\(\int \frac {(c+d x^2)^2}{(a+b x^2)^{5/2} (e+f x^2)^2} \, dx\) [368]

Optimal result

Integrand size = 30, antiderivative size = 345 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=\frac {d^2 x}{3 a f^2 \left (a+b x^2\right )^{3/2}}+\frac {b (d e-c f) (a f (7 d e-3 c f)-2 b e (d e+c f)) x}{6 a e f^2 (b e-a f)^2 \left (a+b x^2\right )^{3/2}}+\frac {2 d^2 x}{3 a^2 f^2 \sqrt {a+b x^2}}-\frac {b (d e-c f) \left (a^2 f^2 (23 d e-3 c f)+4 b^2 e^2 (d e+c f)-4 a b e f (3 d e+4 c f)\right ) x}{6 a^2 e f^2 (b e-a f)^3 \sqrt {a+b x^2}}-\frac {(d e-c f)^2 x}{2 e f (b e-a f) \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}+\frac {(d e-c f) (2 b e (d e-3 c f)+a f (3 d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{2 e^{3/2} (b e-a f)^{7/2}} \] Output:

1/3*d^2*x/a/f^2/(b*x^2+a)^(3/2)+1/6*b*(-c*f+d*e)*(a*f*(-3*c*f+7*d*e)-2*b*e 
*(c*f+d*e))*x/a/e/f^2/(-a*f+b*e)^2/(b*x^2+a)^(3/2)+2/3*d^2*x/a^2/f^2/(b*x^ 
2+a)^(1/2)-1/6*b*(-c*f+d*e)*(a^2*f^2*(-3*c*f+23*d*e)+4*b^2*e^2*(c*f+d*e)-4 
*a*b*e*f*(4*c*f+3*d*e))*x/a^2/e/f^2/(-a*f+b*e)^3/(b*x^2+a)^(1/2)-1/2*(-c*f 
+d*e)^2*x/e/f/(-a*f+b*e)/(b*x^2+a)^(3/2)/(f*x^2+e)+1/2*(-c*f+d*e)*(2*b*e*( 
-3*c*f+d*e)+a*f*(c*f+3*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)^(7/2)
 

Mathematica [A] (verified)

Time = 2.11 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.06 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=\frac {x \left (-4 b^4 c^2 e^2 x^2 \left (e+f x^2\right )-2 a b^3 c e \left (e+f x^2\right ) \left (3 c e+2 d e x^2-8 c f x^2\right )+3 a^4 f \left (-2 c d e f+c^2 f^2+d^2 e \left (3 e+2 f x^2\right )\right )+a^2 b^2 \left (d^2 e^2 x^2 \left (8 e+11 f x^2\right )-2 c d e f x^2 \left (10 e+13 f x^2\right )+3 c^2 f \left (6 e^2+6 e f x^2+f^2 x^4\right )\right )+2 a^3 b \left (3 c^2 f^3 x^2-6 c d e f \left (2 e+3 f x^2\right )+d^2 e \left (3 e^2+8 e f x^2+2 f^2 x^4\right )\right )\right )}{6 a^2 e (-b e+a f)^3 \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}+\frac {(d e-c f) (2 b e (d e-3 c f)+a f (3 d e+c f)) \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{2 e^{3/2} (-b e+a f)^{7/2}} \] Input:

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

Output:

(x*(-4*b^4*c^2*e^2*x^2*(e + f*x^2) - 2*a*b^3*c*e*(e + f*x^2)*(3*c*e + 2*d* 
e*x^2 - 8*c*f*x^2) + 3*a^4*f*(-2*c*d*e*f + c^2*f^2 + d^2*e*(3*e + 2*f*x^2) 
) + a^2*b^2*(d^2*e^2*x^2*(8*e + 11*f*x^2) - 2*c*d*e*f*x^2*(10*e + 13*f*x^2 
) + 3*c^2*f*(6*e^2 + 6*e*f*x^2 + f^2*x^4)) + 2*a^3*b*(3*c^2*f^3*x^2 - 6*c* 
d*e*f*(2*e + 3*f*x^2) + d^2*e*(3*e^2 + 8*e*f*x^2 + 2*f^2*x^4))))/(6*a^2*e* 
(-(b*e) + a*f)^3*(a + b*x^2)^(3/2)*(e + f*x^2)) + ((d*e - c*f)*(2*b*e*(d*e 
 - 3*c*f) + a*f*(3*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])])/(2*e^(3/2)*(-(b*e) + a*f)^(7/2))
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.40, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {425, 402, 25, 402, 25, 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.

