\(\int \frac {(c+d x^2)^2}{(a+b x^2)^{5/2} (e+f x^2)^3} \, dx\) [369]

Optimal result

Integrand size = 30, antiderivative size = 654 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\frac {d (8 b d e-8 b c f+3 a d f) x}{12 a e f^2 (b e-a f) \left (a+b x^2\right )^{3/2}}+\frac {b \left (a^2 f^2 \left (7 d^2 e^2+10 c d e f-9 c^2 f^2\right )-8 b^2 e^2 \left (2 d^2 e^2-2 c d e f-c^2 f^2\right )+4 a b e f \left (11 d^2 e^2-24 c d e f+9 c^2 f^2\right )\right ) x}{24 a e^2 f^2 (b e-a f)^3 \left (a+b x^2\right )^{3/2}}+\frac {4 b d (d e-c f) x}{3 a^2 e f^2 (b e-a f) \sqrt {a+b x^2}}+\frac {b \left (8 a b^2 e^2 f \left (12 d^2 e^2-10 c d e f-11 c^2 f^2\right )-a^3 f^3 \left (23 d^2 e^2+26 c d e f-9 c^2 f^2\right )-16 b^3 e^3 \left (2 d^2 e^2-2 c d e f-c^2 f^2\right )-2 a^2 b e f^2 \left (73 d^2 e^2-142 c d e f+21 c^2 f^2\right )\right ) x}{24 a^2 e^2 f^2 (b e-a f)^4 \sqrt {a+b x^2}}-\frac {f x \left (c+d x^2\right )^2}{4 e (b e-a f) \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}-\frac {(d e-c f) (2 b e (3 d e-5 c f)+a f (d e+3 c f)) x}{8 e^2 f (b e-a f)^2 \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}+\frac {\left (8 a b e f \left (3 d^2 e^2-3 c d e f-2 c^2 f^2\right )+a^2 f^2 \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+8 b^2 e^2 \left (d^2 e^2-6 c d e f+6 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{8 e^{5/2} (b e-a f)^{9/2}} \] Output:

1/12*d*(3*a*d*f-8*b*c*f+8*b*d*e)*x/a/e/f^2/(-a*f+b*e)/(b*x^2+a)^(3/2)+1/24 
*b*(a^2*f^2*(-9*c^2*f^2+10*c*d*e*f+7*d^2*e^2)-8*b^2*e^2*(-c^2*f^2-2*c*d*e* 
f+2*d^2*e^2)+4*a*b*e*f*(9*c^2*f^2-24*c*d*e*f+11*d^2*e^2))*x/a/e^2/f^2/(-a* 
f+b*e)^3/(b*x^2+a)^(3/2)+4/3*b*d*(-c*f+d*e)*x/a^2/e/f^2/(-a*f+b*e)/(b*x^2+ 
a)^(1/2)+1/24*b*(8*a*b^2*e^2*f*(-11*c^2*f^2-10*c*d*e*f+12*d^2*e^2)-a^3*f^3 
*(-9*c^2*f^2+26*c*d*e*f+23*d^2*e^2)-16*b^3*e^3*(-c^2*f^2-2*c*d*e*f+2*d^2*e 
^2)-2*a^2*b*e*f^2*(21*c^2*f^2-142*c*d*e*f+73*d^2*e^2))*x/a^2/e^2/f^2/(-a*f 
+b*e)^4/(b*x^2+a)^(1/2)-1/4*f*x*(d*x^2+c)^2/e/(-a*f+b*e)/(b*x^2+a)^(3/2)/( 
f*x^2+e)^2-1/8*(-c*f+d*e)*(2*b*e*(-5*c*f+3*d*e)+a*f*(3*c*f+d*e))*x/e^2/f/( 
-a*f+b*e)^2/(b*x^2+a)^(3/2)/(f*x^2+e)+1/8*(8*a*b*e*f*(-2*c^2*f^2-3*c*d*e*f 
+3*d^2*e^2)+a^2*f^2*(3*c^2*f^2+2*c*d*e*f+3*d^2*e^2)+8*b^2*e^2*(6*c^2*f^2-6 
*c*d*e*f+d^2*e^2))*arctanh((-a*f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/2))/e^( 
5/2)/(-a*f+b*e)^(9/2)
 

Mathematica [A] (verified)

