\(\int \frac {(c+d x^2)^3}{(a+b x^2)^{5/2} (e+f x^2)^3} \, dx\) [375]

Optimal result

Integrand size = 30, antiderivative size = 615 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\frac {d^3 x}{3 a f^3 \left (a+b x^2\right )^{3/2}}-\frac {b (d e-c f) \left (3 a^2 f^2 \left (17 d^2 e^2-6 c d e f-3 c^2 f^2\right )-12 a b e f \left (2 d^2 e^2+5 c d e f-3 c^2 f^2\right )+8 b^2 e^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) x}{24 a e^2 f^3 (b e-a f)^3 \left (a+b x^2\right )^{3/2}}+\frac {2 d^3 x}{3 a^2 f^3 \sqrt {a+b x^2}}-\frac {b (d e-c f) \left (6 a^2 b e f^2 \left (15 d^2 e^2+40 c d e f-7 c^2 f^2\right )-3 a^3 f^3 \left (49 d^2 e^2-6 c d e f-3 c^2 f^2\right )+16 b^3 e^3 \left (d^2 e^2+c d e f+c^2 f^2\right )-8 a b^2 e^2 f \left (8 d^2 e^2+8 c d e f+11 c^2 f^2\right )\right ) x}{24 a^2 e^2 f^3 (b e-a f)^4 \sqrt {a+b x^2}}+\frac {(d e-c f)^3 x}{4 e f^2 (b e-a f) \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}+\frac {(d e-c f)^2 (3 a f (3 d e+c f)-2 b e (d e+5 c f)) x}{8 e^2 f^2 (b e-a f)^2 \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}+\frac {(d e-c f) \left (24 b^2 c e^2 (d e-2 c f)-4 a b e \left (5 d^2 e^2-13 c d e f-4 c^2 f^2\right )-3 a^2 f \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{8 e^{5/2} (b e-a f)^{9/2}} \] Output:

1/3*d^3*x/a/f^3/(b*x^2+a)^(3/2)-1/24*b*(-c*f+d*e)*(3*a^2*f^2*(-3*c^2*f^2-6 
*c*d*e*f+17*d^2*e^2)-12*a*b*e*f*(-3*c^2*f^2+5*c*d*e*f+2*d^2*e^2)+8*b^2*e^2 
*(c^2*f^2+c*d*e*f+d^2*e^2))*x/a/e^2/f^3/(-a*f+b*e)^3/(b*x^2+a)^(3/2)+2/3*d 
^3*x/a^2/f^3/(b*x^2+a)^(1/2)-1/24*b*(-c*f+d*e)*(6*a^2*b*e*f^2*(-7*c^2*f^2+ 
40*c*d*e*f+15*d^2*e^2)-3*a^3*f^3*(-3*c^2*f^2-6*c*d*e*f+49*d^2*e^2)+16*b^3* 
e^3*(c^2*f^2+c*d*e*f+d^2*e^2)-8*a*b^2*e^2*f*(11*c^2*f^2+8*c*d*e*f+8*d^2*e^ 
2))*x/a^2/e^2/f^3/(-a*f+b*e)^4/(b*x^2+a)^(1/2)+1/4*(-c*f+d*e)^3*x/e/f^2/(- 
a*f+b*e)/(b*x^2+a)^(3/2)/(f*x^2+e)^2+1/8*(-c*f+d*e)^2*(3*a*f*(c*f+3*d*e)-2 
*b*e*(5*c*f+d*e))*x/e^2/f^2/(-a*f+b*e)^2/(b*x^2+a)^(3/2)/(f*x^2+e)+1/8*(-c 
*f+d*e)*(24*b^2*c*e^2*(-2*c*f+d*e)-4*a*b*e*(-4*c^2*f^2-13*c*d*e*f+5*d^2*e^ 
2)-3*a^2*f*(c^2*f^2+2*c*d*e*f+5*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 = 16.34 (sec) , antiderivative size = 340, normalized size of antiderivative = 0.55 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\frac {1}{24} \left (x \sqrt {a+b x^2} \left (\frac {8 (-b c+a d)^3}{a (-b e+a f)^3 \left (a+b x^2\right )^2}+\frac {8 (b c-a d)^2 \left (2 b^2 c e+2 a^2 d f+a b (7 d e-11 c f)\right )}{a^2 (b e-a f)^4 \left (a+b x^2\right )}+\frac {6 (d e-c f)^3}{e (b e-a f)^3 \left (e+f x^2\right )^2}+\frac {3 (d e-c f)^2 (2 b e (d e-7 c f)+3 a f (3 d e+c f))}{e^2 (b e-a f)^4 \left (e+f x^2\right )}\right )-\frac {3 (d e-c f) \left (24 b^2 c e^2 (-d e+2 c f)+4 a b e \left (5 d^2 e^2-13 c d e f-4 c^2 f^2\right )+3 a^2 f \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {-b e+a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{e^{5/2} (-b e+a f)^{9/2}}\right ) \] Input:

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

Output:

(x*Sqrt[a + b*x^2]*((8*(-(b*c) + a*d)^3)/(a*(-(b*e) + a*f)^3*(a + b*x^2)^2 
) + (8*(b*c - a*d)^2*(2*b^2*c*e + 2*a^2*d*f + a*b*(7*d*e - 11*c*f)))/(a^2* 
(b*e - a*f)^4*(a + b*x^2)) + (6*(d*e - c*f)^3)/(e*(b*e - a*f)^3*(e + f*x^2 
)^2) + (3*(d*e - c*f)^2*(2*b*e*(d*e - 7*c*f) + 3*a*f*(3*d*e + c*f)))/(e^2* 
(b*e - a*f)^4*(e + f*x^2))) - (3*(d*e - c*f)*(24*b^2*c*e^2*(-(d*e) + 2*c*f 
) + 4*a*b*e*(5*d^2*e^2 - 13*c*d*e*f - 4*c^2*f^2) + 3*a^2*f*(5*d^2*e^2 + 2* 
c*d*e*f + c^2*f^2))*ArcTan[(Sqrt[-(b*e) + a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2] 
)])/(e^(5/2)*(-(b*e) + a*f)^(9/2)))/24
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1253\) vs. \(2(615)=1230\).

Time = 1.61 (sec) , antiderivative size = 1253, normalized size of antiderivative = 2.04, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.567, Rules used = {425, 425, 402, 25, 402, 25, 27, 291, 221, 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.