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {d \left (\frac {\frac {x \left (2 a^2 d f+a b (d e-5 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} (b e-a f)}-\frac {3 a f (d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} (b e-a f)^{3/2}}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {f \left (\frac {\int -\frac {3 a^2 (2 b e (2 d e-3 c f)+a f (d e+c f))}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (a^2 f (13 d e-3 c f)+2 a b e (d e-8 c f)+4 b^2 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{a (b e-a f)}+\frac {x \left (4 a^2 d f+a b (d e-7 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d \left (\frac {\frac {x \left (2 a^2 d f+a b (d e-5 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} (b e-a f)}-\frac {3 a f (d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} (b e-a f)^{3/2}}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {f \left (\frac {x \sqrt {a+b x^2} \left (a^2 f (13 d e-3 c f)+2 a b e (d e-8 c f)+4 b^2 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 (a f (c f+d e)+2 b e (2 d e-3 c f)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}\right )}{a (b e-a f)}+\frac {x \left (4 a^2 d f+a b (d e-7 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {d \left (\frac {\frac {x \left (2 a^2 d f+a b (d e-5 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} (b e-a f)}-\frac {3 a f (d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} (b e-a f)^{3/2}}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {f \left (\frac {x \sqrt {a+b x^2} \left (a^2 f (13 d e-3 c f)+2 a b e (d e-8 c f)+4 b^2 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 (a f (c f+d e)+2 b e (2 d e-3 c f)) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}\right )}{a (b e-a f)}+\frac {x \left (4 a^2 d f+a b (d e-7 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {d \left (\frac {\frac {x \left (2 a^2 d f+a b (d e-5 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} (b e-a f)}-\frac {3 a f (d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} (b e-a f)^{3/2}}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {f \left (\frac {x \sqrt {a+b x^2} \left (a^2 f (13 d e-3 c f)+2 a b e (d e-8 c f)+4 b^2 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (c f+d e)+2 b e (2 d e-3 c f))}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{a (b e-a f)}+\frac {x \left (4 a^2 d f+a b (d e-7 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{f}\)

Input:

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

Output:

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

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 = 1.15 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.06

method	result	size
pseudoelliptic	\(\frac {-a^{2} \left (\left (c \,f^{2}+3 d e f \right ) a -6 b c e f +2 b d \,e^{2}\right ) \left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (c f -d e \right ) \left (f \,x^{2}+e \right ) \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\sqrt {\left (a f -b e \right ) e}\, x \left (\left (3 d^{2} e^{2}-2 d f \left (-x^{2} d +c \right ) e +c^{2} f^{2}\right ) f \,a^{4}+2 \left (d^{2} e^{3}-4 d \left (-\frac {2 x^{2} d}{3}+c \right ) f \,e^{2}-6 d \,x^{2} f^{2} \left (-\frac {x^{2} d}{9}+c \right ) e +f^{3} x^{2} c^{2}\right ) b \,a^{3}+6 \left (\frac {4 d^{2} e^{3} x^{2}}{9}+f \left (\frac {11}{18} d^{2} x^{4}-\frac {10}{9} c d \,x^{2}+c^{2}\right ) e^{2}+c \,f^{2} x^{2} \left (-\frac {13 x^{2} d}{9}+c \right ) e +\frac {c^{2} f^{3} x^{4}}{6}\right ) b^{2} a^{2}-2 c \left (e \left (\frac {2 x^{2} d}{3}+c \right )-\frac {8 c f \,x^{2}}{3}\right ) b^{3} \left (f \,x^{2}+e \right ) e a -\frac {4 b^{4} c^{2} e^{2} x^{2} \left (f \,x^{2}+e \right )}{3}\right )}{2 \sqrt {\left (a f -b e \right ) e}\, \left (b \,x^{2}+a \right )^{\frac {3}{2}} e \left (f \,x^{2}+e \right ) \left (a f -b e \right )^{3} a^{2}}\)	\(365\)
default	\(\text {Expression too large to display}\)	\(3579\)

Input:

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

Output:

1/2*(-a^2*((c*f^2+3*d*e*f)*a-6*b*c*e*f+2*b*d*e^2)*(b*x^2+a)^(3/2)*(c*f-d*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)*x*((3*d^2*e^2-2*d*f*(-d*x^2+c)*e+c^2*f^2)*f*a^4+2*(d^2*e^3-4*d*(-2/3 
*x^2*d+c)*f*e^2-6*d*x^2*f^2*(-1/9*x^2*d+c)*e+f^3*x^2*c^2)*b*a^3+6*(4/9*d^2 
*e^3*x^2+f*(11/18*d^2*x^4-10/9*c*d*x^2+c^2)*e^2+c*f^2*x^2*(-13/9*x^2*d+c)* 
e+1/6*c^2*f^3*x^4)*b^2*a^2-2*c*(e*(2/3*x^2*d+c)-8/3*c*f*x^2)*b^3*(f*x^2+e) 
*e*a-4/3*b^4*c^2*e^2*x^2*(f*x^2+e)))/((a*f-b*e)*e)^(1/2)/(b*x^2+a)^(3/2)/e 
/(f*x^2+e)/(a*f-b*e)^3/a^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1226 vs. \(2 (314) = 628\).

Time = 7.71 (sec) , antiderivative size = 2492, normalized size of antiderivative = 7.22 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/24*(3*(2*a^4*b*d^2*e^4 - a^5*c^2*e*f^3 + (2*a^2*b^3*d^2*e^3*f - a^3*b^2 
*c^2*f^4 - (8*a^2*b^3*c*d - 3*a^3*b^2*d^2)*e^2*f^2 + 2*(3*a^2*b^3*c^2 - a^ 
3*b^2*c*d)*e*f^3)*x^6 - (8*a^4*b*c*d - 3*a^5*d^2)*e^3*f + 2*(3*a^4*b*c^2 - 
 a^5*c*d)*e^2*f^2 + (2*a^2*b^3*d^2*e^4 - 2*a^4*b*c^2*f^4 - (8*a^2*b^3*c*d 
- 7*a^3*b^2*d^2)*e^3*f + 6*(a^2*b^3*c^2 - 3*a^3*b^2*c*d + a^4*b*d^2)*e^2*f 
^2 + (11*a^3*b^2*c^2 - 4*a^4*b*c*d)*e*f^3)*x^4 + (4*a^3*b^2*d^2*e^4 - a^5* 
c^2*f^4 - 8*(2*a^3*b^2*c*d - a^4*b*d^2)*e^3*f + 3*(4*a^3*b^2*c^2 - 4*a^4*b 
*c*d + a^5*d^2)*e^2*f^2 + 2*(2*a^4*b*c^2 - a^5*c*d)*e*f^3)*x^2)*sqrt(b*e^2 
 - a*e*f)*log(((8*b^2*e^2 - 8*a*b*e*f + a^2*f^2)*x^4 + a^2*e^2 + 2*(4*a*b* 
e^2 - 3*a^2*e*f)*x^2 + 4*((2*b*e - a*f)*x^3 + a*e*x)*sqrt(b*e^2 - a*e*f)*s 
qrt(b*x^2 + a))/(f^2*x^4 + 2*e*f*x^2 + e^2)) + 4*((3*a^3*b^2*c^2*e*f^4 + ( 
4*b^5*c^2 + 4*a*b^4*c*d - 11*a^2*b^3*d^2)*e^4*f - (20*a*b^4*c^2 - 22*a^2*b 
^3*c*d - 7*a^3*b^2*d^2)*e^3*f^2 + (13*a^2*b^3*c^2 - 26*a^3*b^2*c*d + 4*a^4 
*b*d^2)*e^2*f^3)*x^5 + 2*(3*a^4*b*c^2*e*f^4 + 2*(b^5*c^2 + a*b^4*c*d - 2*a 
^2*b^3*d^2)*e^5 - (7*a*b^4*c^2 - 8*a^2*b^3*c*d + 4*a^3*b^2*d^2)*e^4*f - (4 
*a^2*b^3*c^2 - 8*a^3*b^2*c*d - 5*a^4*b*d^2)*e^3*f^2 + 3*(2*a^3*b^2*c^2 - 6 
*a^4*b*c*d + a^5*d^2)*e^2*f^3)*x^3 + 3*(a^5*c^2*e*f^4 + 2*(a*b^4*c^2 - a^3 
*b^2*d^2)*e^5 - (8*a^2*b^3*c^2 - 8*a^3*b^2*c*d + a^4*b*d^2)*e^4*f + 3*(2*a 
^3*b^2*c^2 - 2*a^4*b*c*d + a^5*d^2)*e^3*f^2 - (a^4*b*c^2 + 2*a^5*c*d)*e^2* 
f^3)*x)*sqrt(b*x^2 + a))/(a^4*b^4*e^7 - 4*a^5*b^3*e^6*f + 6*a^6*b^2*e^5...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1116 vs. \(2 (314) = 628\).