Time = 16.16 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.53 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\frac {1}{24} \left (x \sqrt {a+b x^2} \left (-\frac {8 b (b c-a d)^2}{a (-b e+a f)^3 \left (a+b x^2\right )^2}+\frac {8 b (b c-a d) \left (2 b^2 c e+5 a^2 d f+a b (4 d e-11 c f)\right )}{a^2 (b e-a f)^4 \left (a+b x^2\right )}-\frac {6 f (d e-c f)^2}{e (b e-a f)^3 \left (e+f x^2\right )^2}+\frac {3 f (-d e+c f) (2 b e (3 d e-7 c f)+a f (5 d e+3 c f))}{e^2 (b e-a f)^4 \left (e+f x^2\right )}\right )+\frac {3 \left (8 a b e f \left (3 d^2 e^2-3 c d e f-2 c^2 f^2\right )+a^2 f^2 \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+8 b^2 e^2 \left (d^2 e^2-6 c d e f+6 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {-b e+a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{e^{5/2} (-b e+a f)^{9/2}}\right ) \] Input:

Integrate[(c + d*x^2)^2/((a + b*x^2)^(5/2)*(e + f*x^2)^3),x]
 

Output:

(x*Sqrt[a + b*x^2]*((-8*b*(b*c - a*d)^2)/(a*(-(b*e) + a*f)^3*(a + b*x^2)^2 
) + (8*b*(b*c - a*d)*(2*b^2*c*e + 5*a^2*d*f + a*b*(4*d*e - 11*c*f)))/(a^2* 
(b*e - a*f)^4*(a + b*x^2)) - (6*f*(d*e - c*f)^2)/(e*(b*e - a*f)^3*(e + f*x 
^2)^2) + (3*f*(-(d*e) + c*f)*(2*b*e*(3*d*e - 7*c*f) + a*f*(5*d*e + 3*c*f)) 
)/(e^2*(b*e - a*f)^4*(e + f*x^2))) + (3*(8*a*b*e*f*(3*d^2*e^2 - 3*c*d*e*f 
- 2*c^2*f^2) + a^2*f^2*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2) + 8*b^2*e^2*(d^ 
2*e^2 - 6*c*d*e*f + 6*c^2*f^2))*ArcTan[(Sqrt[-(b*e) + a*f]*x)/(Sqrt[e]*Sqr 
t[a + b*x^2])])/(e^(5/2)*(-(b*e) + a*f)^(9/2)))/24
 

Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 751, normalized size of antiderivative = 1.15, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {425, 402, 25, 402, 25, 27, 402, 27, 291, 221, 402, 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.

\(\displaystyle \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx\)

