\(\int \frac {(c+d x^2)^3}{(a+b x^2)^{5/2} (e+f x^2)^2} \, dx\) [374]

Optimal result

Integrand size = 30, antiderivative size = 384 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=-\frac {d^2 (2 b d e-3 b c f+a d f) x}{3 a b f^3 \left (a+b x^2\right )^{3/2}}-\frac {b (d e-c f)^2 (3 a f (3 d e-c f)-2 b e (2 d e+c f)) x}{6 a e f^3 (b e-a f)^2 \left (a+b x^2\right )^{3/2}}-\frac {d^2 (4 b d e-6 b c f-a d f) x}{3 a^2 b f^3 \sqrt {a+b x^2}}+\frac {b (d e-c f)^2 \left (3 a^2 f^2 (11 d e-c f)+4 b^2 e^2 (2 d e+c f)-2 a b e f (13 d e+8 c f)\right ) x}{6 a^2 e f^3 (b e-a f)^3 \sqrt {a+b x^2}}+\frac {(d e-c f)^3 x}{2 e f^2 (b e-a f) \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}+\frac {(d e-c f)^2 (6 b c e-5 a d e-a c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{2 e^{3/2} (b e-a f)^{7/2}} \] Output:

-1/3*d^2*(a*d*f-3*b*c*f+2*b*d*e)*x/a/b/f^3/(b*x^2+a)^(3/2)-1/6*b*(-c*f+d*e 
)^2*(3*a*f*(-c*f+3*d*e)-2*b*e*(c*f+2*d*e))*x/a/e/f^3/(-a*f+b*e)^2/(b*x^2+a 
)^(3/2)-1/3*d^2*(-a*d*f-6*b*c*f+4*b*d*e)*x/a^2/b/f^3/(b*x^2+a)^(1/2)+1/6*b 
*(-c*f+d*e)^2*(3*a^2*f^2*(-c*f+11*d*e)+4*b^2*e^2*(c*f+2*d*e)-2*a*b*e*f*(8* 
c*f+13*d*e))*x/a^2/e/f^3/(-a*f+b*e)^3/(b*x^2+a)^(1/2)+1/2*(-c*f+d*e)^3*x/e 
/f^2/(-a*f+b*e)/(b*x^2+a)^(3/2)/(f*x^2+e)+1/2*(-c*f+d*e)^2*(-a*c*f-5*a*d*e 
+6*b*c*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 = 2.51 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.13 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=\frac {x \left (-4 b^4 c^3 e^2 x^2 \left (e+f x^2\right )-2 a b^3 c^2 e \left (e+f x^2\right ) \left (3 c e+3 d e x^2-8 c f x^2\right )+3 a^2 b^2 \left (-d^3 e^3 x^4+c d^2 e^2 x^2 \left (8 e+11 f x^2\right )-c^2 d e f x^2 \left (10 e+13 f x^2\right )+c^3 f \left (6 e^2+6 e f x^2+f^2 x^4\right )\right )+a^4 \left (-9 c^2 d e f^2+3 c^3 f^3+9 c d^2 e f \left (3 e+2 f x^2\right )+d^3 e \left (-15 e^2-10 e f x^2+2 f^2 x^4\right )\right )+2 a^3 b \left (3 c^3 f^3 x^2-9 c^2 d e f \left (2 e+3 f x^2\right )-d^3 e^2 x^2 \left (10 e+7 f x^2\right )+3 c d^2 e \left (3 e^2+8 e f x^2+2 f^2 x^4\right )\right )\right )}{6 a^2 e (-b e+a f)^3 \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}-\frac {(d e-c f)^2 (-6 b c e+5 a d e+a c f) \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{2 e^{3/2} (-b e+a f)^{7/2}} \] Input:

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

Output:

