\(\int \frac {\sqrt {a+b x^2}}{(c+d x^2)^3 (e+f x^2)^2} \, dx\) [288]

Optimal result

Integrand size = 30, antiderivative size = 378 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^3 \left (e+f x^2\right )^2} \, dx=\frac {d^2 x \sqrt {a+b x^2}}{4 c (d e-c f)^2 \left (c+d x^2\right )^2}-\frac {d^2 (a d (3 d e-11 c f)-2 b c (d e-5 c f)) x \sqrt {a+b x^2}}{8 c^2 (b c-a d) (d e-c f)^3 \left (c+d x^2\right )}-\frac {f^3 x \sqrt {a+b x^2}}{2 e (d e-c f)^3 \left (e+f x^2\right )}-\frac {d \left (24 b^2 c^4 f^2-4 a b c d \left (d^2 e^2-4 c d e f+15 c^2 f^2\right )+a^2 d^2 \left (3 d^2 e^2-14 c d e f+35 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{3/2} (d e-c f)^4}+\frac {f^2 \left (6 b d e^2-a f (7 d e-c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{2 e^{3/2} \sqrt {b e-a f} (d e-c f)^4} \] Output:

1/4*d^2*x*(b*x^2+a)^(1/2)/c/(-c*f+d*e)^2/(d*x^2+c)^2-1/8*d^2*(a*d*(-11*c*f 
+3*d*e)-2*b*c*(-5*c*f+d*e))*x*(b*x^2+a)^(1/2)/c^2/(-a*d+b*c)/(-c*f+d*e)^3/ 
(d*x^2+c)-1/2*f^3*x*(b*x^2+a)^(1/2)/e/(-c*f+d*e)^3/(f*x^2+e)-1/8*d*(24*b^2 
*c^4*f^2-4*a*b*c*d*(15*c^2*f^2-4*c*d*e*f+d^2*e^2)+a^2*d^2*(35*c^2*f^2-14*c 
*d*e*f+3*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)^(3/2)/(-c*f+d*e)^4+1/2*f^2*(6*b*d*e^2-a*f*(-c*f+7*d*e))*ar 
ctanh((-a*f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/2))/e^(3/2)/(-a*f+b*e)^(1/2) 
/(-c*f+d*e)^4
 

Mathematica [A] (verified)

Time = 11.69 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^3 \left (e+f x^2\right )^2} \, dx=\frac {1}{8} \left (x \sqrt {a+b x^2} \left (\frac {2 d^2}{c (d e-c f)^2 \left (c+d x^2\right )^2}+\frac {d^2 (a d (3 d e-11 c f)+2 b c (-d e+5 c f))}{c^2 (b c-a d) (-d e+c f)^3 \left (c+d x^2\right )}-\frac {4 f^3}{e (d e-c f)^3 \left (e+f x^2\right )}\right )+\frac {d \left (24 b^2 c^4 f^2-4 a b c d \left (d^2 e^2-4 c d e f+15 c^2 f^2\right )+a^2 d^2 \left (3 d^2 e^2-14 c d e f+35 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {-b c+a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{c^{5/2} (-b c+a d)^{3/2} (d e-c f)^4}+\frac {4 f^2 \left (6 b d e^2+a f (-7 d e+c f)\right ) \arctan \left (\frac {\sqrt {-b e+a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{e^{3/2} \sqrt {-b e+a f} (d e-c f)^4}\right ) \] Input:

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

Output:

(x*Sqrt[a + b*x^2]*((2*d^2)/(c*(d*e - c*f)^2*(c + d*x^2)^2) + (d^2*(a*d*(3 
*d*e - 11*c*f) + 2*b*c*(-(d*e) + 5*c*f)))/(c^2*(b*c - a*d)*(-(d*e) + c*f)^ 
3*(c + d*x^2)) - (4*f^3)/(e*(d*e - c*f)^3*(e + f*x^2))) + (d*(24*b^2*c^4*f 
^2 - 4*a*b*c*d*(d^2*e^2 - 4*c*d*e*f + 15*c^2*f^2) + a^2*d^2*(3*d^2*e^2 - 1 
4*c*d*e*f + 35*c^2*f^2))*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b 
*x^2])])/(c^(5/2)*(-(b*c) + a*d)^(3/2)*(d*e - c*f)^4) + (4*f^2*(6*b*d*e^2 
+ a*f*(-7*d*e + c*f))*ArcTan[(Sqrt[-(b*e) + a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^ 
2])])/(e^(3/2)*Sqrt[-(b*e) + a*f]*(d*e - c*f)^4))/8
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1281\) vs. \(2(378)=756\).