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {d \left (\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}-\frac {\int -\frac {4 (b c-a d) f x^2+2 b c e+a d e-3 a c f}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}dx}{3 a (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\int -\frac {6 (b c-a d) f x^2+2 b c e+a d e-3 a c f}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^3}dx}{3 a (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {d \left (\frac {\int \frac {4 (b c-a d) f x^2+2 b c e+a d e-3 a c f}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}dx}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\int \frac {6 (b c-a d) f x^2+2 b c e+a d e-3 a c f}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^3}dx}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {d \left (\frac {\frac {x \left (4 a^2 d f+a b (d e-7 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}-\frac {\int -\frac {f \left (2 \left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x^2+a (2 b c e-5 a d e+3 a c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{a (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {x \left (6 a^2 d f+a b (d e-9 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\int -\frac {f \left (4 \left (6 d f a^2+b (d e-9 c f) a+2 b^2 c e\right ) x^2+a (4 b c e-7 a d e+3 a c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{a (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {d \left (\frac {\frac {\int \frac {f \left (2 \left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x^2+a (2 b c e-5 a d e+3 a c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{a (b e-a f)}+\frac {x \left (4 a^2 d f+a b (d e-7 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {\int \frac {f \left (4 \left (6 d f a^2+b (d e-9 c f) a+2 b^2 c e\right ) x^2+a (4 b c e-7 a d e+3 a c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{a (b e-a f)}+\frac {x \left (6 a^2 d f+a b (d e-9 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d \left (\frac {\frac {f \int \frac {2 \left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x^2+a (2 b c e-5 a d e+3 a c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{a (b e-a f)}+\frac {x \left (4 a^2 d f+a b (d e-7 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {f \int \frac {4 \left (6 d f a^2+b (d e-9 c f) a+2 b^2 c e\right ) x^2+a (4 b c e-7 a d e+3 a c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{a (b e-a f)}+\frac {x \left (6 a^2 d f+a b (d e-9 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {d \left (\frac {\frac {f \left (\frac {\int -\frac {3 a^2 (2 b e (2 d e-3 c f)+a f (d e+c f))}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (a^2 f (13 d e-3 c f)+2 a b e (d e-8 c f)+4 b^2 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{a (b e-a f)}+\frac {x \left (4 a^2 d f+a b (d e-7 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {f \left (\frac {\int \frac {2 b \left (f (31 d e-3 c f) a^2+4 b e (d e-10 c f) a+8 b^2 c e^2\right ) x^2+a \left (-3 f (d e+3 c f) a^2-4 b e (8 d e-9 c f) a+8 b^2 c e^2\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (a^2 f (31 d e-3 c f)+4 a b e (d e-10 c f)+8 b^2 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{a (b e-a f)}+\frac {x \left (6 a^2 d f+a b (d e-9 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d \left (\frac {\frac {f \left (\frac {x \sqrt {a+b x^2} \left (a^2 f (13 d e-3 c f)+2 a b e (d e-8 c f)+4 b^2 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 (a f (c f+d e)+2 b e (2 d e-3 c f)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}\right )}{a (b e-a f)}+\frac {x \left (4 a^2 d f+a b (d e-7 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {f \left (\frac {\int \frac {2 b \left (f (31 d e-3 c f) a^2+4 b e (d e-10 c f) a+8 b^2 c e^2\right ) x^2+a \left (-3 f (d e+3 c f) a^2-4 b e (8 d e-9 c f) a+8 b^2 c e^2\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (a^2 f (31 d e-3 c f)+4 a b e (d e-10 c f)+8 b^2 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{a (b e-a f)}+\frac {x \left (6 a^2 d f+a b (d e-9 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {d \left (\frac {\frac {f \left (\frac {x \sqrt {a+b x^2} \left (a^2 f (13 d e-3 c f)+2 a b e (d e-8 c f)+4 b^2 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 (a f (c f+d e)+2 b e (2 d e-3 c f)) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}\right )}{a (b e-a f)}+\frac {x \left (4 a^2 d f+a b (d e-7 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {f \left (\frac {\int \frac {2 b \left (f (31 d e-3 c f) a^2+4 b e (d e-10 c f) a+8 b^2 c e^2\right ) x^2+a \left (-3 f (d e+3 c f) a^2-4 b e (8 d e-9 c f) a+8 b^2 c e^2\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (a^2 f (31 d e-3 c f)+4 a b e (d e-10 c f)+8 b^2 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{a (b e-a f)}+\frac {x \left (6 a^2 d f+a b (d e-9 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {d \left (\frac {\frac {f \left (\frac {x \sqrt {a+b x^2} \left (a^2 f (13 d e-3 c f)+2 a b e (d e-8 c f)+4 b^2 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (c f+d e)+2 b e (2 d e-3 c f))}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{a (b e-a f)}+\frac {x \left (4 a^2 d f+a b (d e-7 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {f \left (\frac {\int \frac {2 b \left (f (31 d e-3 c f) a^2+4 b e (d e-10 c f) a+8 b^2 c e^2\right ) x^2+a \left (-3 f (d e+3 c f) a^2-4 b e (8 d e-9 c f) a+8 b^2 c e^2\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (a^2 f (31 d e-3 c f)+4 a b e (d e-10 c f)+8 b^2 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{a (b e-a f)}+\frac {x \left (6 a^2 d f+a b (d e-9 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {d \left (\frac {\frac {f \left (\frac {x \sqrt {a+b x^2} \left (a^2 f (13 d e-3 c f)+2 a b e (d e-8 c f)+4 b^2 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (c f+d e)+2 b e (2 d e-3 c f))}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{a (b e-a f)}+\frac {x \left (4 a^2 d f+a b (d e-7 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {f \left (\frac {\frac {\int -\frac {3 a^2 \left (24 b^2 (d e-2 c f) e^2+4 a b f (3 d e+4 c f) e-a^2 f^2 (d e+3 c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (3 a^3 f^2 (3 c f+d e)+2 a^2 b e f (47 d e-21 c f)+8 a b^2 e^2 (d e-11 c f)+16 b^3 c e^3\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (a^2 f (31 d e-3 c f)+4 a b e (d e-10 c f)+8 b^2 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{a (b e-a f)}+\frac {x \left (6 a^2 d f+a b (d e-9 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d \left (\frac {\frac {f \left (\frac {x \sqrt {a+b x^2} \left (a^2 f (13 d e-3 c f)+2 a b e (d e-8 c f)+4 b^2 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (c f+d e)+2 b e (2 d e-3 c f))}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{a (b e-a f)}+\frac {x \left (4 a^2 d f+a b (d e-7 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {f \left (\frac {\frac {x \sqrt {a+b x^2} \left (3 a^3 f^2 (3 c f+d e)+2 a^2 b e f (47 d e-21 c f)+8 a b^2 e^2 (d e-11 c f)+16 b^3 c e^3\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 \left (-a^2 f^2 (3 c f+d e)+4 a b e f (4 c f+3 d e)+24 b^2 e^2 (d e-2 c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (a^2 f (31 d e-3 c f)+4 a b e (d e-10 c f)+8 b^2 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{a (b e-a f)}+\frac {x \left (6 a^2 d f+a b (d e-9 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {d \left (\frac {\frac {f \left (\frac {x \sqrt {a+b x^2} \left (a^2 f (13 d e-3 c f)+2 a b e (d e-8 c f)+4 b^2 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (c f+d e)+2 b e (2 d e-3 c f))}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{a (b e-a f)}+\frac {x \left (4 a^2 d f+a b (d e-7 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {f \left (\frac {\frac {x \sqrt {a+b x^2} \left (3 a^3 f^2 (3 c f+d e)+2 a^2 b e f (47 d e-21 c f)+8 a b^2 e^2 (d e-11 c f)+16 b^3 c e^3\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 \left (-a^2 f^2 (3 c f+d e)+4 a b e f (4 c f+3 d e)+24 b^2 e^2 (d e-2 c f)\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (a^2 f (31 d e-3 c f)+4 a b e (d e-10 c f)+8 b^2 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{a (b e-a f)}+\frac {x \left (6 a^2 d f+a b (d e-9 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {d \left (\frac {\frac {f \left (\frac {x \sqrt {a+b x^2} \left (a^2 f (13 d e-3 c f)+2 a b e (d e-8 c f)+4 b^2 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (c f+d e)+2 b e (2 d e-3 c f))}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{a (b e-a f)}+\frac {x \left (4 a^2 d f+a b (d e-7 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {x \left (6 a^2 d f+a b (d e-9 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}+\frac {f \left (\frac {x \sqrt {a+b x^2} \left (a^2 f (31 d e-3 c f)+4 a b e (d e-10 c f)+8 b^2 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}+\frac {\frac {x \sqrt {a+b x^2} \left (3 a^3 f^2 (3 c f+d e)+2 a^2 b e f (47 d e-21 c f)+8 a b^2 e^2 (d e-11 c f)+16 b^3 c e^3\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (-a^2 f^2 (3 c f+d e)+4 a b e f (4 c f+3 d e)+24 b^2 e^2 (d e-2 c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\)