Time = 0.44 (sec) , antiderivative size = 1116, normalized size of antiderivative = 3.23 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

1/3*(2*(b^8*c^2*e^4 + a*b^7*c*d*e^4 - 2*a^2*b^6*d^2*e^4 - 7*a*b^7*c^2*e^3* 
f + 2*a^2*b^6*c*d*e^3*f + 5*a^3*b^5*d^2*e^3*f + 15*a^2*b^6*c^2*e^2*f^2 - 1 
2*a^3*b^5*c*d*e^2*f^2 - 3*a^4*b^4*d^2*e^2*f^2 - 13*a^3*b^5*c^2*e*f^3 + 14* 
a^4*b^4*c*d*e*f^3 - a^5*b^3*d^2*e*f^3 + 4*a^4*b^4*c^2*f^4 - 5*a^5*b^3*c*d* 
f^4 + a^6*b^2*d^2*f^4)*x^2/(a^2*b^7*e^6 - 6*a^3*b^6*e^5*f + 15*a^4*b^5*e^4 
*f^2 - 20*a^5*b^4*e^3*f^3 + 15*a^6*b^3*e^2*f^4 - 6*a^7*b^2*e*f^5 + a^8*b*f 
^6) + 3*(a*b^7*c^2*e^4 - a^3*b^5*d^2*e^4 - 6*a^2*b^6*c^2*e^3*f + 4*a^3*b^5 
*c*d*e^3*f + 2*a^4*b^4*d^2*e^3*f + 12*a^3*b^5*c^2*e^2*f^2 - 12*a^4*b^4*c*d 
*e^2*f^2 - 10*a^4*b^4*c^2*e*f^3 + 12*a^5*b^3*c*d*e*f^3 - 2*a^6*b^2*d^2*e*f 
^3 + 3*a^5*b^3*c^2*f^4 - 4*a^6*b^2*c*d*f^4 + a^7*b*d^2*f^4)/(a^2*b^7*e^6 - 
 6*a^3*b^6*e^5*f + 15*a^4*b^5*e^4*f^2 - 20*a^5*b^4*e^3*f^3 + 15*a^6*b^3*e^ 
2*f^4 - 6*a^7*b^2*e*f^5 + a^8*b*f^6))*x/(b*x^2 + a)^(3/2) - 1/2*(2*b^(3/2) 
*d^2*e^3 - 8*b^(3/2)*c*d*e^2*f + 3*a*sqrt(b)*d^2*e^2*f + 6*b^(3/2)*c^2*e*f 
^2 - 2*a*sqrt(b)*c*d*e*f^2 - a*sqrt(b)*c^2*f^3)*arctan(1/2*((sqrt(b)*x - s 
qrt(b*x^2 + a))^2*f + 2*b*e - a*f)/sqrt(-b^2*e^2 + a*b*e*f))/((b^3*e^4 - 3 
*a*b^2*e^3*f + 3*a^2*b*e^2*f^2 - a^3*e*f^3)*sqrt(-b^2*e^2 + a*b*e*f)) - (2 
*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^(3/2)*d^2*e^3 - 4*(sqrt(b)*x - sqrt(b*x 
^2 + a))^2*b^(3/2)*c*d*e^2*f - (sqrt(b)*x - sqrt(b*x^2 + a))^2*a*sqrt(b)*d 
^2*e^2*f + 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^(3/2)*c^2*e*f^2 + 2*(sqrt(b 
)*x - sqrt(b*x^2 + a))^2*a*sqrt(b)*c*d*e*f^2 - (sqrt(b)*x - sqrt(b*x^2 ...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.67 (sec) , antiderivative size = 8023, normalized size of antiderivative = 23.26 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 3*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*f**4 - 3*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*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**6*b*c*d*e**2*f**3 - 6*sqrt(e)*sqrt(a*f - b*e)*a 
tan((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*d*e*f**4*x**2 + 9*sqrt(e)*sqrt(a*f - b*e)*atan((sqr 
t(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt 
(b)))*a**6*b*d**2*e**3*f**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** 
6*b*d**2*e**2*f**3*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**5*b* 
*2*c**2*e**2*f**3 - 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**5*b**2*c** 
2*f**5*x**4 - 48*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*s 
qrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**5*b**2*c*d*e**3 
*f**2 - 60*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**5*b**2*c*d*e**2*f**3* 
x**2 - 12*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(...