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {d \left (\frac {d \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2}}+\frac {\frac {\left (2 d f a^2+b (d e-5 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a}}-\frac {3 a f (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{b e-a f}}{3 a (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}+\frac {\frac {\left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {f \int \frac {2 \left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x^2+a (2 b c e-5 a d e+3 a c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}+\frac {\frac {\left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {f \int \frac {2 \left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x^2+a (2 b c e-5 a d e+3 a c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {\left (6 d f a^2+b (d e-9 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \int \frac {4 \left (6 d f a^2+b (d e-9 c f) a+2 b^2 c e\right ) x^2+a (4 b c e-7 a d e+3 a c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {d \left (\frac {d \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2}}+\frac {\frac {\left (2 d f a^2+b (d e-5 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a}}-\frac {3 a f (d e-c f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{b e-a f}}{3 a (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}+\frac {\frac {\left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {f \int \frac {2 \left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x^2+a (2 b c e-5 a d e+3 a c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}+\frac {\frac {\left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {f \int \frac {2 \left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x^2+a (2 b c e-5 a d e+3 a c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {\left (6 d f a^2+b (d e-9 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \int \frac {4 \left (6 d f a^2+b (d e-9 c f) a+2 b^2 c e\right ) x^2+a (4 b c e-7 a d e+3 a c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {d \left (\frac {d \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2}}+\frac {\frac {\left (2 d f a^2+b (d e-5 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a}}-\frac {3 a f (d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} (b e-a f)^{3/2}}}{3 a (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}+\frac {\frac {\left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {f \left (\frac {\left (f (13 d e-3 c f) a^2+2 b e (d e-8 c f) a+4 b^2 c e^2\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\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)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}+\frac {\frac {\left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {f \left (\frac {\left (f (13 d e-3 c f) a^2+2 b e (d e-8 c f) a+4 b^2 c e^2\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\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)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {\left (6 d f a^2+b (d e-9 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \left (\frac {\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 ) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\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)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d \left (\frac {d \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2}}+\frac {\frac {\left (2 d f a^2+b (d e-5 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a}}-\frac {3 a f (d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} (b e-a f)^{3/2}}}{3 a (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}+\frac {\frac {\left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {f \left (\frac {\left (f (13 d e-3 c f) a^2+2 b e (d e-8 c f) a+4 b^2 c e^2\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 (2 b e (2 d e-3 c f)+a f (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)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}+\frac {\frac {\left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {f \left (\frac {\left (f (13 d e-3 c f) a^2+2 b e (d e-8 c f) a+4 b^2 c e^2\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 (2 b e (2 d e-3 c f)+a f (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)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {\left (6 d f a^2+b (d e-9 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \left (\frac {\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 ) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\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)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {d \left (\frac {d \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2}}+\frac {\frac {\left (2 d f a^2+b (d e-5 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a}}-\frac {3 a f (d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} (b e-a f)^{3/2}}}{3 a (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}+\frac {\frac {\left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {f \left (\frac {\left (f (13 d e-3 c f) a^2+2 b e (d e-8 c f) a+4 b^2 c e^2\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 (2 b e (2 d e-3 c f)+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}}}{2 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}+\frac {\frac {\left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {f \left (\frac {\left (f (13 d e-3 c f) a^2+2 b e (d e-8 c f) a+4 b^2 c e^2\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 (2 b e (2 d e-3 c f)+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}}}{2 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {\left (6 d f a^2+b (d e-9 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \left (\frac {\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 ) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\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)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {d \left (\frac {d \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2}}+\frac {\frac {\left (2 d f a^2+b (d e-5 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a}}-\frac {3 a f (d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} (b e-a f)^{3/2}}}{3 a (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}+\frac {\frac {\left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {f \left (\frac {\left (f (13 d e-3 c f) a^2+2 b e (d e-8 c f) a+4 b^2 c e^2\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 (2 b e (2 d e-3 c f)+a f (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}}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}+\frac {\frac {\left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {f \left (\frac {\left (f (13 d e-3 c f) a^2+2 b e (d e-8 c f) a+4 b^2 c e^2\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 (2 b e (2 d e-3 c f)+a f (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}}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {\left (6 d f a^2+b (d e-9 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \left (\frac {\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 ) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\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)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {d \left (\frac {d \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2}}+\frac {\frac {\left (2 d f a^2+b (d e-5 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a}}-\frac {3 a f (d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} (b e-a f)^{3/2}}}{3 a (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}+\frac {\frac {\left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {f \left (\frac {\left (f (13 d e-3 c f) a^2+2 b e (d e-8 c f) a+4 b^2 c e^2\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 (2 b e (2 d e-3 c f)+a f (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}}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}+\frac {\frac {\left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {f \left (\frac {\left (f (13 d e-3 c f) a^2+2 b e (d e-8 c f) a+4 b^2 c e^2\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 (2 b e (2 d e-3 c f)+a f (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}}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {\left (6 d f a^2+b (d e-9 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \left (\frac {\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 ) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (3 f^2 (d e+3 c f) a^3+2 b e f (47 d e-21 c f) a^2+8 b^2 e^2 (d e-11 c f) a+16 b^3 c e^3\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\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)}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d \left (\frac {d \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2}}+\frac {\frac {\left (2 d f a^2+b (d e-5 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a}}-\frac {3 a f (d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} (b e-a f)^{3/2}}}{3 a (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}+\frac {\frac {\left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {f \left (\frac {\left (f (13 d e-3 c f) a^2+2 b e (d e-8 c f) a+4 b^2 c e^2\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 (2 b e (2 d e-3 c f)+a f (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}}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}+\frac {\frac {\left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {f \left (\frac {\left (f (13 d e-3 c f) a^2+2 b e (d e-8 c f) a+4 b^2 c e^2\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 (2 b e (2 d e-3 c f)+a f (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}}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {\left (6 d f a^2+b (d e-9 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \left (\frac {\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 ) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (3 f^2 (d e+3 c f) a^3+2 b e f (47 d e-21 c f) a^2+8 b^2 e^2 (d e-11 c f) a+16 b^3 c e^3\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\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 ) \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)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {d \left (\frac {d \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2}}+\frac {\frac {\left (2 d f a^2+b (d e-5 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a}}-\frac {3 a f (d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} (b e-a f)^{3/2}}}{3 a (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}+\frac {\frac {\left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {f \left (\frac {\left (f (13 d e-3 c f) a^2+2 b e (d e-8 c f) a+4 b^2 c e^2\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 (2 b e (2 d e-3 c f)+a f (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}}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}+\frac {\frac {\left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {f \left (\frac {\left (f (13 d e-3 c f) a^2+2 b e (d e-8 c f) a+4 b^2 c e^2\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 (2 b e (2 d e-3 c f)+a f (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}}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {\left (6 d f a^2+b (d e-9 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \left (\frac {\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 ) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (3 f^2 (d e+3 c f) a^3+2 b e f (47 d e-21 c f) a^2+8 b^2 e^2 (d e-11 c f) a+16 b^3 c e^3\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\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 ) \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)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {d \left (\frac {d \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2}}+\frac {\frac {\left (2 d f a^2+b (d e-5 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a}}-\frac {3 a f (d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} (b e-a f)^{3/2}}}{3 a (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}+\frac {\frac {\left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {f \left (\frac {\left (f (13 d e-3 c f) a^2+2 b e (d e-8 c f) a+4 b^2 c e^2\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 (2 b e (2 d e-3 c f)+a f (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}}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}+\frac {\frac {\left (4 d f a^2+b (d e-7 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {f \left (\frac {\left (f (13 d e-3 c f) a^2+2 b e (d e-8 c f) a+4 b^2 c e^2\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 (2 b e (2 d e-3 c f)+a f (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}}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {\left (6 d f a^2+b (d e-9 c f) a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \left (\frac {\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 ) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (3 f^2 (d e+3 c f) a^3+2 b e f (47 d e-21 c f) a^2+8 b^2 e^2 (d e-11 c f) a+16 b^3 c e^3\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\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 ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{f}\right )}{f}\)

