^5*x^6*(b*x^2+a)*f^8+25/34*a^4*(-7/5*b*x^2+a)*(b*x^2+a 
)*x^4*e*f^7+a^3*(b*x^2+a)*(33/17*b^2*x^4-107/34*a*b*x^2+a^2)*x^2*e^2*f^6-4 
8/17*a^3*(b*x^2+a)*b*x^2*(-3/2*b*x^2+a)*e^3*f^5-12/17*a^2*b*e^4*(b*x^2+a)* 
(3*b^2*x^4-6*a*b*x^2+a^2)*f^4+36/17*(-1/3*b^2*x^4+a^2)*b^2*(-4/3*b^2*x^...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 14609 vs. \(2 (898) = 1796\).

Time = 82.01 (sec) , antiderivative size = 14609, normalized size of antiderivative = 15.49 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

b^6*x/((a*b^6*c^3*e^3 - 3*a^2*b^5*c^2*d*e^3 + 3*a^3*b^4*c*d^2*e^3 - a^4*b^ 
3*d^3*e^3 - 3*a^2*b^5*c^3*e^2*f + 9*a^3*b^4*c^2*d*e^2*f - 9*a^4*b^3*c*d^2* 
e^2*f + 3*a^5*b^2*d^3*e^2*f + 3*a^3*b^4*c^3*e*f^2 - 9*a^4*b^3*c^2*d*e*f^2 
+ 9*a^5*b^2*c*d^2*e*f^2 - 3*a^6*b*d^3*e*f^2 - a^4*b^3*c^3*f^3 + 3*a^5*b^2* 
c^2*d*f^3 - 3*a^6*b*c*d^2*f^3 + a^7*d^3*f^3)*sqrt(b*x^2 + a)) + 3/8*(8*b^( 
5/2)*c^2*d^6*e^2 - 4*a*b^(3/2)*c*d^7*e^2 + a^2*sqrt(b)*d^8*e^2 - 32*b^(5/2 
)*c^3*d^5*e*f + 28*a*b^(3/2)*c^2*d^6*e*f - 6*a^2*sqrt(b)*c*d^7*e*f + 40*b^ 
(5/2)*c^4*d^4*f^2 - 56*a*b^(3/2)*c^3*d^5*f^2 + 21*a^2*sqrt(b)*c^2*d^6*f^2) 
*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^3*c^5*d^5*e^5 - 3*a*b^2*c^4*d^6*e^5 + 3*a^2*b*c^3*d^7*e^ 
5 - a^3*c^2*d^8*e^5 - 5*b^3*c^6*d^4*e^4*f + 15*a*b^2*c^5*d^5*e^4*f - 15*a^ 
2*b*c^4*d^6*e^4*f + 5*a^3*c^3*d^7*e^4*f + 10*b^3*c^7*d^3*e^3*f^2 - 30*a*b^ 
2*c^6*d^4*e^3*f^2 + 30*a^2*b*c^5*d^5*e^3*f^2 - 10*a^3*c^4*d^6*e^3*f^2 - 10 
*b^3*c^8*d^2*e^2*f^3 + 30*a*b^2*c^7*d^3*e^2*f^3 - 30*a^2*b*c^6*d^4*e^2*f^3 
 + 10*a^3*c^5*d^5*e^2*f^3 + 5*b^3*c^9*d*e*f^4 - 15*a*b^2*c^8*d^2*e*f^4 + 1 
5*a^2*b*c^7*d^3*e*f^4 - 5*a^3*c^6*d^4*e*f^4 - b^3*c^10*f^5 + 3*a*b^2*c^9*d 
*f^5 - 3*a^2*b*c^8*d^2*f^5 + a^3*c^7*d^3*f^5)*sqrt(-b^2*c^2 + a*b*c*d)) - 
3/8*(40*b^(5/2)*d^2*e^4*f^4 - 32*b^(5/2)*c*d*e^3*f^5 - 56*a*b^(3/2)*d^2*e^ 
3*f^5 + 8*b^(5/2)*c^2*e^2*f^6 + 28*a*b^(3/2)*c*d*e^2*f^6 + 21*a^2*sqrt(b)* 
d^2*e^2*f^6 - 4*a*b^(3/2)*c^2*e*f^7 - 6*a^2*sqrt(b)*c*d*e*f^7 + a^2*sqr...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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