\(\int \frac {1}{(a+b x^2)^{3/2} (c+d x^2)^2 (e+f x^2)^2} \, dx\) [355]

Optimal result
Mathematica [C] (warning: unable to verify)
Rubi [F]
Maple [A] (verified)
Fricas [F(-1)]
Sympy [F(-1)]
Maxima [F]
Giac [B] (verification not implemented)
Mupad [F(-1)]
Reduce [F]

Optimal result

Integrand size = 30, antiderivative size = 418 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\frac {b \left (2 b^3 c e (d e-c f)^2+a^3 d^2 f^2 (d e+c f)-2 a^2 b d f \left (d^2 e^2+c^2 f^2\right )+a b^2 \left (d^3 e^3+c^3 f^3\right )\right ) x}{2 a c (b c-a d)^2 e (b e-a f)^2 (d e-c f)^2 \sqrt {a+b x^2}}-\frac {d^3 x}{2 c (b c-a d) (d e-c f)^2 \sqrt {a+b x^2} \left (c+d x^2\right )}-\frac {f^3 x}{2 e (b e-a f) (d e-c f)^2 \sqrt {a+b x^2} \left (e+f x^2\right )}+\frac {d^3 (a d (d e-5 c f)-4 b c (d e-2 c f)) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{5/2} (d e-c f)^3}-\frac {f^3 (4 b e (2 d e-c f)-a f (5 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{2 e^{3/2} (b e-a f)^{5/2} (d e-c f)^3} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 19.17 (sec) , antiderivative size = 2255, normalized size of antiderivative = 5.39 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\text {Result too large to show} \] Input:

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

Output:

(-2*d^2*f*x*(-15*c*Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))] - 10*d*x^2*Sqrt 
[((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 15*c*ArcTanh[Sqrt[((b*c - a*d)*x^2)/ 
(c*(a + b*x^2))]] + 10*d*x^2*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2) 
)]] + 2*c*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(5/2)*Hypergeometric2F1[2, 5 
/2, 7/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 2*d*x^2*(((b*c - a*d)*x^2)/( 
c*(a + b*x^2)))^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, ((b*c - a*d)*x^2)/(c* 
(a + b*x^2))]))/(5*a*c^2*(d*e - c*f)^3*(((b*c - a*d)*x^2)/(c*(a + b*x^2))) 
^(3/2)*Sqrt[a + b*x^2]*(1 + (b*x^2)/a)) - (2*d*f^2*x*(-15*e*Sqrt[((b*e - a 
*f)*x^2)/(e*(a + b*x^2))] - 10*f*x^2*Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2) 
)] + 15*e*ArcTanh[Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]] + 10*f*x^2*ArcT 
anh[Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]] + 2*e*(((b*e - a*f)*x^2)/(e*( 
a + b*x^2)))^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, ((b*e - a*f)*x^2)/(e*(a 
+ b*x^2))] + 2*f*x^2*(((b*e - a*f)*x^2)/(e*(a + b*x^2)))^(5/2)*Hypergeomet 
ric2F1[2, 5/2, 7/2, ((b*e - a*f)*x^2)/(e*(a + b*x^2))]))/(5*a*e^2*(-(d*e) 
+ c*f)^3*(((b*e - a*f)*x^2)/(e*(a + b*x^2)))^(3/2)*Sqrt[a + b*x^2]*(1 + (b 
*x^2)/a)) + (d^2*x*(-2625*Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))] - (5250* 
d*x^2*Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))])/c - (2310*d^2*x^4*Sqrt[((b* 
c - a*d)*x^2)/(c*(a + b*x^2))])/c^2 + 70*(((b*c - a*d)*x^2)/(c*(a + b*x^2) 
))^(3/2) + (560*d*x^2*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(3/2))/c + (280* 
d^2*x^4*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(3/2))/c^2 + 2625*ArcTanh[S...
 

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.

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

\(\Big \downarrow \) 426

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 301

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 426

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

Input:

Int[1/((a + b*x^2)^(3/2)*(c + d*x^2)^2*(e + f*x^2)^2),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 398
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) 
, x_Symbol] :> Simp[f/b   Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ 
b   Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} 
, x]
 

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 426
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ 
)^2)^(r_), x_Symbol] :> Simp[b/(b*c - a*d)   Int[(a + b*x^2)^p*(c + d*x^2)^ 
(q + 1)*(e + f*x^2)^r, x], x] - Simp[d/(b*c - a*d)   Int[(a + b*x^2)^(p + 1 
)*(c + d*x^2)^q*(e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && 
 ILtQ[p, 0] && LeQ[q, -1]
 
Maple [A] (verified)

