\(\int \frac {A+C x^2}{(a+b x)^{7/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [37]

Optimal result

Integrand size = 44, antiderivative size = 1617 \[ \int \frac {A+C x^2}{(a+b x)^{7/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx =\text {Too large to display} \] Output:

-2/5*(A*b^2+C*a^2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/(-a*d+b*c)/(- 
a*f+b*e)/(-a*h+b*g)/(b*x+a)^(5/2)+4/15*(a^4*C*d*f*h+2*A*b^4*(c*e*h+c*f*g+d 
*e*g)+a^3*b*C*(c*f*h+d*e*h+d*f*g)+3*a^2*b^2*(2*A*d*f*h-C*(c*e*h+c*f*g+d*e* 
g))-a*b^3*(4*A*d*(e*h+f*g)-c*(-4*A*f*h+5*C*e*g)))*(d*x+c)^(1/2)*(f*x+e)^(1 
/2)*(h*x+g)^(1/2)/(-a*d+b*c)^2/(-a*f+b*e)^2/(-a*h+b*g)^2/(b*x+a)^(3/2)-2/1 
5*(-c*h+d*g)*(4*a*(A*b^2+C*a^2)*d*f*h*(3/2*b^2*c*e*g-3/2*a*b*(c*e*h+c*f*g+ 
d*e*g)+a^2*(c*f*h+d*e*h+d*f*g))+b*(3*a^2*d*f*h+2*b^2*(c*e*h+c*f*g+d*e*g)-3 
*a*b*(c*f*h+d*e*h+d*f*g))*(4*A*b^2*(c*e*h+c*f*g+d*e*g)+5*a*b*(-A*c*f*h-A*d 
*e*h-A*d*f*g+C*c*e*g)+a^2*(5*A*d*f*h-C*(c*e*h+c*f*g+d*e*g)))+2*a*(3*a*d*f* 
h-b*(c*f*h+d*e*h+d*f*g))*(2*A*b^3*(c*e*h+c*f*g+d*e*g)+a^3*C*(c*f*h+d*e*h+d 
*f*g)+a^2*b*(5*A*d*f*h-3*C*(c*e*h+c*f*g+d*e*g))-a*b^2*(4*A*d*(e*h+f*g)-c*( 
-4*A*f*h+5*C*e*g)))+b*(3*b*c*e*g-a*(c*e*h+c*f*g+d*e*g))*(2*a^3*C*d*f*h+2*a 
^2*b*C*(c*f*h+d*e*h+d*f*g)+a*b^2*(7*A*d*f*h-5*C*(c*e*h+c*f*g+d*e*g))-b^3*( 
3*A*d*(e*h+f*g)-c*(-3*A*f*h+5*C*e*g))))*(f*x+e)^(1/2)*(-(-a*d+b*c)*(h*x+g) 
/(-c*h+d*g)/(b*x+a))^(1/2)*EllipticE((-a*f+b*e)^(1/2)*(d*x+c)^(1/2)/(-c*f+ 
d*e)^(1/2)/(b*x+a)^(1/2),((-c*f+d*e)*(-a*h+b*g)/(-a*f+b*e)/(-c*h+d*g))^(1/ 
2))/(-a*d+b*c)^3/(-a*f+b*e)^(5/2)/(-c*f+d*e)^(1/2)/(-a*h+b*g)^3/(-(-a*d+b* 
c)*(f*x+e)/(-c*f+d*e)/(b*x+a))^(1/2)/(h*x+g)^(1/2)-2/15*(5*a^4*d*f*h*(3*A* 
d*f*h^2+c*C*h*(-e*h+f*g)+C*d*g*(e*h+2*f*g))+a*b^3*(A*d^2*g*(-11*e^2*h^2+3* 
e*f*g*h+8*f^2*g^2)-c^2*h*(10*C*e*g*(e*h+2*f*g)+A*f*h*(19*e*h+11*f*g))-c...
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(38948\) vs. \(2(1617)=3234\).

