\(\int \frac {\sqrt {a+b x^2}}{(c+d x^2)^2 (e+f x^2)^3} \, dx\) [286]

Optimal result

Integrand size = 30, antiderivative size = 377 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\frac {d^3 x \sqrt {a+b x^2}}{2 c (d e-c f)^3 \left (c+d x^2\right )}+\frac {f^2 x \sqrt {a+b x^2}}{4 e (d e-c f)^2 \left (e+f x^2\right )^2}-\frac {f^2 (a f (11 d e-3 c f)-2 b e (5 d e-c f)) x \sqrt {a+b x^2}}{8 e^2 (b e-a f) (d e-c f)^3 \left (e+f x^2\right )}+\frac {d^2 \left (6 b c^2 f+a d (d e-7 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} \sqrt {b c-a d} (d e-c f)^4}-\frac {f \left (24 b^2 d^2 e^4-4 a b e f \left (15 d^2 e^2-4 c d e f+c^2 f^2\right )+a^2 f^2 \left (35 d^2 e^2-14 c d e f+3 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{8 e^{5/2} (b e-a f)^{3/2} (d e-c f)^4} \] Output:

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

Mathematica [A] (verified)

Time = 11.65 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\frac {1}{8} \left (x \sqrt {a+b x^2} \left (-\frac {4 d^3}{c (-d e+c f)^3 \left (c+d x^2\right )}+\frac {2 f^2}{e (d e-c f)^2 \left (e+f x^2\right )^2}+\frac {f^2 (2 b e (5 d e-c f)+a f (-11 d e+3 c f))}{e^2 (b e-a f) (d e-c f)^3 \left (e+f x^2\right )}\right )+\frac {4 d^2 \left (6 b c^2 f+a d (d e-7 c f)\right ) \arctan \left (\frac {\sqrt {-b c+a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{c^{3/2} \sqrt {-b c+a d} (d e-c f)^4}+\frac {f \left (24 b^2 d^2 e^4-4 a b e f \left (15 d^2 e^2-4 c d e f+c^2 f^2\right )+a^2 f^2 \left (35 d^2 e^2-14 c d e f+3 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {-b e+a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{e^{5/2} (-b e+a f)^{3/2} (d e-c f)^4}\right ) \] Input:

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

Output:

(x*Sqrt[a + b*x^2]*((-4*d^3)/(c*(-(d*e) + c*f)^3*(c + d*x^2)) + (2*f^2)/(e 
*(d*e - c*f)^2*(e + f*x^2)^2) + (f^2*(2*b*e*(5*d*e - c*f) + a*f*(-11*d*e + 
 3*c*f)))/(e^2*(b*e - a*f)*(d*e - c*f)^3*(e + f*x^2))) + (4*d^2*(6*b*c^2*f 
 + a*d*(d*e - 7*c*f))*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^ 
2])])/(c^(3/2)*Sqrt[-(b*c) + a*d]*(d*e - c*f)^4) + (f*(24*b^2*d^2*e^4 - 4* 
a*b*e*f*(15*d^2*e^2 - 4*c*d*e*f + c^2*f^2) + a^2*f^2*(35*d^2*e^2 - 14*c*d* 
e*f + 3*c^2*f^2))*ArcTan[(Sqrt[-(b*e) + a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])] 
)/(e^(5/2)*(-(b*e) + a*f)^(3/2)*(d*e - c*f)^4))/8
 

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.

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 407

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 426

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 407

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 426

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 402

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 25

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

rule 27

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

rule 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 = 1.83 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.30

method	result	size
pseudoelliptic	\(-\frac {3 \left (\left (8 b^{2} d^{2} e^{4}-20 a b \,d^{2} e^{3} f +\frac {35 a \left (a d +\frac {16 b c}{35}\right ) d \,f^{2} e^{2}}{3}-\frac {14 a \left (a d +\frac {2 b c}{7}\right ) c \,f^{3} e}{3}+a^{2} c^{2} f^{4}\right ) \sqrt {\left (a d -b c \right ) c}\, c \left (x^{2} d +c \right ) \left (f \,x^{2}+e \right )^{2} f \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )-\frac {28 \left (\left (-\frac {a \,d^{2} e}{7}+c f \left (a d -\frac {6 b c}{7}\right )\right ) d^{2} \left (x^{2} d +c \right ) \left (a f -b e \right ) \left (f \,x^{2}+e \right )^{2} e^{2} \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )+\frac {5 \sqrt {\left (a d -b c \right ) c}\, \left (\frac {4 b \,d^{3} e^{5}}{5}-\frac {4 d^{3} f \left (-2 b \,x^{2}+a \right ) e^{4}}{5}-\frac {8 d \left (-\frac {3 b \,c^{2}}{2}-\frac {3 b c d \,x^{2}}{2}+d^{2} x^{2} \left (-\frac {b \,x^{2}}{2}+a \right )\right ) f^{2} e^{3}}{5}-\frac {13 \left (\frac {4 c^{3} b}{13}+d \left (-\frac {6 b \,x^{2}}{13}+a \right ) c^{2}+d^{2} x^{2} \left (-\frac {10 b \,x^{2}}{13}+a \right ) c +\frac {4 a \,d^{3} x^{4}}{13}\right ) f^{3} e^{2}}{5}+\left (c \left (-\frac {2 b \,x^{2}}{5}+a \right )-\frac {11 a d \,x^{2}}{5}\right ) c \left (x^{2} d +c \right ) f^{4} e +\frac {3 a \,c^{2} f^{5} x^{2} \left (x^{2} d +c \right )}{5}\right ) \left (c f -d e \right ) \sqrt {b \,x^{2}+a}\, x}{28}\right ) \sqrt {\left (a f -b e \right ) e}}{3}\right )}{8 \sqrt {\left (a d -b c \right ) c}\, \sqrt {\left (a f -b e \right ) e}\, \left (x^{2} d +c \right ) \left (c f -d e \right )^{4} c \left (f \,x^{2}+e \right )^{2} \left (a f -b e \right ) e^{2}}\)	\(490\)
default	\(\text {Expression too large to display}\)	\(6570\)

