\(\int \frac {(a+b x^2)^{3/2} (c+d x^2)^3}{(e+f x^2)^2} \, dx\) [305]

Optimal result

Integrand size = 30, antiderivative size = 375 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx=\frac {d \left (a^2 d^2 f^2-10 a b d f (2 d e-3 c f)+24 b^2 (d e-c f)^2\right ) x \sqrt {a+b x^2}}{16 b f^4}-\frac {d^2 (12 b d e-18 b c f-7 a d f) x^3 \sqrt {a+b x^2}}{24 f^3}+\frac {b d^3 x^5 \sqrt {a+b x^2}}{6 f^2}+\frac {(b e-a f) (d e-c f)^3 x \sqrt {a+b x^2}}{2 e f^4 \left (e+f x^2\right )}-\frac {\left (a^3 d^3 f^3+6 a^2 b d^2 f^2 (2 d e-3 c f)-72 a b^2 d f (d e-c f)^2+16 b^3 (d e-c f)^2 (4 d e-c f)\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{3/2} f^5}+\frac {\sqrt {b e-a f} (d e-c f)^2 (2 b e (4 d e-c f)-a f (5 d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{2 e^{3/2} f^5} \] Output:

1/16*d*(a^2*d^2*f^2-10*a*b*d*f*(-3*c*f+2*d*e)+24*b^2*(-c*f+d*e)^2)*x*(b*x^ 
2+a)^(1/2)/b/f^4-1/24*d^2*(-7*a*d*f-18*b*c*f+12*b*d*e)*x^3*(b*x^2+a)^(1/2) 
/f^3+1/6*b*d^3*x^5*(b*x^2+a)^(1/2)/f^2+1/2*(-a*f+b*e)*(-c*f+d*e)^3*x*(b*x^ 
2+a)^(1/2)/e/f^4/(f*x^2+e)-1/16*(a^3*d^3*f^3+6*a^2*b*d^2*f^2*(-3*c*f+2*d*e 
)-72*a*b^2*d*f*(-c*f+d*e)^2+16*b^3*(-c*f+d*e)^2*(-c*f+4*d*e))*arctanh(b^(1 
/2)*x/(b*x^2+a)^(1/2))/b^(3/2)/f^5+1/2*(-a*f+b*e)^(1/2)*(-c*f+d*e)^2*(2*b* 
e*(-c*f+4*d*e)-a*f*(c*f+5*d*e))*arctanh((-a*f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+ 
a)^(1/2))/e^(3/2)/f^5
 

Mathematica [A] (verified)

Time = 3.67 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx=\frac {\frac {f x \sqrt {a+b x^2} \left (3 a^2 d^3 e f^2 \left (e+f x^2\right )+2 a b f \left (-36 c^2 d e f^2+12 c^3 f^3+9 c d^2 e f \left (9 e+5 f x^2\right )+d^3 e \left (-42 e^2-23 e f x^2+7 f^2 x^4\right )\right )+4 b^2 e \left (-6 c^3 f^3+18 c^2 d f^2 \left (2 e+f x^2\right )+9 c d^2 f \left (-6 e^2-3 e f x^2+f^2 x^4\right )+2 d^3 \left (12 e^3+6 e^2 f x^2-2 e f^2 x^4+f^3 x^6\right )\right )\right )}{b e \left (e+f x^2\right )}+\frac {24 \sqrt {-b e+a f} (d e-c f)^2 (2 b e (4 d e-c f)-a f (5 d e+c f)) \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{e^{3/2}}+\frac {3 \left (a^3 d^3 f^3+6 a^2 b d^2 f^2 (2 d e-3 c f)-72 a b^2 d f (d e-c f)^2+16 b^3 (d e-c f)^2 (4 d e-c f)\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{3/2}}}{48 f^5} \] Input:

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

Output:

