\(\int \frac {(a+b x^2)^{3/2}}{(c+d x^2)^2 (e+f x^2)^2} \, dx\) [312]

Optimal result

Integrand size = 30, antiderivative size = 266 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=-\frac {d (b c-a d) x \sqrt {a+b x^2}}{2 c (d e-c f)^2 \left (c+d x^2\right )}-\frac {f (b e-a f) x \sqrt {a+b x^2}}{2 e (d e-c f)^2 \left (e+f x^2\right )}-\frac {\sqrt {b c-a d} (a d (d e-5 c f)+2 b c (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} (d e-c f)^3}-\frac {\sqrt {b e-a f} (a f (5 d e-c f)-2 b e (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{2 e^{3/2} (d e-c f)^3} \] Output:

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

Mathematica [A] (verified)

Time = 2.27 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.13 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\frac {1}{2} \left (\frac {x \sqrt {a+b x^2} \left (a \left (c^2 f^2+c d f^2 x^2+d^2 e \left (e+f x^2\right )\right )-b c e \left (c f+d \left (e+2 f x^2\right )\right )\right )}{c e (d e-c f)^2 \left (c+d x^2\right ) \left (e+f x^2\right )}+\frac {\sqrt {-b c+a d} (a d (d e-5 c f)+2 b c (d e+c f)) \arctan \left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {-b c+a d}}\right )}{c^{3/2} (-d e+c f)^3}+\frac {\sqrt {-b e+a f} (a f (-5 d e+c f)+2 b e (d e+c f)) \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{e^{3/2} (d e-c f)^3}\right ) \] Input:

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

Output:

((x*Sqrt[a + b*x^2]*(a*(c^2*f^2 + c*d*f^2*x^2 + d^2*e*(e + f*x^2)) - b*c*e 
*(c*f + d*(e + 2*f*x^2))))/(c*e*(d*e - c*f)^2*(c + d*x^2)*(e + f*x^2)) + ( 
Sqrt[-(b*c) + a*d]*(a*d*(d*e - 5*c*f) + 2*b*c*(d*e + c*f))*ArcTan[(-(d*x*S 
qrt[a + b*x^2]) + Sqrt[b]*(c + d*x^2))/(Sqrt[c]*Sqrt[-(b*c) + a*d])])/(c^( 
3/2)*(-(d*e) + c*f)^3) + (Sqrt[-(b*e) + a*f]*(a*f*(-5*d*e + c*f) + 2*b*e*( 
d*e + c*f))*ArcTan[(-(f*x*Sqrt[a + b*x^2]) + Sqrt[b]*(e + f*x^2))/(Sqrt[e] 
*Sqrt[-(b*e) + a*f])])/(e^(3/2)*(d*e - c*f)^3))/2
 

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.

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 301

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 426

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 25

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

rule 27

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

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 221

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

rule 224

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

rule 291

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

rule 301

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

rule 398

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

rule 401

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

rule 402

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

rule 421

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

rule 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 = 1.66 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.24

method	result	size
pseudoelliptic	\(\frac {-5 \left (a d -b c \right ) \sqrt {\left (a f -b e \right ) e}\, \left (-\frac {2 b \,c^{2} f}{5}+d \left (a f -\frac {2 b e}{5}\right ) c -\frac {a \,d^{2} e}{5}\right ) \left (x^{2} d +c \right ) \left (f \,x^{2}+e \right ) e \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )+\left (-c \left (\left (a \,f^{2}+2 b e f \right ) c -5 a d e f +2 b d \,e^{2}\right ) \left (x^{2} d +c \right ) \left (a f -b e \right ) \left (f \,x^{2}+e \right ) \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\sqrt {\left (a f -b e \right ) e}\, \left (c f -d e \right ) \left (\left (a \,f^{2}-b e f \right ) c^{2}+d \left (a \,f^{2} x^{2}-2 b e f \,x^{2}-b \,e^{2}\right ) c +a \,d^{2} e \left (f \,x^{2}+e \right )\right ) \sqrt {b \,x^{2}+a}\, x \right ) \sqrt {\left (a d -b c \right ) c}}{2 \sqrt {\left (a d -b c \right ) c}\, \sqrt {\left (a f -b e \right ) e}\, \left (x^{2} d +c \right ) \left (c f -d e \right )^{3} c e \left (f \,x^{2}+e \right )}\)	\(331\)
default	\(\text {Expression too large to display}\)	\(7078\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1331 vs. \(2 (235) = 470\).

Time = 1.98 (sec) , antiderivative size = 1331, normalized size of antiderivative = 5.00 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*(2*b^(5/2)*c^2*d*e - a*b^(3/2)*c*d^2*e - a^2*sqrt(b)*d^3*e + 2*b^(5/2) 
*c^3*f - 7*a*b^(3/2)*c^2*d*f + 5*a^2*sqrt(b)*c*d^2*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^3 
*e^3 - 3*c^2*d^2*e^2*f + 3*c^3*d*e*f^2 - c^4*f^3)*sqrt(-b^2*c^2 + a*b*c*d) 
) - 1/2*(2*b^(5/2)*d*e^3 + 2*b^(5/2)*c*e^2*f - 7*a*b^(3/2)*d*e^2*f - a*b^( 
3/2)*c*e*f^2 + 5*a^2*sqrt(b)*d*e*f^2 - a^2*sqrt(b)*c*f^3)*arctan(1/2*((sqr 
t(b)*x - sqrt(b*x^2 + a))^2*f + 2*b*e - a*f)/sqrt(-b^2*e^2 + a*b*e*f))/((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^2*e^2 + a*b* 
e*f)) - (2*(sqrt(b)*x - sqrt(b*x^2 + a))^6*b^(5/2)*c*d*e^2 + 2*(sqrt(b)*x 
- sqrt(b*x^2 + a))^6*b^(5/2)*c^2*e*f - 6*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a 
*b^(3/2)*c*d*e*f + (sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*sqrt(b)*d^2*e*f + ( 
sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*sqrt(b)*c*d*f^2 + 16*(sqrt(b)*x - sqrt( 
b*x^2 + a))^4*b^(7/2)*c^2*e^2 - 16*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b^(5/ 
2)*c*d*e^2 + 4*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(3/2)*d^2*e^2 - 16*(s 
qrt(b)*x - sqrt(b*x^2 + a))^4*a*b^(5/2)*c^2*e*f + 14*(sqrt(b)*x - sqrt(b*x 
^2 + a))^4*a^2*b^(3/2)*c*d*e*f - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^3*sqr 
t(b)*d^2*e*f + 4*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(3/2)*c^2*f^2 - 3*( 
sqrt(b)*x - sqrt(b*x^2 + a))^4*a^3*sqrt(b)*c*d*f^2 + 6*(sqrt(b)*x - sqrt(b 
*x^2 + a))^2*a^2*b^(5/2)*c*d*e^2 - 4*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^3*b 
^(3/2)*d^2*e^2 + 6*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*b^(5/2)*c^2*e*f ...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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