(x*(-4*b^4*c^3*e^2*x^2*(e + f*x^2) - 2*a*b^3*c^2*e*(e + f*x^2)*(3*c*e + 3* 
d*e*x^2 - 8*c*f*x^2) + 3*a^2*b^2*(-(d^3*e^3*x^4) + c*d^2*e^2*x^2*(8*e + 11 
*f*x^2) - c^2*d*e*f*x^2*(10*e + 13*f*x^2) + c^3*f*(6*e^2 + 6*e*f*x^2 + f^2 
*x^4)) + a^4*(-9*c^2*d*e*f^2 + 3*c^3*f^3 + 9*c*d^2*e*f*(3*e + 2*f*x^2) + d 
^3*e*(-15*e^2 - 10*e*f*x^2 + 2*f^2*x^4)) + 2*a^3*b*(3*c^3*f^3*x^2 - 9*c^2* 
d*e*f*(2*e + 3*f*x^2) - d^3*e^2*x^2*(10*e + 7*f*x^2) + 3*c*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*(-6*b*c*e + 5*a*d*e + a*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 = 1.20 (sec) , antiderivative size = 753, normalized size of antiderivative = 1.96, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.867, Rules used = {425, 419, 25, 398, 224, 219, 291, 221, 401, 25, 27, 298, 224, 219, 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.

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 419

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 298

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(d*((((b*c - a*d)*(b*e - a*f)*x*(c + d*x^2))/(3*a*(a + b*x^2)^(3/2)) + ((( 
b*c - a*d)*(2*b*c*e + 3*a*d*e - 5*a*c*f)*x)/(a*Sqrt[a + b*x^2]) + (3*a*d*( 
d*e - c*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/Sqrt[b])/(3*a))/(b*e - a* 
f)^2 - (f*(d*e - c*f)*((d*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*f 
) - ((d*e - c*f)*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/( 
Sqrt[e]*f*Sqrt[b*e - a*f])))/(b*e - a*f)^2))/f - ((d*e - c*f)*((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))/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 219

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

rule 221

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

rule 224

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

rule 291

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

rule 298

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

rule 398

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

rule 401

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

rule 402

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

rule 419

Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( 
x_)^2), x_Symbol] :> Simp[b*((b*e - a*f)/(b*c - a*d)^2)   Int[(c + d*x^2)^( 
q + 2)*((e + f*x^2)^(r - 1)/(a + b*x^2)), x], x] - Simp[1/(b*c - a*d)^2   I 
nt[(c + d*x^2)^q*(e + f*x^2)^(r - 1)*(2*b*c*d*e - a*d^2*e - b*c^2*f + d^2*( 
b*e - a*f)*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[q, -1] && Gt 
Q[r, 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 = 1.28 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.10

method	result	size
pseudoelliptic	\(\frac {-a^{2} \left (c f -d e \right )^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (f \,x^{2}+e \right ) \left (\left (c f +5 d e \right ) a -6 b c e \right ) \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\left (\left (-5 e^{3} d^{3}+9 \left (-\frac {10 x^{2} d}{27}+c \right ) d^{2} f \,e^{2}-3 d \left (-\frac {2}{9} d^{2} x^{4}-2 c d \,x^{2}+c^{2}\right ) f^{2} e +c^{3} f^{3}\right ) a^{4}+2 \left (\left (-\frac {10}{3} x^{2} d^{3}+3 c \,d^{2}\right ) e^{3}-6 d \left (\frac {7}{18} d^{2} x^{4}-\frac {4}{3} c d \,x^{2}+c^{2}\right ) f \,e^{2}-9 \left (-\frac {2 x^{2} d}{9}+c \right ) c d \,x^{2} f^{2} e +x^{2} f^{3} c^{3}\right ) b \,a^{3}+6 \left (\frac {4 d^{2} \left (-\frac {x^{2} d}{8}+c \right ) x^{2} e^{3}}{3}+c f \left (\frac {11}{6} d^{2} x^{4}-\frac {5}{3} c d \,x^{2}+c^{2}\right ) e^{2}+c^{2} f^{2} x^{2} \left (-\frac {13 x^{2} d}{6}+c \right ) e +\frac {c^{3} f^{3} x^{4}}{6}\right ) b^{2} a^{2}-2 \left (e \left (x^{2} d +c \right )-\frac {8 c f \,x^{2}}{3}\right ) c^{2} b^{3} \left (f \,x^{2}+e \right ) e a -\frac {4 b^{4} c^{3} e^{2} x^{2} \left (f \,x^{2}+e \right )}{3}\right ) \sqrt {\left (a f -b e \right ) e}\, x}{2 \sqrt {\left (a f -b e \right ) e}\, \left (b \,x^{2}+a \right )^{\frac {3}{2}} e \left (f \,x^{2}+e \right ) \left (a f -b e \right )^{3} a^{2}}\)	\(421\)
default	\(\text {Expression too large to display}\)	\(3733\)