Input:

Int[(c + d*x^2)^2/((a + b*x^2)^(5/2)*(e + f*x^2)^3),x]
 

Output:

(d*(((b*c - a*d)*x)/(3*a*(b*e - a*f)*(a + b*x^2)^(3/2)*(e + f*x^2)) + (((2 
*b^2*c*e + 4*a^2*d*f + a*b*(d*e - 7*c*f))*x)/(a*(b*e - a*f)*Sqrt[a + b*x^2 
]*(e + f*x^2)) + (f*(((4*b^2*c*e^2 + 2*a*b*e*(d*e - 8*c*f) + a^2*f*(13*d*e 
 - 3*c*f))*x*Sqrt[a + b*x^2])/(2*e*(b*e - a*f)*(e + f*x^2)) - (3*a^2*(2*b* 
e*(2*d*e - 3*c*f) + a*f*(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))))/(a*(b*e - a*f)))/(3*a*( 
b*e - a*f))))/f - ((d*e - c*f)*(((b*c - a*d)*x)/(3*a*(b*e - a*f)*(a + b*x^ 
2)^(3/2)*(e + f*x^2)^2) + (((2*b^2*c*e + 6*a^2*d*f + a*b*(d*e - 9*c*f))*x) 
/(a*(b*e - a*f)*Sqrt[a + b*x^2]*(e + f*x^2)^2) + (f*(((8*b^2*c*e^2 + 4*a*b 
*e*(d*e - 10*c*f) + a^2*f*(31*d*e - 3*c*f))*x*Sqrt[a + b*x^2])/(4*e*(b*e - 
 a*f)*(e + f*x^2)^2) + (((16*b^3*c*e^3 + 2*a^2*b*e*f*(47*d*e - 21*c*f) + 8 
*a*b^2*e^2*(d*e - 11*c*f) + 3*a^3*f^2*(d*e + 3*c*f))*x*Sqrt[a + b*x^2])/(2 
*e*(b*e - a*f)*(e + f*x^2)) - (3*a^2*(24*b^2*e^2*(d*e - 2*c*f) - a^2*f^2*( 
d*e + 3*c*f) + 4*a*b*e*f*(3*d*e + 4*c*f))*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqr 
t[e]*Sqrt[a + b*x^2])])/(2*e^(3/2)*(b*e - a*f)^(3/2)))/(4*e*(b*e - a*f)))) 
/(a*(b*e - a*f)))/(3*a*(b*e - a*f))))/f
 