Input:

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

Output:

(d*((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])])/(S 
qrt[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))/f - ((d*e - c*f)*((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*Sq...
 

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 = 1.30 (sec) , antiderivative size = 746, normalized size of antiderivative = 1.21

method	result	size
pseudoelliptic	\(\frac {-\frac {3 a^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (c f -d e \right ) \left (f \,x^{2}+e \right )^{2} \left (a^{2} f \left (c^{2} f^{2}+2 c d e f +5 d^{2} e^{2}\right )-\frac {16 \left (c^{2} f^{2}+\frac {13}{4} c d e f -\frac {5}{4} d^{2} e^{2}\right ) b e a}{3}+16 b^{2} c^{2} e^{2} f -8 b^{2} d \,e^{3} c \right ) \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )}{8}+\frac {5 \left (\left (3 e^{4} d^{3}-\frac {9 d^{2} \left (-\frac {25 x^{2} d}{9}+c \right ) f \,e^{3}}{5}-\frac {3 d \left (-\frac {8}{3} d^{2} x^{4}+5 c d \,x^{2}+c^{2}\right ) f^{2} e^{2}}{5}+c^{2} f^{3} \left (\frac {3 x^{2} d}{5}+c \right ) e +\frac {3 c^{3} f^{4} x^{2}}{5}\right ) f \,a^{5}-\frac {16 \left (-\frac {5 d^{3} e^{5}}{4}+\frac {9 d^{2} \left (-\frac {20 x^{2} d}{27}+c \right ) f \,e^{4}}{2}-\frac {9 d \left (\frac {67}{54} d^{2} x^{4}-\frac {11}{3} c d \,x^{2}+c^{2}\right ) f^{2} e^{3}}{4}+f^{3} \left (-\frac {1}{3} d^{3} x^{6}+\frac {39}{8} c \,d^{2} x^{4}-\frac {3}{2} c^{2} d \,x^{2}+c^{3}\right ) e^{2}+\frac {c^{2} \left (-\frac {3 x^{2} d}{2}+c \right ) x^{2} f^{4} e}{4}-\frac {3 c^{3} f^{5} x^{4}}{8}\right ) b \,a^{4}}{5}-\frac {32 b^{2} \left (\left (-\frac {5}{6} x^{2} d^{3}+\frac {3}{4} c \,d^{2}\right ) e^{5}-\frac {9 d \left (\frac {145}{216} d^{2} x^{4}-\frac {17}{9} c d \,x^{2}+c^{2}\right ) f \,e^{4}}{4}-\frac {27 d \left (\frac {83}{648} d^{2} x^{4}-\frac {149}{216} c d \,x^{2}+c^{2}\right ) x^{2} f^{2} e^{3}}{4}+f^{3} x^{2} c \left (\frac {55}{32} d^{2} x^{4}-\frac {129}{32} c d \,x^{2}+c^{2}\right ) e^{2}+\frac {23 c^{2} \left (-\frac {3 x^{2} d}{23}+c \right ) x^{4} f^{4} e}{32}-\frac {3 c^{3} f^{5} x^{6}}{32}\right ) a^{3}}{5}-\frac {32 b^{3} e \left (d^{2} x^{2} \left (-\frac {x^{2} d}{8}+c \right ) e^{4}+f \left (-\frac {1}{16} d^{3} x^{6}+\frac {11}{4} c \,d^{2} x^{4}-2 c^{2} d \,x^{2}+c^{3}\right ) e^{3}+2 c \left (\frac {25}{32} d^{2} x^{4}-\frac {41}{16} c d \,x^{2}+c^{2}\right ) x^{2} f^{2} e^{2}+\frac {3 c^{2} \left (-\frac {47 x^{2} d}{24}+c \right ) x^{4} f^{3} e}{2}+\frac {7 c^{3} f^{4} x^{6}}{16}\right ) a^{2}}{5}+\frac {8 c^{2} \left (e \left (x^{2} d +c \right )-\frac {11 c f \,x^{2}}{3}\right ) b^{4} \left (f \,x^{2}+e \right )^{2} e^{2} a}{5}+\frac {16 b^{5} c^{3} e^{3} x^{2} \left (f \,x^{2}+e \right )^{2}}{15}\right ) \sqrt {\left (a f -b e \right ) e}\, x}{8}}{\left (f \,x^{2}+e \right )^{2} e^{2} \left (a f -b e \right )^{4} \sqrt {\left (a f -b e \right ) e}\, \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2}}\)	\(746\)
default	\(\text {Expression too large to display}\)	\(7377\)