Input:

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

Output:

1/2*(-a^2*(c*f-d*e)^2*(b*x^2+a)^(3/2)*(f*x^2+e)*((c*f+5*d*e)*a-6*b*c*e)*ar 
ctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2))+((-5*e^3*d^3+9*(-10/27*x^2*d 
+c)*d^2*f*e^2-3*d*(-2/9*d^2*x^4-2*c*d*x^2+c^2)*f^2*e+c^3*f^3)*a^4+2*((-10/ 
3*x^2*d^3+3*c*d^2)*e^3-6*d*(7/18*d^2*x^4-4/3*c*d*x^2+c^2)*f*e^2-9*(-2/9*x^ 
2*d+c)*c*d*x^2*f^2*e+x^2*f^3*c^3)*b*a^3+6*(4/3*d^2*(-1/8*x^2*d+c)*x^2*e^3+ 
c*f*(11/6*d^2*x^4-5/3*c*d*x^2+c^2)*e^2+c^2*f^2*x^2*(-13/6*x^2*d+c)*e+1/6*c 
^3*f^3*x^4)*b^2*a^2-2*(e*(d*x^2+c)-8/3*c*f*x^2)*c^2*b^3*(f*x^2+e)*e*a-4/3* 
b^4*c^3*e^2*x^2*(f*x^2+e))*((a*f-b*e)*e)^(1/2)*x)/((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. 1444 vs. \(2 (352) = 704\).

Time = 19.50 (sec) , antiderivative size = 2928, normalized size of antiderivative = 7.62 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/24*(3*(a^5*c^3*e*f^3 + (a^3*b^2*c^3*f^4 - (6*a^2*b^3*c*d^2 - 5*a^3*b^2 
*d^3)*e^3*f + 3*(4*a^2*b^3*c^2*d - 3*a^3*b^2*c*d^2)*e^2*f^2 - 3*(2*a^2*b^3 
*c^3 - a^3*b^2*c^2*d)*e*f^3)*x^6 - (6*a^4*b*c*d^2 - 5*a^5*d^3)*e^4 + 3*(4* 
a^4*b*c^2*d - 3*a^5*c*d^2)*e^3*f - 3*(2*a^4*b*c^3 - a^5*c^2*d)*e^2*f^2 + ( 
2*a^4*b*c^3*f^4 - (6*a^2*b^3*c*d^2 - 5*a^3*b^2*d^3)*e^4 + (12*a^2*b^3*c^2* 
d - 21*a^3*b^2*c*d^2 + 10*a^4*b*d^3)*e^3*f - 3*(2*a^2*b^3*c^3 - 9*a^3*b^2* 
c^2*d + 6*a^4*b*c*d^2)*e^2*f^2 - (11*a^3*b^2*c^3 - 6*a^4*b*c^2*d)*e*f^3)*x 
^4 + (a^5*c^3*f^4 - 2*(6*a^3*b^2*c*d^2 - 5*a^4*b*d^3)*e^4 + (24*a^3*b^2*c^ 
2*d - 24*a^4*b*c*d^2 + 5*a^5*d^3)*e^3*f - 3*(4*a^3*b^2*c^3 - 6*a^4*b*c^2*d 
 + 3*a^5*c*d^2)*e^2*f^2 - (4*a^4*b*c^3 - 3*a^5*c^2*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) 
*sqrt(b*x^2 + a))/(f^2*x^4 + 2*e*f*x^2 + e^2)) - 4*((3*a^2*b^3*d^3*e^5 + 3 
*a^3*b^2*c^3*e*f^4 + (4*b^5*c^3 + 6*a*b^4*c^2*d - 33*a^2*b^3*c*d^2 + 11*a^ 
3*b^2*d^3)*e^4*f - (20*a*b^4*c^3 - 33*a^2*b^3*c^2*d - 21*a^3*b^2*c*d^2 + 1 
6*a^4*b*d^3)*e^3*f^2 + (13*a^2*b^3*c^3 - 39*a^3*b^2*c^2*d + 12*a^4*b*c*d^2 
 + 2*a^5*d^3)*e^2*f^3)*x^5 + 2*(3*a^4*b*c^3*e*f^4 + (2*b^5*c^3 + 3*a*b^4*c 
^2*d - 12*a^2*b^3*c*d^2 + 10*a^3*b^2*d^3)*e^5 - (7*a*b^4*c^3 - 12*a^2*b^3* 
c^2*d + 12*a^3*b^2*c*d^2 + 5*a^4*b*d^3)*e^4*f - (4*a^2*b^3*c^3 - 12*a^3*b^ 
2*c^2*d - 15*a^4*b*c*d^2 + 5*a^5*d^3)*e^3*f^2 + 3*(2*a^3*b^2*c^3 - 9*a^...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