Input:

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

Output:

-3/8/((a*d-b*c)*c)^(1/2)*((8*b^2*d^2*e^4-20*a*b*d^2*e^3*f+35/3*a*(a*d+16/3 
5*b*c)*d*f^2*e^2-14/3*a*(a*d+2/7*b*c)*c*f^3*e+a^2*c^2*f^4)*((a*d-b*c)*c)^( 
1/2)*c*(d*x^2+c)*(f*x^2+e)^2*f*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1 
/2))-28/3*((-1/7*a*d^2*e+c*f*(a*d-6/7*b*c))*d^2*(d*x^2+c)*(a*f-b*e)*(f*x^2 
+e)^2*e^2*arctan(c*(b*x^2+a)^(1/2)/x/((a*d-b*c)*c)^(1/2))+5/28*((a*d-b*c)* 
c)^(1/2)*(4/5*b*d^3*e^5-4/5*d^3*f*(-2*b*x^2+a)*e^4-8/5*d*(-3/2*b*c^2-3/2*b 
*c*d*x^2+d^2*x^2*(-1/2*b*x^2+a))*f^2*e^3-13/5*(4/13*c^3*b+d*(-6/13*b*x^2+a 
)*c^2+d^2*x^2*(-10/13*b*x^2+a)*c+4/13*a*d^3*x^4)*f^3*e^2+(c*(-2/5*b*x^2+a) 
-11/5*a*d*x^2)*c*(d*x^2+c)*f^4*e+3/5*a*c^2*f^5*x^2*(d*x^2+c))*(c*f-d*e)*(b 
*x^2+a)^(1/2)*x)*((a*f-b*e)*e)^(1/2))/((a*f-b*e)*e)^(1/2)/(d*x^2+c)/(c*f-d 
*e)^4/c/(f*x^2+e)^2/(a*f-b*e)/e^2
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1461 vs. \(2 (342) = 684\).

Time = 3.98 (sec) , antiderivative size = 1461, normalized size of antiderivative = 3.88 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*(a*sqrt(b)*d^4*e + 6*b^(3/2)*c^2*d^2*f - 7*a*sqrt(b)*c*d^3*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))/((c*d^4*e^4 - 4*c^2*d^3*e^3*f + 6*c^3*d^2*e^2*f^2 - 4*c^4*d*e*f^3 + 
c^5*f^4)*sqrt(-b^2*c^2 + a*b*c*d)) + 1/8*(24*b^(5/2)*d^2*e^4*f - 60*a*b^(3 
/2)*d^2*e^3*f^2 + 16*a*b^(3/2)*c*d*e^2*f^3 + 35*a^2*sqrt(b)*d^2*e^2*f^3 - 
4*a*b^(3/2)*c^2*e*f^4 - 14*a^2*sqrt(b)*c*d*e*f^4 + 3*a^2*sqrt(b)*c^2*f^5)* 
arctan(1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2*f + 2*b*e - a*f)/sqrt(-b^2*e^2 
 + a*b*e*f))/((b*d^4*e^7 - 4*b*c*d^3*e^6*f - a*d^4*e^6*f + 6*b*c^2*d^2*e^5 
*f^2 + 4*a*c*d^3*e^5*f^2 - 4*b*c^3*d*e^4*f^3 - 6*a*c^2*d^2*e^4*f^3 + b*c^4 
*e^3*f^4 + 4*a*c^3*d*e^3*f^4 - a*c^4*e^2*f^5)*sqrt(-b^2*e^2 + a*b*e*f)) + 
(2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^(3/2)*c*d^2 - (sqrt(b)*x - sqrt(b*x^2 
 + a))^2*a*sqrt(b)*d^3 + a^2*sqrt(b)*d^3)/((c*d^3*e^3 - 3*c^2*d^2*e^2*f + 
3*c^3*d*e*f^2 - c^4*f^3)*((sqrt(b)*x - sqrt(b*x^2 + a))^4*d + 4*(sqrt(b)*x 
 - sqrt(b*x^2 + a))^2*b*c - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*d + a^2*d) 
) + 1/4*(16*(sqrt(b)*x - sqrt(b*x^2 + a))^6*b^(5/2)*d*e^3*f^2 - 28*(sqrt(b 
)*x - sqrt(b*x^2 + a))^6*a*b^(3/2)*d*e^2*f^3 + 4*(sqrt(b)*x - sqrt(b*x^2 + 
 a))^6*a*b^(3/2)*c*e*f^4 + 11*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*sqrt(b)* 
d*e*f^4 - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*sqrt(b)*c*f^5 + 80*(sqrt(b 
)*x - sqrt(b*x^2 + a))^4*b^(7/2)*d*e^4*f - 16*(sqrt(b)*x - sqrt(b*x^2 + a) 
)^4*b^(7/2)*c*e^3*f^2 - 168*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b^(5/2)*d...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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