Time = 1.97 (sec) , antiderivative size = 1281, normalized size of antiderivative = 3.39, number of steps used = 30, number of rules used = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.967, Rules used = {425, 426, 421, 25, 291, 221, 402, 25, 27, 291, 221, 402, 25, 27, 291, 221, 407, 291, 221, 426, 421, 25, 291, 221, 402, 25, 27, 291, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 426

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 407

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 426

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(b*((d*((d*(-1/2*(d*(d*e - c*f)*x*Sqrt[a + b*x^2])/(c*(b*c - a*d)*(c + d*x 
^2)) - ((a*d*(d*e - 3*c*f) - 2*b*c*(d*e - 2*c*f))*ArcTanh[(Sqrt[b*c - a*d] 
*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(2*c^(3/2)*(b*c - a*d)^(3/2))))/(d*e - c*f 
)^2 + (f^2*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(Sqrt[e 
]*Sqrt[b*e - a*f]*(d*e - c*f)^2)))/(d*e - c*f) - (f*((d^2*ArcTanh[(Sqrt[b* 
c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sqrt[c]*Sqrt[b*c - a*d]*(d*e - c* 
f)^2) - (f*(-1/2*(f*(d*e - c*f)*x*Sqrt[a + b*x^2])/(e*(b*e - a*f)*(e + f*x 
^2)) + ((2*b*e*(2*d*e - c*f) - a*f*(3*d*e - c*f))*ArcTanh[(Sqrt[b*e - a*f] 
*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(2*e^(3/2)*(b*e - a*f)^(3/2))))/(d*e - c*f 
)^2))/(d*e - c*f)))/d - ((b*c - a*d)*((d*((d*(-1/4*(d*(d*e - c*f)*x*Sqrt[a 
 + b*x^2])/(c*(b*c - a*d)*(c + d*x^2)^2) - (-1/2*(d*(a*d*(3*d*e - 7*c*f) - 
 2*b*c*(3*d*e - 5*c*f))*x*Sqrt[a + b*x^2])/(c*(b*c - a*d)*(c + d*x^2)) - ( 
(a^2*d^2*(3*d*e - 7*c*f) - 4*a*b*c*d*(2*d*e - 5*c*f) + 8*b^2*c^2*(d*e - 2* 
c*f))*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(2*c^(3/2)*( 
b*c - a*d)^(3/2)))/(4*c*(b*c - a*d))))/(d*e - c*f)^2 + (f^2*((d*ArcTanh[(S 
qrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sqrt[c]*Sqrt[b*c - a*d]*(d* 
e - c*f)) - (f*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(Sq 
rt[e]*Sqrt[b*e - a*f]*(d*e - c*f))))/(d*e - c*f)^2))/(d*e - c*f) - (f*((d* 
((d*(-1/2*(d*(d*e - c*f)*x*Sqrt[a + b*x^2])/(c*(b*c - a*d)*(c + d*x^2)) - 
((a*d*(d*e - 3*c*f) - 2*b*c*(d*e - 2*c*f))*ArcTanh[(Sqrt[b*c - a*d]*x)/...
 

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 221

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

rule 291

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

rule 402

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

rule 407

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

rule 421

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

rule 425

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

rule 426

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

Maple [A] (verified)

Time = 1.86 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.30

method	result	size
pseudoelliptic	\(-\frac {\frac {35 d \left (x^{2} d +c \right )^{2} \left (\frac {24 b^{2} c^{4} f^{2}}{35}-\frac {12 a b \,c^{3} d \,f^{2}}{7}+a \,d^{2} f \left (a f +\frac {16 b e}{35}\right ) c^{2}-\frac {2 a \,d^{3} \left (a f +\frac {2 b e}{7}\right ) e c}{5}+\frac {3 a^{2} e^{2} d^{4}}{35}\right ) \sqrt {\left (a f -b e \right ) e}\, \left (f \,x^{2}+e \right ) e \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )}{4}+\sqrt {\left (a d -b c \right ) c}\, \left (\left (a d -b c \right ) c^{2} \left (a c \,f^{2}+\left (-7 a e f +6 b \,e^{2}\right ) d \right ) \left (x^{2} d +c \right )^{2} \left (f \,x^{2}+e \right ) f^{2} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )-\left (-b \,c^{5} f^{3}+d \,f^{3} \left (-2 b \,x^{2}+a \right ) c^{4}+2 d^{2} \left (-\frac {3 b \,e^{2}}{2}-\frac {3 b e f \,x^{2}}{2}+f^{2} x^{2} \left (-\frac {b \,x^{2}}{2}+a \right )\right ) f \,c^{3}+\frac {13 d^{3} \left (\frac {4 b \,e^{3}}{13}+f \left (-\frac {6 b \,x^{2}}{13}+a \right ) e^{2}+f^{2} x^{2} \left (-\frac {10 b \,x^{2}}{13}+a \right ) e +\frac {4 a \,f^{3} x^{4}}{13}\right ) c^{2}}{4}-\frac {5 d^{4} \left (\left (-\frac {2 b \,x^{2}}{5}+a \right ) e -\frac {11 a f \,x^{2}}{5}\right ) \left (f \,x^{2}+e \right ) e c}{4}-\frac {3 a \,d^{5} e^{2} x^{2} \left (f \,x^{2}+e \right )}{4}\right ) \left (c f -d e \right ) \sqrt {\left (a f -b e \right ) e}\, \sqrt {b \,x^{2}+a}\, x \right )}{2 \sqrt {\left (a d -b c \right ) c}\, \sqrt {\left (a f -b e \right ) e}\, \left (x^{2} d +c \right )^{2} \left (a d -b c \right ) \left (c f -d e \right )^{4} c^{2} e \left (f \,x^{2}+e \right )}\)	\(492\)
default	\(\text {Expression too large to display}\)	\(6568\)