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 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]
 

Maple [A] (verified)

Time = 1.18 (sec) , antiderivative size = 603, normalized size of antiderivative = 0.92

method	result	size
pseudoelliptic	\(\frac {-\frac {3 a^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (f^{2} \left (c^{2} f^{2}+\frac {2}{3} c d e f +d^{2} e^{2}\right ) a^{2}-\frac {16 \left (c^{2} f^{2}+\frac {3}{2} c d e f -\frac {3}{2} d^{2} e^{2}\right ) b f e a}{3}+16 \left (c^{2} f^{2}-c d e f +\frac {1}{6} d^{2} e^{2}\right ) b^{2} e^{2}\right ) \left (f \,x^{2}+e \right )^{2} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )}{8}+\frac {5 \left (\left (c f -d e \right ) f^{2} \left (\frac {3 d \,e^{2}}{5}+f \left (x^{2} d +c \right ) e +\frac {3 c \,f^{2} x^{2}}{5}\right ) a^{5}-\frac {16 \left (\frac {3 e^{4} d^{2}}{2}-\frac {3 \left (-\frac {11 x^{2} d}{6}+c \right ) d f \,e^{3}}{2}+f^{2} \left (\frac {13}{8} d^{2} x^{4}-c d \,x^{2}+c^{2}\right ) e^{2}+\frac {c \,f^{3} x^{2} \left (-x^{2} d +c \right ) e}{4}-\frac {3 c^{2} f^{4} x^{4}}{8}\right ) b f \,a^{4}}{5}-\frac {32 b^{2} \left (\frac {d^{2} e^{5}}{4}-\frac {3 \left (-\frac {17 x^{2} d}{18}+c \right ) d f \,e^{4}}{2}-\frac {9 d \,x^{2} f^{2} \left (-\frac {149 x^{2} d}{432}+c \right ) e^{3}}{2}+f^{3} x^{2} \left (\frac {55}{96} d^{2} x^{4}-\frac {43}{16} c d \,x^{2}+c^{2}\right ) e^{2}+\frac {23 c \left (-\frac {2 x^{2} d}{23}+c \right ) x^{4} f^{4} e}{32}-\frac {3 c^{2} f^{5} x^{6}}{32}\right ) a^{3}}{5}-\frac {32 \left (\frac {d^{2} e^{4} x^{2}}{3}+f \left (\frac {11}{12} d^{2} x^{4}-\frac {4}{3} c d \,x^{2}+c^{2}\right ) e^{3}+2 x^{2} f^{2} \left (\frac {25}{96} d^{2} x^{4}-\frac {41}{24} c d \,x^{2}+c^{2}\right ) e^{2}+\frac {3 c \left (-\frac {47 x^{2} d}{36}+c \right ) x^{4} f^{3} e}{2}+\frac {7 c^{2} f^{4} x^{6}}{16}\right ) b^{3} e \,a^{2}}{5}+\frac {8 c \,b^{4} \left (f \,x^{2}+e \right )^{2} e^{2} \left (e \left (\frac {2 x^{2} d}{3}+c \right )-\frac {11 c f \,x^{2}}{3}\right ) a}{5}+\frac {16 b^{5} c^{2} e^{3} x^{2} \left (f \,x^{2}+e \right )^{2}}{15}\right ) \sqrt {\left (a f -b e \right ) e}\, x}{8}}{\left (a f -b e \right )^{4} e^{2} \sqrt {\left (a f -b e \right ) e}\, \left (f \,x^{2}+e \right )^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2}}\)	\(603\)
default	\(\text {Expression too large to display}\)	\(7261\)