Time = 51.31 (sec) , antiderivative size = 38948, normalized size of antiderivative = 24.09 \[ \int \frac {A+C x^2}{(a+b x)^{7/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Result too large to show} \] Input:

Integrate[(A + C*x^2)/((a + b*x)^(7/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g 
+ h*x]),x]
 

Output:

Result too large to show
 

Rubi [A] (warning: unable to verify)

Time = 12.97 (sec) , antiderivative size = 2609, normalized size of antiderivative = 1.61, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2108, 25, 2107, 25, 2102, 25, 2105, 27, 188, 194, 321, 327}

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.

\(\displaystyle \int \frac {A+C x^2}{(a+b x)^{7/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\)

\(\Big \downarrow \) 2108

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2107

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2102

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2105

\(\displaystyle -\frac {2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (C a^2+A b^2\right )}{5 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{5/2}}-\frac {-\frac {4 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (C d f h a^4+b C (d f g+d e h+c f h) a^3+3 b^2 (2 A d f h-C (d e g+c f g+c e h)) a^2-b^3 (4 A d (f g+e h)-c (5 C e g-4 A f h)) a+2 A b^4 (d e g+c f g+c e h)\right )}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {-\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (4 a \left (C a^2+A b^2\right ) d f h \left ((d f g+d e h+c f h) a^2-\frac {3}{2} b (d e g+c f g+c e h) a+\frac {3}{2} b^2 c e g\right )+b \left (3 d f h a^2-3 b (d f g+d e h+c f h) a+2 b^2 (d e g+c f g+c e h)\right ) \left ((5 A d f h-C (d e g+c f g+c e h)) a^2+5 b (c C e g-A d f g-A d e h-A c f h) a+4 A b^2 (d e g+c f g+c e h)\right )+2 a (3 a d f h-b (d f g+d e h+c f h)) \left (C (d f g+d e h+c f h) a^3+b (5 A d f h-3 C (d e g+c f g+c e h)) a^2-b^2 (4 A d (f g+e h)-c (5 C e g-4 A f h)) a+2 A b^3 (d e g+c f g+c e h)\right )+b (3 b c e g-a (d e g+c f g+c e h)) \left (2 C d f h a^3+2 b C (d f g+d e h+c f h) a^2+b^2 (7 A d f h-5 C (d e g+c f g+c e h)) a-b^3 (3 A d (f g+e h)-c (5 C e g-3 A f h))\right )\right )}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}-\frac {-\left ((d e-c f) (d g-c h) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}}dx \left (4 a \left (C a^2+A b^2\right ) d f h \left ((d f g+d e h+c f h) a^2-\frac {3}{2} b (d e g+c f g+c e h) a+\frac {3}{2} b^2 c e g\right )+b \left (3 d f h a^2-3 b (d f g+d e h+c f h) a+2 b^2 (d e g+c f g+c e h)\right ) \left ((5 A d f h-C (d e g+c f g+c e h)) a^2+5 b (c C e g-A d f g-A d e h-A c f h) a+4 A b^2 (d e g+c f g+c e h)\right )+2 a (3 a d f h-b (d f g+d e h+c f h)) \left (C (d f g+d e h+c f h) a^3+b (5 A d f h-3 C (d e g+c f g+c e h)) a^2-b^2 (4 A d (f g+e h)-c (5 C e g-4 A f h)) a+2 A b^3 (d e g+c f g+c e h)\right )+b (3 b c e g-a (d e g+c f g+c e h)) \left (2 C d f h a^3+2 b C (d