integrate((d*x^2 + c)^3/((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. 1390 vs. \(2 (352) = 704\).

Time = 0.39 (sec) , antiderivative size = 1390, normalized size of antiderivative = 3.62 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/3*((2*b^8*c^3*e^4 + 3*a*b^7*c^2*d*e^4 - 12*a^2*b^6*c*d^2*e^4 + 7*a^3*b^5 
*d^3*e^4 - 14*a*b^7*c^3*e^3*f + 6*a^2*b^6*c^2*d*e^3*f + 30*a^3*b^5*c*d^2*e 
^3*f - 22*a^4*b^4*d^3*e^3*f + 30*a^2*b^6*c^3*e^2*f^2 - 36*a^3*b^5*c^2*d*e^ 
2*f^2 - 18*a^4*b^4*c*d^2*e^2*f^2 + 24*a^5*b^3*d^3*e^2*f^2 - 26*a^3*b^5*c^3 
*e*f^3 + 42*a^4*b^4*c^2*d*e*f^3 - 6*a^5*b^3*c*d^2*e*f^3 - 10*a^6*b^2*d^3*e 
*f^3 + 8*a^4*b^4*c^3*f^4 - 15*a^5*b^3*c^2*d*f^4 + 6*a^6*b^2*c*d^2*f^4 + a^ 
7*b*d^3*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^3*e^4 - 3*a^3*b^5*c*d^2*e^4 + 2*a^4*b^4*d^3*e^4 - 6*a^2*b^6*c^3*e^3 
*f + 6*a^3*b^5*c^2*d*e^3*f + 6*a^4*b^4*c*d^2*e^3*f - 6*a^5*b^3*d^3*e^3*f + 
 12*a^3*b^5*c^3*e^2*f^2 - 18*a^4*b^4*c^2*d*e^2*f^2 + 6*a^6*b^2*d^3*e^2*f^2 
 - 10*a^4*b^4*c^3*e*f^3 + 18*a^5*b^3*c^2*d*e*f^3 - 6*a^6*b^2*c*d^2*e*f^3 - 
 2*a^7*b*d^3*e*f^3 + 3*a^5*b^3*c^3*f^4 - 6*a^6*b^2*c^2*d*f^4 + 3*a^7*b*c*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*(6*b^(3/2)*c*d^2*e^3 - 5*a*sqrt(b)*d^3*e^3 - 12*b^(3/2)*c^2*d* 
e^2*f + 9*a*sqrt(b)*c*d^2*e^2*f + 6*b^(3/2)*c^3*e*f^2 - 3*a*sqrt(b)*c^2*d* 
e*f^2 - a*sqrt(b)*c^3*f^3)*arctan(1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2*f + 
 2*b*e - a*f)/sqrt(-b^2*e^2 + a*b*e*f))/((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...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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