Input:

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

Output:

-1/2/((a*d-b*c)*c)^(1/2)*(35/4*d*(d*x^2+c)^2*(24/35*b^2*c^4*f^2-12/7*a*b*c 
^3*d*f^2+a*d^2*f*(a*f+16/35*b*e)*c^2-2/5*a*d^3*(a*f+2/7*b*e)*e*c+3/35*a^2* 
e^2*d^4)*((a*f-b*e)*e)^(1/2)*(f*x^2+e)*e*arctan(c*(b*x^2+a)^(1/2)/x/((a*d- 
b*c)*c)^(1/2))+((a*d-b*c)*c)^(1/2)*((a*d-b*c)*c^2*(a*c*f^2+(-7*a*e*f+6*b*e 
^2)*d)*(d*x^2+c)^2*(f*x^2+e)*f^2*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^ 
(1/2))-(-b*c^5*f^3+d*f^3*(-2*b*x^2+a)*c^4+2*d^2*(-3/2*b*e^2-3/2*b*e*f*x^2+ 
f^2*x^2*(-1/2*b*x^2+a))*f*c^3+13/4*d^3*(4/13*b*e^3+f*(-6/13*b*x^2+a)*e^2+f 
^2*x^2*(-10/13*b*x^2+a)*e+4/13*a*f^3*x^4)*c^2-5/4*d^4*((-2/5*b*x^2+a)*e-11 
/5*a*f*x^2)*(f*x^2+e)*e*c-3/4*a*d^5*e^2*x^2*(f*x^2+e))*(c*f-d*e)*((a*f-b*e 
)*e)^(1/2)*(b*x^2+a)^(1/2)*x))/((a*f-b*e)*e)^(1/2)/(d*x^2+c)^2/(a*d-b*c)/( 
c*f-d*e)^4/c^2/e/(f*x^2+e)
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1462 vs. \(2 (343) = 686\).

Time = 3.85 (sec) , antiderivative size = 1462, normalized size of antiderivative = 3.87 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^3 \left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/8*(4*a*b^(3/2)*c*d^4*e^2 - 3*a^2*sqrt(b)*d^5*e^2 - 16*a*b^(3/2)*c^2*d^3 
*e*f + 14*a^2*sqrt(b)*c*d^4*e*f - 24*b^(5/2)*c^4*d*f^2 + 60*a*b^(3/2)*c^3* 
d^2*f^2 - 35*a^2*sqrt(b)*c^2*d^3*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*c^3*d^4*e^4 - a*c^2 
*d^5*e^4 - 4*b*c^4*d^3*e^3*f + 4*a*c^3*d^4*e^3*f + 6*b*c^5*d^2*e^2*f^2 - 6 
*a*c^4*d^3*e^2*f^2 - 4*b*c^6*d*e*f^3 + 4*a*c^5*d^2*e*f^3 + b*c^7*f^4 - a*c 
^6*d*f^4)*sqrt(-b^2*c^2 + a*b*c*d)) - 1/2*(6*b^(3/2)*d*e^2*f^2 - 7*a*sqrt( 
b)*d*e*f^3 + a*sqrt(b)*c*f^4)*arctan(1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2* 
f + 2*b*e - a*f)/sqrt(-b^2*e^2 + a*b*e*f))/((d^4*e^5 - 4*c*d^3*e^4*f + 6*c 
^2*d^2*e^3*f^2 - 4*c^3*d*e^2*f^3 + c^4*e*f^4)*sqrt(-b^2*e^2 + a*b*e*f)) - 
(2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^(3/2)*e*f^2 - (sqrt(b)*x - sqrt(b*x^2 
 + a))^2*a*sqrt(b)*f^3 + a^2*sqrt(b)*f^3)/((d^3*e^4 - 3*c*d^2*e^3*f + 3*c^ 
2*d*e^2*f^2 - c^3*e*f^3)*((sqrt(b)*x - sqrt(b*x^2 + a))^4*f + 4*(sqrt(b)*x 
 - sqrt(b*x^2 + a))^2*b*e - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*f + a^2*f) 
) - 1/4*(4*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(3/2)*c*d^4*e - 3*(sqrt(b)* 
x - sqrt(b*x^2 + a))^6*a^2*sqrt(b)*d^5*e + 16*(sqrt(b)*x - sqrt(b*x^2 + a) 
)^6*b^(5/2)*c^3*d^2*f - 28*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(3/2)*c^2*d 
^3*f + 11*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*sqrt(b)*c*d^4*f - 16*(sqrt(b 
)*x - sqrt(b*x^2 + a))^4*b^(7/2)*c^3*d^2*e + 40*(sqrt(b)*x - sqrt(b*x^2 + 
a))^4*a*b^(5/2)*c^2*d^3*e - 30*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(3...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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