f g+d e h+c f h) a^2+b^2 (7 A d f h-5 C (d e g+c f g+c e h)) a-b^3 (3 A d (f g+e h)-c (5 C e g-3 A f h))\right )\right )\right )-\frac {2 d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (4 a \left (C a^2+A b^2\right ) d f h \left ((d f g+d e h+c f h) a^2-\frac {3}{2} b (d e g+c f g+c e h) a+\frac {3}{2} b^2 c e g\right )+b \left (3 d f h a^2-3 b (d f g+d e h+c f h) a+2 b^2 (d e g+c f g+c e h)\right ) \left ((5 A d f h-C (d e g+c f g+c e h)) a^2+5 b (c C e g-A d f g-A d e h-A c f h) a+4 A b^2 (d e g+c f g+c e h)\right )+2 a (3 a d f h-b (d f g+d e h+c f h)) \left (C (d f g+d e h+c f h) a^3+b (5 A d f h-3 C (d e g+c f g+c e h)) a^2-b^2 (4 A d (f g+e h)-c (5 C e g-4 A f h)) a+2 A b^3 (d e g+c f g+c e h)\right )+b (3 b c e g-a (d e g+c f g+c e h)) \left (2 C d f h a^3+2 b C (d f g+d e h+c f h) a^2+b^2 (7 A d f h-5 C (d e g+c f g+c e h)) a-b^3 (3 A d (f g+e h)-c (5 C e g-3 A f h))\right )\right )}{\sqrt {c+d x}}+\frac {\int \frac {2 b d f (b e-a f) h (b g-a h) \left (5 d f h \left (3 A d^2 f h-C \left (-2 f h c^2-d f g c-d e h c+d^2 e g\right )\right ) a^4-b \left (30 A f h (f g+e h) d^3+C \left (2 f^2 h^2 c^3+27 d f h (f g+e h) c^2+d^2 \left (f^2 g^2+4 e f h g+e^2 h^2\right ) c-d^3 e g (f g+e h)\right )\right ) a^3-3 b^2 \left (5 A d \left (-\left (\left (f^2 g^2+3 e f h g+e^2 h^2\right ) d^2\right )-c f h (f g+e h) d+c^2 f^2 h^2\right )-C \left (2 f h (f g+e h) c^3+d \left (3 f^2 g^2+17 e f h g+3 e^2 h^2\right ) c^2+d^2 e g (f g+e h) c+d^3 e^2 g^2\right )\right ) a^2-b^3 \left (2 f h (5 C e g-4 A f h) c^3+d (f g+e h) (20 C e g-3 A f h) c^2+d^2 \left (10 C e^2 g^2+11 A f^2 g^2+14 A e f h g+11 A e^2 h^2\right ) c+19 A d^3 e g (f g+e h)\right ) a+b^4 \left (-4 A f h (f g+e h) c^3+d \left (15 C e^2 g^2+4 A f^2 g^2+A e f h g+4 A e^2 h^2\right ) c^2+3 A d^2 e g (f g+e h) c+8 A d^3 e^2 g^2\right )\right )}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{2 b d f h}}{(b c-a d) (b e-a f) (b g-a h)}}{3 (b c-a d) (b e-a f) (b g-a h)}}{5 (b c-a d) (b e-a f) (b g-a h)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (C a^2+A b^2\right )}{5 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{5/2}}-\frac {-\frac {4 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (C d f h a^4+b C (d f g+d e h+c f h) a^3+3 b^2 (2 A d f h-C (d e g+c f g+c e h)) a^2-b^3 (4 A d (f g+e h)-c (5 C e g-4 A f h)) a+2 A b^4 (d e g+c f g+c e h)\right )}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {-\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (4 a \left (C a^2+A b^2\right ) d f h \left ((d f g+d e h+c f h) a^2-\frac {3}{2} b (d e g+c f g+c e h) a+\frac {3}{2} b^2 c e g\right )+b \left (3 d f h a^2-3 b (d f g+d e h+c f h) a+2 b^2 (d e g+c f g+c e h)\right ) \left ((5 A d f h-C (d e g+c f g+c e h)) a^2+5 b (c C e g-A d f g-A d e h-A c f h) a+4 A b^2 (d e g+c f g+c e h)\right )+2 a (3 a d f h-b (d f g+d e h+c f h)) \left (C (d f g+d e h+c f h) a^3+b (5 A d f h-3 C (d e g+c f g+c e h)) a^2-b^2 (4 A d (f