Input:

int((d*x^2+c)^3/(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)*(c*f-d*e)*(f*x^2+e)^2*(a^2*f*(c^2*f^2+2*c*d* 
e*f+5*d^2*e^2)-16/3*(c^2*f^2+13/4*c*d*e*f-5/4*d^2*e^2)*b*e*a+16*b^2*c^2*e^ 
2*f-8*b^2*d*e^3*c)*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2))+((3*e^4 
*d^3-9/5*d^2*(-25/9*x^2*d+c)*f*e^3-3/5*d*(-8/3*d^2*x^4+5*c*d*x^2+c^2)*f^2* 
e^2+c^2*f^3*(3/5*x^2*d+c)*e+3/5*c^3*f^4*x^2)*f*a^5-16/5*(-5/4*d^3*e^5+9/2* 
d^2*(-20/27*x^2*d+c)*f*e^4-9/4*d*(67/54*d^2*x^4-11/3*c*d*x^2+c^2)*f^2*e^3+ 
f^3*(-1/3*d^3*x^6+39/8*c*d^2*x^4-3/2*c^2*d*x^2+c^3)*e^2+1/4*c^2*(-3/2*x^2* 
d+c)*x^2*f^4*e-3/8*c^3*f^5*x^4)*b*a^4-32/5*b^2*((-5/6*x^2*d^3+3/4*c*d^2)*e 
^5-9/4*d*(145/216*d^2*x^4-17/9*c*d*x^2+c^2)*f*e^4-27/4*d*(83/648*d^2*x^4-1 
49/216*c*d*x^2+c^2)*x^2*f^2*e^3+f^3*x^2*c*(55/32*d^2*x^4-129/32*c*d*x^2+c^ 
2)*e^2+23/32*c^2*(-3/23*x^2*d+c)*x^4*f^4*e-3/32*c^3*f^5*x^6)*a^3-32/5*b^3* 
e*(d^2*x^2*(-1/8*x^2*d+c)*e^4+f*(-1/16*d^3*x^6+11/4*c*d^2*x^4-2*c^2*d*x^2+ 
c^3)*e^3+2*c*(25/32*d^2*x^4-41/16*c*d*x^2+c^2)*x^2*f^2*e^2+3/2*c^2*(-47/24 
*x^2*d+c)*x^4*f^3*e+7/16*c^3*f^4*x^6)*a^2+8/5*c^2*(e*(d*x^2+c)-11/3*c*f*x^ 
2)*b^4*(f*x^2+e)^2*e^2*a+16/15*b^5*c^3*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. 2513 vs. \(2 (579) = 1158\).

