\(\int \frac {\sqrt {a+b x^2}}{(c+d x^2)^2 (e+f x^2)^2} \, dx\) [285]

Optimal result

Integrand size = 30, antiderivative size = 249 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\frac {d^2 x \sqrt {a+b x^2}}{2 c (d e-c f)^2 \left (c+d x^2\right )}+\frac {f^2 x \sqrt {a+b x^2}}{2 e (d e-c f)^2 \left (e+f x^2\right )}+\frac {d \left (4 b c^2 f+a d (d e-5 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} \sqrt {b c-a d} (d e-c f)^3}-\frac {f \left (4 b d e^2-a f (5 d e-c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{2 e^{3/2} \sqrt {b e-a f} (d e-c f)^3} \] Output:

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

Mathematica [A] (verified)

Time = 10.66 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\frac {(d e-c f) x \sqrt {a+b x^2} \left (\frac {d^2}{c^2+c d x^2}+\frac {f^2}{e^2+e f x^2}\right )+\frac {d \left (4 b c^2 f+a d (d e-5 c f)\right ) \arctan \left (\frac {\sqrt {-b c+a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{c^{3/2} \sqrt {-b c+a d}}-\frac {f \left (4 b d e^2+a f (-5 d e+c f)\right ) \arctan \left (\frac {\sqrt {-b e+a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{e^{3/2} \sqrt {-b e+a f}}}{2 (d e-c f)^3} \] Input:

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

Output:

((d*e - c*f)*x*Sqrt[a + b*x^2]*(d^2/(c^2 + c*d*x^2) + f^2/(e^2 + e*f*x^2)) 
 + (d*(4*b*c^2*f + a*d*(d*e - 5*c*f))*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[ 
c]*Sqrt[a + b*x^2])])/(c^(3/2)*Sqrt[-(b*c) + a*d]) - (f*(4*b*d*e^2 + a*f*( 
-5*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]))/(2*(d*e - c*f)^3)
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(650\) vs. \(2(249)=498\).

Time = 0.96 (sec) , antiderivative size = 650, normalized size of antiderivative = 2.61, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {425, 421, 291, 221, 402, 27, 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.

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 426

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(b*((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 - ((b*c - a*d)*((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
 

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 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 = 1.55 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.12

method	result	size
pseudoelliptic	\(\frac {-5 \left (a c d f -\frac {1}{5} a \,d^{2} e -\frac {4}{5} b \,c^{2} f \right ) \sqrt {\left (a f -b e \right ) e}\, d \left (x^{2} d +c \right ) \left (f \,x^{2}+e \right ) e \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )+\sqrt {\left (a d -b c \right ) c}\, \left (-c f \left (f \,x^{2}+e \right ) \left (x^{2} d +c \right ) \left (a c \,f^{2}-5 a d e f +4 b d \,e^{2}\right ) \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\left (c^{2} f^{2}+c d \,f^{2} x^{2}+d^{2} e \left (f \,x^{2}+e \right )\right ) \sqrt {\left (a f -b e \right ) e}\, \left (c f -d e \right ) \sqrt {b \,x^{2}+a}\, x \right )}{2 \sqrt {\left (a d -b c \right ) c}\, \sqrt {\left (a f -b e \right ) e}\, \left (x^{2} d +c \right ) \left (c f -d e \right )^{3} c e \left (f \,x^{2}+e \right )}\)	\(278\)
default	\(\text {Expression too large to display}\)	\(4270\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

integrate(sqrt(b*x^2 + a)/((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. 1081 vs. \(2 (217) = 434\).

Time = 1.99 (sec) , antiderivative size = 1081, normalized size of antiderivative = 4.34 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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