g+e h)-c (5 C e g-4 A f h)) a+2 A b^3 (d e g+c f g+c e h)\right )+b (3 b c e g-a (d e g+c f g+c e h)) \left (2 C d f h a^3+2 b C (d f g+d e h+c f h) a^2+b^2 (7 A d f h-5 C (d e g+c f g+c e h)) a-b^3 (3 A d (f g+e h)-c (5 C e g-3 A f h))\right )\right )}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}-\frac {-\left ((d e-c f) (d g-c h) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}}dx \left (4 a \left (C a^2+A b^2\right ) d f h \left ((d f g+d e h+c f h) a^2-\frac {3}{2} b (d e g+c f g+c e h) a+\frac {3}{2} b^2 c e g\right )+b \left (3 d f h a^2-3 b (d f g+d e h+c f h) a+2 b^2 (d e g+c f g+c e h)\right ) \left ((5 A d f h-C (d e g+c f g+c e h)) a^2+5 b (c C e g-A d f g-A d e h-A c f h) a+4 A b^2 (d e g+c f g+c e h)\right )+2 a (3 a d f h-b (d f g+d e h+c f h)) \left (C (d f g+d e h+c f h) a^3+b (5 A d f h-3 C (d e g+c f g+c e h)) a^2-b^2 (4 A d (f g+e h)-c (5 C e g-4 A f h)) a+2 A b^3 (d e g+c f g+c e h)\right )+b (3 b c e g-a (d e g+c f g+c e h)) \left (2 C d f h a^3+2 b C (d f g+d e h+c f h) a^2+b^2 (7 A d f h-5 C (d e g+c f g+c e h)) a-b^3 (3 A d (f g+e h)-c (5 C e g-3 A f h))\right )\right )\right )-\frac {2 d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (4 a \left (C a^2+A b^2\right ) d f h \left ((d f g+d e h+c f h) a^2-\frac {3}{2} b (d e g+c f g+c e h) a+\frac {3}{2} b^2 c e g\right )+b \left (3 d f h a^2-3 b (d f g+d e h+c f h) a+2 b^2 (d e g+c f g+c e h)\right ) \left ((5 A d f h-C (d e g+c f g+c e h)) a^2+5 b (c C e g-A d f g-A d e h-A c f h) a+4 A b^2 (d e g+c f g+c e h)\right )+2 a (3 a d f h-b (d f g+d e h+c f h)) \left (C (d f g+d e h+c f h) a^3+b (5 A d f h-3 C (d e g+c f g+c e h)) a^2-b^2 (4 A d (f g+e h)-c (5 C e g-4 A f h)) a+2 A b^3 (d e g+c f g+c e h)\right )+b (3 b c e g-a (d e g+c f g+c e h)) \left (2 C d f h a^3+2 b C (d f g+d e h+c f h) a^2+b^2 (7 A d f h-5 C (d e g+c f g+c e h)) a-b^3 (3 A d (f g+e h)-c (5 C e g-3 A f h))\right )\right )}{\sqrt {c+d x}}+(b e-a f) (b g-a h) \left (5 d f h \left (3 A d^2 f h-C \left (-2 f h c^2-d f g c-d e h c+d^2 e g\right )\right ) a^4-b \left (30 A f h (f g+e h) d^3+C \left (2 f^2 h^2 c^3+27 d f h (f g+e h) c^2+d^2 \left (f^2 g^2+4 e f h g+e^2 h^2\right ) c-d^3 e g (f g+e h)\right )\right ) a^3-3 b^2 \left (5 A d \left (-\left (\left (f^2 g^2+3 e f h g+e^2 h^2\right ) d^2\right )-c f h (f g+e h) d+c^2 f^2 h^2\right )-C \left (2 f h (f g+e h) c^3+d \left (3 f^2 g^2+17 e f h g+3 e^2 h^2\right ) c^2+d^2 e g (f g+e h) c+d^3 e^2 g^2\right )\right ) a^2-b^3 \left (2 f h (5 C e g-4 A f h) c^3+d (f g+e h) (20 C e g-3 A f h) c^2+d^2 \left (10 C e^2 g^2+11 A f^2 g^2+14 A e f h g+11 A e^2 h^2\right ) c+19 A d^3 e g (f g+e h)\right ) a+b^4 \left (-4 A f h (f g+e h) c^3+d \left (15 C e^2 g^2+4 A f^2 g^2+A e f h g+4 A e^2 h^2\right ) c^2+3 A d^2 e g (f g+e h) c+8 A d^3 e^2 g^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{(b c-a d) (b e-a f) (b g-a h)}}{3 (b c-a d) (b e-a f) (b g-a h)}}{5 (b c-a d) (b e-a f) (b g-a h)}\)