Input:

int((d*x^2+c)^2/(b*x^2+a)^(5/2)/(f*x^2+e)^3,x,method=_RETURNVERBOSE)
 

Output:

5/8*(-3/5*a^2*(b*x^2+a)^(3/2)*(f^2*(c^2*f^2+2/3*c*d*e*f+d^2*e^2)*a^2-16/3* 
(c^2*f^2+3/2*c*d*e*f-3/2*d^2*e^2)*b*f*e*a+16*(c^2*f^2-c*d*e*f+1/6*d^2*e^2) 
*b^2*e^2)*(f*x^2+e)^2*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2))+((c* 
f-d*e)*f^2*(3/5*d*e^2+f*(d*x^2+c)*e+3/5*c*f^2*x^2)*a^5-16/5*(3/2*e^4*d^2-3 
/2*(-11/6*x^2*d+c)*d*f*e^3+f^2*(13/8*d^2*x^4-c*d*x^2+c^2)*e^2+1/4*c*f^3*x^ 
2*(-d*x^2+c)*e-3/8*c^2*f^4*x^4)*b*f*a^4-32/5*b^2*(1/4*d^2*e^5-3/2*(-17/18* 
x^2*d+c)*d*f*e^4-9/2*d*x^2*f^2*(-149/432*x^2*d+c)*e^3+f^3*x^2*(55/96*d^2*x 
^4-43/16*c*d*x^2+c^2)*e^2+23/32*c*(-2/23*x^2*d+c)*x^4*f^4*e-3/32*c^2*f^5*x 
^6)*a^3-32/5*(1/3*d^2*e^4*x^2+f*(11/12*d^2*x^4-4/3*c*d*x^2+c^2)*e^3+2*x^2* 
f^2*(25/96*d^2*x^4-41/24*c*d*x^2+c^2)*e^2+3/2*c*(-47/36*x^2*d+c)*x^4*f^3*e 
+7/16*c^2*f^4*x^6)*b^3*e*a^2+8/5*c*b^4*(f*x^2+e)^2*e^2*(e*(2/3*x^2*d+c)-11 
/3*c*f*x^2)*a+16/15*b^5*c^2*e^3*x^2*(f*x^2+e)^2)*((a*f-b*e)*e)^(1/2)*x)/(( 
a*f-b*e)*e)^(1/2)/(b*x^2+a)^(3/2)/(f*x^2+e)^2/e^2/(a*f-b*e)^4/a^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2084 vs. \(2 (618) = 1236\).

Time = 34.31 (sec) , antiderivative size = 4208, normalized size of antiderivative = 6.43 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((d*x^2+c)^2/(b*x^2+a)^(5/2)/(f*x^2+e)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate((d*x**2+c)**2/(b*x**2+a)**(5/2)/(f*x**2+e)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}^{3}} \,d x } \] Input:

integrate((d*x^2+c)^2/(b*x^2+a)^(5/2)/(f*x^2+e)^3,x, algorithm="maxima")
 

Output:

integrate((d*x^2 + c)^2/((b*x^2 + a)^(5/2)*(f*x^2 + e)^3), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2249 vs. \(2 (618) = 1236\).

Time = 0.46 (sec) , antiderivative size = 2249, normalized size of antiderivative = 3.44 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((d*x^2+c)^2/(b*x^2+a)^(5/2)/(f*x^2+e)^3,x, algorithm="giac")
 

Output:

1/3*((2*b^10*c^2*e^5 + 2*a*b^9*c*d*e^5 - 4*a^2*b^8*d^2*e^5 - 19*a*b^9*c^2* 
e^4*f + 8*a^2*b^8*c*d*e^4*f + 11*a^3*b^7*d^2*e^4*f + 56*a^2*b^8*c^2*e^3*f^ 
2 - 52*a^3*b^7*c*d*e^3*f^2 - 4*a^4*b^6*d^2*e^3*f^2 - 74*a^3*b^7*c^2*e^2*f^ 
3 + 88*a^4*b^6*c*d*e^2*f^3 - 14*a^5*b^5*d^2*e^2*f^3 + 46*a^4*b^6*c^2*e*f^4 
 - 62*a^5*b^5*c*d*e*f^4 + 16*a^6*b^4*d^2*e*f^4 - 11*a^5*b^5*c^2*f^5 + 16*a 
^6*b^4*c*d*f^5 - 5*a^7*b^3*d^2*f^5)*x^2/(a^2*b^9*e^8 - 8*a^3*b^8*e^7*f + 2 
8*a^4*b^7*e^6*f^2 - 56*a^5*b^6*e^5*f^3 + 70*a^6*b^5*e^4*f^4 - 56*a^7*b^4*e 
^3*f^5 + 28*a^8*b^3*e^2*f^6 - 8*a^9*b^2*e*f^7 + a^10*b*f^8) + 3*(a*b^9*c^2 
*e^5 - a^3*b^7*d^2*e^5 - 8*a^2*b^8*c^2*e^4*f + 6*a^3*b^7*c*d*e^4*f + 2*a^4 
*b^6*d^2*e^4*f + 22*a^3*b^7*c^2*e^3*f^2 - 24*a^4*b^6*c*d*e^3*f^2 + 2*a^5*b 
^5*d^2*e^3*f^2 - 28*a^4*b^6*c^2*e^2*f^3 + 36*a^5*b^5*c*d*e^2*f^3 - 8*a^6*b 
^4*d^2*e^2*f^3 + 17*a^5*b^5*c^2*e*f^4 - 24*a^6*b^4*c*d*e*f^4 + 7*a^7*b^3*d 
^2*e*f^4 - 4*a^6*b^4*c^2*f^5 + 6*a^7*b^3*c*d*f^5 - 2*a^8*b^2*d^2*f^5)/(a^2 
*b^9*e^8 - 8*a^3*b^8*e^7*f + 28*a^4*b^7*e^6*f^2 - 56*a^5*b^6*e^5*f^3 + 70* 
a^6*b^5*e^4*f^4 - 56*a^7*b^4*e^3*f^5 + 28*a^8*b^3*e^2*f^6 - 8*a^9*b^2*e*f^ 
7 + a^10*b*f^8))*x/(b*x^2 + a)^(3/2) - 1/8*(8*b^(5/2)*d^2*e^4 - 48*b^(5/2) 
*c*d*e^3*f + 24*a*b^(3/2)*d^2*e^3*f + 48*b^(5/2)*c^2*e^2*f^2 - 24*a*b^(3/2 
)*c*d*e^2*f^2 + 3*a^2*sqrt(b)*d^2*e^2*f^2 - 16*a*b^(3/2)*c^2*e*f^3 + 2*a^2 
*sqrt(b)*c*d*e*f^3 + 3*a^2*sqrt(b)*c^2*f^4)*arctan(1/2*((sqrt(b)*x - sqrt( 
b*x^2 + a))^2*f + 2*b*e - a*f)/sqrt(-b^2*e^2 + a*b*e*f))/((b^4*e^6 - 4*...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\int \frac {{\left (d\,x^2+c\right )}^2}{{\left (b\,x^2+a\right )}^{5/2}\,{\left (f\,x^2+e\right )}^3} \,d x \] Input:

int((c + d*x^2)^2/((a + b*x^2)^(5/2)*(e + f*x^2)^3),x)
 

Output:

int((c + d*x^2)^2/((a + b*x^2)^(5/2)*(e + f*x^2)^3), x)
 

Reduce [B] (verification not implemented)

Time = 1.58 (sec) , antiderivative size = 13481, normalized size of antiderivative = 20.61 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((d*x^2+c)^2/(b*x^2+a)^(5/2)/(f*x^2+e)^3,x)
 

Output:

( - 9*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x 
**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**7*c**2*e**2*f**5 - 18*sqrt 
(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqr 
t(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**7*c**2*e*f**6*x**2 - 9*sqrt(e)*sqrt( 
a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt 
(b)*x)/(sqrt(e)*sqrt(b)))*a**7*c**2*f**7*x**4 - 6*sqrt(e)*sqrt(a*f - b*e)* 
atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqr 
t(e)*sqrt(b)))*a**7*c*d*e**3*f**4 - 12*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt( 
a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b 
)))*a**7*c*d*e**2*f**5*x**2 - 6*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b 
*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a** 
7*c*d*e*f**6*x**4 - 9*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt 
(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**7*d**2*e** 
4*f**3 - 18*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a 
 + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**7*d**2*e**3*f**4*x** 
2 - 9*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x 
**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**7*d**2*e**2*f**5*x**4 + 12 
0*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) 
 - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**6*b*c**2*e**3*f**4 + 222*sqrt( 
e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - s...