((f*x*Sqrt[a + b*x^2]*(3*a^2*d^3*e*f^2*(e + f*x^2) + 2*a*b*f*(-36*c^2*d*e* 
f^2 + 12*c^3*f^3 + 9*c*d^2*e*f*(9*e + 5*f*x^2) + d^3*e*(-42*e^2 - 23*e*f*x 
^2 + 7*f^2*x^4)) + 4*b^2*e*(-6*c^3*f^3 + 18*c^2*d*f^2*(2*e + f*x^2) + 9*c* 
d^2*f*(-6*e^2 - 3*e*f*x^2 + f^2*x^4) + 2*d^3*(12*e^3 + 6*e^2*f*x^2 - 2*e*f 
^2*x^4 + f^3*x^6))))/(b*e*(e + f*x^2)) + (24*Sqrt[-(b*e) + a*f]*(d*e - c*f 
)^2*(2*b*e*(4*d*e - c*f) - a*f*(5*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])])/e^(3/2) + (3*(a^3 
*d^3*f^3 + 6*a^2*b*d^2*f^2*(2*d*e - 3*c*f) - 72*a*b^2*d*f*(d*e - c*f)^2 + 
16*b^3*(d*e - c*f)^2*(4*d*e - c*f))*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/b 
^(3/2))/(48*f^5)
 

Rubi [F]

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.

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 420

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

\(\Big \downarrow \) 318

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 420

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 420

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

\(\Big \downarrow \) 318

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 420

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 398

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

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 299

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

rule 318

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S 
imp[1/(b*(2*(p + q) + 1))   Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b 
*c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 
 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G 
tQ[q, 1] && NeQ[2*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntBinomialQ[a, b, c, 
d, 2, p, q, x]
 

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 403

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

rule 420

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

rule 425

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

Maple [A] (verified)

Time = 1.32 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.22

method	result	size
pseudoelliptic	\(-\frac {-2 b^{\frac {5}{2}} \left (-4 b d \,e^{2}+\left (b c f +\frac {5}{2} a d f \right ) e +\frac {a c \,f^{2}}{2}\right ) \left (-a f +b e \right ) \left (c f -d e \right )^{2} \left (f \,x^{2}+e \right ) \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\sqrt {\left (a f -b e \right ) e}\, \left (\frac {b \left (f \,x^{2}+e \right ) e \left (64 b^{3} d^{3} e^{3}-72 b^{2} d^{2} f \left (a d +2 b c \right ) e^{2}+12 b d \,f^{2} \left (a^{2} d^{2}+12 a b c d +8 b^{2} c^{2}\right ) e +f^{3} \left (a^{3} d^{3}-18 a^{2} b c \,d^{2}-72 a \,b^{2} c^{2} d -16 b^{3} c^{3}\right )\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{8}+b^{\frac {3}{2}} \left (-4 b^{2} d^{3} e^{4}+\frac {7 \left (\left (-\frac {4 x^{2} d}{7}+\frac {18 c}{7}\right ) b +a d \right ) d^{2} b f \,e^{3}}{2}-\frac {d \,f^{2} \left (\left (-\frac {16}{3} d^{2} x^{4}-36 c d \,x^{2}+48 c^{2}\right ) b^{2}+54 a d \left (-\frac {23 x^{2} d}{81}+c \right ) b +a^{2} d^{2}\right ) e^{2}}{8}-\frac {\left (\left (\frac {8}{3} d^{3} x^{6}+12 c \,d^{2} x^{4}+24 c^{2} d \,x^{2}-8 c^{3}\right ) b^{2}-24 a d \left (-\frac {7}{36} d^{2} x^{4}-\frac {5}{4} c d \,x^{2}+c^{2}\right ) b +a^{2} d^{3} x^{2}\right ) f^{3} e}{8}-a b \,c^{3} f^{4}\right ) \sqrt {b \,x^{2}+a}\, x f \right )}{2 \sqrt {\left (a f -b e \right ) e}\, b^{\frac {5}{2}} f^{5} e \left (f \,x^{2}+e \right )}\)	\(457\)
risch	\(\text {Expression too large to display}\)	\(1732\)
default	\(\text {Expression too large to display}\)	\(3693\)

Input:

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

Output:

-1/2*(-2*b^(5/2)*(-4*b*d*e^2+(b*c*f+5/2*a*d*f)*e+1/2*a*c*f^2)*(-a*f+b*e)*( 
c*f-d*e)^2*(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)*(1/8*b*(f*x^2+e)*e*(64*b^3*d^3*e^3-72*b^2*d^2*f*(a*d+2*b*c) 
*e^2+12*b*d*f^2*(a^2*d^2+12*a*b*c*d+8*b^2*c^2)*e+f^3*(a^3*d^3-18*a^2*b*c*d 
^2-72*a*b^2*c^2*d-16*b^3*c^3))*arctanh((b*x^2+a)^(1/2)/x/b^(1/2))+b^(3/2)* 
(-4*b^2*d^3*e^4+7/2*((-4/7*x^2*d+18/7*c)*b+a*d)*d^2*b*f*e^3-1/8*d*f^2*((-1 
6/3*d^2*x^4-36*c*d*x^2+48*c^2)*b^2+54*a*d*(-23/81*x^2*d+c)*b+a^2*d^2)*e^2- 
1/8*((8/3*d^3*x^6+12*c*d^2*x^4+24*c^2*d*x^2-8*c^3)*b^2-24*a*d*(-7/36*d^2*x 
^4-5/4*c*d*x^2+c^2)*b+a^2*d^3*x^2)*f^3*e-a*b*c^3*f^4)*(b*x^2+a)^(1/2)*x*f) 
)/((a*f-b*e)*e)^(1/2)/b^(5/2)/f^5/e/(f*x^2+e)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 861 vs. \(2 (339) = 678\).

Time = 72.70 (sec) , antiderivative size = 3557, normalized size of antiderivative = 9.49 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1167 vs. \(2 (339) = 678\).

Time = 0.19 (sec) , antiderivative size = 1167, normalized size of antiderivative = 3.11 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

1/48*(2*(4*b*d^3*x^2/f^2 - (12*b^5*d^3*e*f^11 - 18*b^5*c*d^2*f^12 - 7*a*b^ 
4*d^3*f^12)/(b^4*f^14))*x^2 + 3*(24*b^5*d^3*e^2*f^10 - 48*b^5*c*d^2*e*f^11 
 - 20*a*b^4*d^3*e*f^11 + 24*b^5*c^2*d*f^12 + 30*a*b^4*c*d^2*f^12 + a^2*b^3 
*d^3*f^12)/(b^4*f^14))*sqrt(b*x^2 + a)*x + 1/2*(8*b^(5/2)*d^3*e^5 - 18*b^( 
5/2)*c*d^2*e^4*f - 13*a*b^(3/2)*d^3*e^4*f + 12*b^(5/2)*c^2*d*e^3*f^2 + 27* 
a*b^(3/2)*c*d^2*e^3*f^2 + 5*a^2*sqrt(b)*d^3*e^3*f^2 - 2*b^(5/2)*c^3*e^2*f^ 
3 - 15*a*b^(3/2)*c^2*d*e^2*f^3 - 9*a^2*sqrt(b)*c*d^2*e^2*f^3 + a*b^(3/2)*c 
^3*e*f^4 + 3*a^2*sqrt(b)*c^2*d*e*f^4 + a^2*sqrt(b)*c^3*f^5)*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))/ 
(sqrt(-b^2*e^2 + a*b*e*f)*e*f^5) + 1/32*(64*b^3*d^3*e^3 - 144*b^3*c*d^2*e^ 
2*f - 72*a*b^2*d^3*e^2*f + 96*b^3*c^2*d*e*f^2 + 144*a*b^2*c*d^2*e*f^2 + 12 
*a^2*b*d^3*e*f^2 - 16*b^3*c^3*f^3 - 72*a*b^2*c^2*d*f^3 - 18*a^2*b*c*d^2*f^ 
3 + a^3*d^3*f^3)*log((sqrt(b)*x - sqrt(b*x^2 + a))^2)/(b^(3/2)*f^5) + (2*( 
sqrt(b)*x - sqrt(b*x^2 + a))^2*b^3*d^3*e^5 - 6*(sqrt(b)*x - sqrt(b*x^2 + a 
))^2*b^3*c*d^2*e^4*f - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*b^2*d^3*e^4*f + 
 6*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^3*c^2*d*e^3*f^2 + 9*(sqrt(b)*x - sqrt 
(b*x^2 + a))^2*a*b^2*c*d^2*e^3*f^2 + (sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*b 
*d^3*e^3*f^2 - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^3*c^3*e^2*f^3 - 9*(sqrt 
(b)*x - sqrt(b*x^2 + a))^2*a*b^2*c^2*d*e^2*f^3 - 3*(sqrt(b)*x - sqrt(b*x^2 
 + a))^2*a^2*b*c*d^2*e^2*f^3 + 3*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*b^2*...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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