\(\Big \downarrow \) 188

\(\displaystyle -\frac {2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (C a^2+A b^2\right )}{5 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{5/2}}-\frac {-\frac {4 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (C d f h a^4+b C (d f g+d e h+c f h) a^3+3 b^2 (2 A d f h-C (d e g+c f g+c e h)) a^2-b^3 (4 A d (f g+e h)-c (5 C e g-4 A f h)) a+2 A b^4 (d e g+c f g+c e h)\right )}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {-\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (4 a \left (C a^2+A b^2\right ) d f h \left ((d f g+d e h+c f h) a^2-\frac {3}{2} b (d e g+c f g+c e h) a+\frac {3}{2} b^2 c e g\right )+b \left (3 d f h a^2-3 b (d f g+d e h+c f h) a+2 b^2 (d e g+c f g+c e h)\right ) \left ((5 A d f h-C (d e g+c f g+c e h)) a^2+5 b (c C e g-A d f g-A d e h-A c f h) a+4 A b^2 (d e g+c f g+c e h)\right )+2 a (3 a d f h-b (d f g+d e h+c f h)) \left (C (d f g+d e h+c f h) a^3+b (5 A d f h-3 C (d e g+c f g+c e h)) a^2-b^2 (4 A d (f g+e h)-c (5 C e g-4 A f h)) a+2 A b^3 (d e g+c f g+c e h)\right )+b (3 b c e g-a (d e g+c f g+c e h)) \left (2 C d f h a^3+2 b C (d f g+d e h+c f h) a^2+b^2 (7 A d f h-5 C (d e g+c f g+c e h)) a-b^3 (3 A d (f g+e h)-c (5 C e g-3 A f h))\right )\right )}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}-\frac {-\left ((d e-c f) (d g-c h) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}}dx \left (4 a \left (C a^2+A b^2\right ) d f h \left ((d f g+d e h+c f h) a^2-\frac {3}{2} b (d e g+c f g+c e h) a+\frac {3}{2} b^2 c e g\right )+b \left (3 d f h a^2-3 b (d f g+d e h+c f h) a+2 b^2 (d e g+c f g+c e h)\right ) \left ((5 A d f h-C (d e g+c f g+c e h)) a^2+5 b (c C e g-A d f g-A d e h-A c f h) a+4 A b^2 (d e g+c f g+c e h)\right )+2 a (3 a d f h-b (d f g+d e h+c f h)) \left (C (d f g+d e h+c f h) a^3+b (5 A d f h-3 C (d e g+c f g+c e h)) a^2-b^2 (4 A d (f g+e h)-c (5 C e g-4 A f h)) a+2 A b^3 (d e g+c f g+c e h)\right )+b (3 b c e g-a (d e g+c f g+c e h)) \left (2 C d f h a^3+2 b C (d f g+d e h+c f h) a^2+b^2 (7 A d f h-5 C (d e g+c f g+c e h)) a-b^3 (3 A d (f g+e h)-c (5 C e g-3 A f