Time = 2.10 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.34

method result size
pseudoelliptic \(\frac {-5 a \sqrt {b \,x^{2}+a}\, \sqrt {\left (a f -b e \right ) e}\, d^{3} \left (x^{2} d +c \right ) \left (a f -b e \right )^{2} \left (d \left (c f -\frac {d e}{5}\right ) a -\frac {8 \left (c f -\frac {d e}{2}\right ) c b}{5}\right ) \left (f \,x^{2}+e \right ) e \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )+\sqrt {\left (a d -b c \right ) c}\, \left (-\left (a d -b c \right )^{2} a \sqrt {b \,x^{2}+a}\, c \left (x^{2} d +c \right ) \left (\left (c \,f^{2}-5 d e f \right ) a -4 b e \left (c f -2 d e \right )\right ) \left (f \,x^{2}+e \right ) f^{3} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\sqrt {\left (a f -b e \right ) e}\, \left (d^{2} \left (c^{2} f^{2}+c d \,f^{2} x^{2}+d^{2} e \left (f \,x^{2}+e \right )\right ) f^{2} a^{4}-2 d \left (c^{3} f^{3}+\frac {c^{2} d \,f^{3} x^{2}}{2}-\frac {c \,d^{2} f^{3} x^{4}}{2}+d^{3} \left (f \,x^{2}+e \right ) \left (-\frac {f \,x^{2}}{2}+e \right ) e \right ) b f \,a^{3}+\left (c^{4} f^{4}-c^{3} d \,f^{4} x^{2}-2 c^{2} d^{2} f^{4} x^{4}+d^{4} e^{2} \left (f \,x^{2}+e \right ) \left (-2 f \,x^{2}+e \right )\right ) b^{2} a^{2}+\left (c^{4} f^{4}+c^{3} d \,f^{4} x^{2}+d^{4} e^{3} \left (f \,x^{2}+e \right )\right ) b^{3} x^{2} a +2 b^{4} c e \left (f \,x^{2}+e \right ) \left (x^{2} d +c \right ) \left (c f -d e \right )^{2}\right ) \left (c f -d e \right ) x \right )}{2 \sqrt {\left (a f -b e \right ) e}\, \sqrt {\left (a d -b c \right ) c}\, \sqrt {b \,x^{2}+a}\, \left (x^{2} d +c \right ) \left (c f -d e \right )^{3} \left (a d -b c \right )^{2} c e \left (f \,x^{2}+e \right ) \left (a f -b e \right )^{2} a}\) \(561\)
default \(\text {Expression too large to display}\) \(4192\)

Input:

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

Output:

1/2*(-5*a*(b*x^2+a)^(1/2)*((a*f-b*e)*e)^(1/2)*d^3*(d*x^2+c)*(a*f-b*e)^2*(d 
*(c*f-1/5*d*e)*a-8/5*(c*f-1/2*d*e)*c*b)*(f*x^2+e)*e*arctan(c*(b*x^2+a)^(1/ 
2)/x/((a*d-b*c)*c)^(1/2))+((a*d-b*c)*c)^(1/2)*(-(a*d-b*c)^2*a*(b*x^2+a)^(1 
/2)*c*(d*x^2+c)*((c*f^2-5*d*e*f)*a-4*b*e*(c*f-2*d*e))*(f*x^2+e)*f^3*arctan 
(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2))+((a*f-b*e)*e)^(1/2)*(d^2*(c^2*f^ 
2+c*d*f^2*x^2+d^2*e*(f*x^2+e))*f^2*a^4-2*d*(c^3*f^3+1/2*c^2*d*f^3*x^2-1/2* 
c*d^2*f^3*x^4+d^3*(f*x^2+e)*(-1/2*f*x^2+e)*e)*b*f*a^3+(c^4*f^4-c^3*d*f^4*x 
^2-2*c^2*d^2*f^4*x^4+d^4*e^2*(f*x^2+e)*(-2*f*x^2+e))*b^2*a^2+(c^4*f^4+c^3* 
d*f^4*x^2+d^4*e^3*(f*x^2+e))*b^3*x^2*a+2*b^4*c*e*(f*x^2+e)*(d*x^2+c)*(c*f- 
d*e)^2)*(c*f-d*e)*x))/((a*f-b*e)*e)^(1/2)/((a*d-b*c)*c)^(1/2)/(b*x^2+a)^(1 
/2)/(d*x^2+c)/(c*f-d*e)^3/(a*d-b*c)^2/c/e/(f*x^2+e)/(a*f-b*e)^2/a
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2677 vs. \(2 (383) = 766\).

Time = 6.91 (sec) , antiderivative size = 2677, normalized size of antiderivative = 6.40 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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