Time = 88.18 (sec) , antiderivative size = 5066, normalized size of antiderivative = 8.24 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

integrate((d*x^2 + c)^3/((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. 2892 vs. \(2 (579) = 1158\).

Time = 0.49 (sec) , antiderivative size = 2892, normalized size of antiderivative = 4.70 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

1/3*((2*b^10*c^3*e^5 + 3*a*b^9*c^2*d*e^5 - 12*a^2*b^8*c*d^2*e^5 + 7*a^3*b^ 
7*d^3*e^5 - 19*a*b^9*c^3*e^4*f + 12*a^2*b^8*c^2*d*e^4*f + 33*a^3*b^7*c*d^2 
*e^4*f - 26*a^4*b^6*d^3*e^4*f + 56*a^2*b^8*c^3*e^3*f^2 - 78*a^3*b^7*c^2*d* 
e^3*f^2 - 12*a^4*b^6*c*d^2*e^3*f^2 + 34*a^5*b^5*d^3*e^3*f^2 - 74*a^3*b^7*c 
^3*e^2*f^3 + 132*a^4*b^6*c^2*d*e^2*f^3 - 42*a^5*b^5*c*d^2*e^2*f^3 - 16*a^6 
*b^4*d^3*e^2*f^3 + 46*a^4*b^6*c^3*e*f^4 - 93*a^5*b^5*c^2*d*e*f^4 + 48*a^6* 
b^4*c*d^2*e*f^4 - a^7*b^3*d^3*e*f^4 - 11*a^5*b^5*c^3*f^5 + 24*a^6*b^4*c^2* 
d*f^5 - 15*a^7*b^3*c*d^2*f^5 + 2*a^8*b^2*d^3*f^5)*x^2/(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) 
+ 3*(a*b^9*c^3*e^5 - 3*a^3*b^7*c*d^2*e^5 + 2*a^4*b^6*d^3*e^5 - 8*a^2*b^8*c 
^3*e^4*f + 9*a^3*b^7*c^2*d*e^4*f + 6*a^4*b^6*c*d^2*e^4*f - 7*a^5*b^5*d^3*e 
^4*f + 22*a^3*b^7*c^3*e^3*f^2 - 36*a^4*b^6*c^2*d*e^3*f^2 + 6*a^5*b^5*c*d^2 
*e^3*f^2 + 8*a^6*b^4*d^3*e^3*f^2 - 28*a^4*b^6*c^3*e^2*f^3 + 54*a^5*b^5*c^2 
*d*e^2*f^3 - 24*a^6*b^4*c*d^2*e^2*f^3 - 2*a^7*b^3*d^3*e^2*f^3 + 17*a^5*b^5 
*c^3*e*f^4 - 36*a^6*b^4*c^2*d*e*f^4 + 21*a^7*b^3*c*d^2*e*f^4 - 2*a^8*b^2*d 
^3*e*f^4 - 4*a^6*b^4*c^3*f^5 + 9*a^7*b^3*c^2*d*f^5 - 6*a^8*b^2*c*d^2*f^5 + 
 a^9*b*d^3*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*(24*b^(5...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 41.22 (sec) , antiderivative size = 17693, normalized size of antiderivative = 28.77 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((d*x^2+c)^3/(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*b*c**3*e**2*f**6 - 18*sq 
rt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - s 
qrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**7*b*c**3*e*f**7*x**2 - 9*sqrt(e)*s 
qrt(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*b*c**3*f**8*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*b*c**2*d*e**3*f**5 - 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)/(sqr 
t(e)*sqrt(b)))*a**7*b*c**2*d*e**2*f**6*x**2 - 9*sqrt(e)*sqrt(a*f - b*e)*at 
an((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt( 
e)*sqrt(b)))*a**7*b*c**2*d*e*f**7*x**4 - 27*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)*s 
qrt(b)))*a**7*b*c*d**2*e**4*f**4 - 54*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*b*c*d**2*e**3*f**5*x**2 - 27*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*b*c*d**2*e**2*f**6*x**4 + 45*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*b*d**3*e**5*f**3 + 90*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*...