h))\right )\right )\right )-\frac {2 d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (4 a \left (C a^2+A b^2\right ) d f h \left ((d f g+d e h+c f h) a^2-\frac {3}{2} b (d e g+c f g+c e h) a+\frac {3}{2} b^2 c e g\right )+b \left (3 d f h a^2-3 b (d f g+d e h+c f h) a+2 b^2 (d e g+c f g+c e h)\right ) \left ((5 A d f h-C (d e g+c f g+c e h)) a^2+5 b (c C e g-A d f g-A d e h-A c f h) a+4 A b^2 (d e g+c f g+c e h)\right )+2 a (3 a d f h-b (d f g+d e h+c f h)) \left (C (d f g+d e h+c f h) a^3+b (5 A d f h-3 C (d e g+c f g+c e h)) a^2-b^2 (4 A d (f g+e h)-c (5 C e g-4 A f h)) a+2 A b^3 (d e g+c f g+c e h)\right )+b (3 b c e g-a (d e g+c f g+c e h)) \left (2 C d f h a^3+2 b C (d f g+d e h+c f h) a^2+b^2 (7 A d f h-5 C (d e g+c f g+c e h)) a-b^3 (3 A d (f g+e h)-c (5 C e g-3 A f h))\right )\right )}{\sqrt {c+d x}}+\frac {2 (b e-a f) (b g-a h) \left (5 d f h \left (3 A d^2 f h-C \left (-2 f h c^2-d f g c-d e h c+d^2 e g\right )\right ) a^4-b \left (30 A f h (f g+e h) d^3+C \left (2 f^2 h^2 c^3+27 d f h (f g+e h) c^2+d^2 \left (f^2 g^2+4 e f h g+e^2 h^2\right ) c-d^3 e g (f g+e h)\right )\right ) a^3-3 b^2 \left (5 A d \left (-\left (\left (f^2 g^2+3 e f h g+e^2 h^2\right ) d^2\right )-c f h (f g+e h) d+c^2 f^2 h^2\right )-C \left (2 f h (f g+e h) c^3+d \left (3 f^2 g^2+17 e f h g+3 e^2 h^2\right ) c^2+d^2 e g (f g+e h) c+d^3 e^2 g^2\right )\right ) a^2-b^3 \left (2 f h (5 C e g-4 A f h) c^3+d (f g+e h) (20 C e g-3 A f h) c^2+d^2 \left (10 C e^2 g^2+11 A f^2 g^2+14 A e f h g+11 A e^2 h^2\right ) c+19 A d^3 e g (f g+e h)\right ) a+b^4 \left (-4 A f h (f g+e h) c^3+d \left (15 C e^2 g^2+4 A f^2 g^2+A e f h g+4 A e^2 h^2\right ) c^2+3 A d^2 e g (f g+e h) c+8 A d^3 e^2 g^2\right )\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \int \frac {1}{\sqrt {\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}+1} \sqrt {1-\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}}}d\frac {\sqrt {e+f x}}{\sqrt {a+b x}}}{(f g-e h) \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}}{(b c-a d) (b e-a f) (b g-a h)}}{3 (b c-a d) (b e-a f) (b g-a h)}}{5 (b c-a d) (b e-a f) (b g-a h)}\)

\(\Big \downarrow \) 194

\(\displaystyle -\frac {2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (C a^2+A b^2\right )}{5 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{5/2}}-\frac {-\frac {4 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (C d f h a^4+b C (d f g+d e h+c f h) a^3+3 b^2 (2 A d f h-C (d e g+c f g+c e h)) a^2-b^3 (4 A d (f g+e h)-c (5 C e g-4 A f h)) a+2 A b^4 (d e g+c f g+c e h)\right )}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {-\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (4 a \left (C a^2+A b^2\right ) d f h \left ((d f g+d e h+c f h) a^2-\frac {3}{2} b (d e g+c f g+c e h) a+\frac {3}{2} b^2 c e g\right )+b \left (3 d f h a^2-3 b (d f g+d e h+c f h) a+2 b^2 (d e g+c f g+c e h)\right ) \left ((5 A d f h-C (d e g+c f g+c e h)) a^2+5 b (c C e g-A d f g-A d e h-A c f h) a+4 A b^2 (d e g+c f g+c e h)\right )+2 a (3 a d f h-b (d f g+d e h+c f h)) \left (C (d f g+d e h+c f h) a^3+b (5 A d f h-3 C (d e g+c f g+c e h)) a^2-b^2 (4 A d (f g+e h)-c (5 C e g-4 A f h)) a+2 A b^3 (d e g+c f g+c e h)\right )+b (3 b c e g-a (d e g+c f g+c e h)) \left (2 C d f h a^3+2 b C (d f g+d e h+c f h) a^2+b^2 (7 A d f h-5 C (d e g+c f g+c e h)) a-b^3 (3 A d (f g+e h)-c (5 C e g-3 A f h))\right )\right )}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}-\frac {\frac {2 (d g-c h) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} \int \frac {\sqrt {1-\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}}}{\sqrt {1-\frac {(d g-c h) (e+f x)}{(f g-e h) (c+d x)}}}d\frac {\sqrt {e+f x}}{\sqrt {c+d x}} \left (4 a \left (C a^2+A b^2\right ) d f h \left ((d f g+d e h+c f h) a^2-\frac {3}{2} b (d e g+c f g+c e h) a+\frac {3}{2} b^2 c e g\right )+b \left (3 d f h a^2-3 b (d f g+d e h+c f h) a+2 b^2 (d e g+c f g+c e h)\right ) \left ((5 A d f h-C (d e g+c f g+c e h)) a^2+5 b (c C e g-A d f g-A d e h-A c f h) a+4 A b^2 (d e g+c f g+c e h)\right )+2 a (3 a d f h-b (d f g+d e h+c f h)) \left (C (d f g+d e h+c f h) a^3+b (5 A d f h-3 C (d e g+c f g+c e h)) a^2-b^2 (4 A d (f g+e h)-c (5 C e g-4 A f h)) a+2 A b^3 (d e g+c f g+c e h)\right )+b (3 b c e g-a (d e g+c f g+c e h)) \left (2 C d f h a^3+2 b C (d f g+d e h+c f h) a^2+b^2 (7 A d f h-5 C (d e g+c f g+c e h)) a-b^3 (3 A d (f g+e h)-c (5 C e g-3 A f h))\right )\right )}{\sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {2 d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (4 a \left (C a^2+A b^2\right ) d f h \left ((d f g+d e h+c f h) a^2-\frac {3}{2} b (d e g+c f g+c e h) a+\frac {3}{2} b^2 c e g\right )+b \left (3 d f h a^2-3 b (d f g+d e h+c f h) a+2 b^2 (d e g+c f g+c e h)\right ) \left ((5 A d f h-C (d e g+c f g+c e h)) a^2+5 b (c C e g-A d f g-A d e h-A c f h) a+4 A b^2 (d e g+c f g+c e h)\right )+2 a (3 a d f h-b (d f g+d e h+c f h)) \left (C (d f g+d e h+c f h) a^3+b (5 A d f h-3 C (d e g+c f g+c e h)) a^2-b^2 (4 A d (f g+e h)-c (5 C e g-4 A f h)) a+2 A b^3 (d e g+c f g+c e h)\right )+b (3 b c e g-a (d e g+c f g+c e h)) \left (2 C d f h a^3+2 b C (d f g+d e h+c f h) a^2+b^2 (7 A d f h-5 C (d e g+c f g+c e h)) a-b^3 (3 A d (f g+e h)-c (5 C e g-3 A f h))\right )\right )}{\sqrt {c+d x}}+\frac {2 (b e-a f) (b g-a h) \left (5 d f h \left (3 A d^2 f h-C \left (-2 f h c^2-d f g c-d e h c+d^2 e g\right )\right ) a^4-b \left (30 A f h (f g+e h) d^3+C \left (2 f^2 h^2 c^3+27 d f h (f g+e h) c^2+d^2 \left (f^2 g^2+4 e f h g+e^2 h^2\right ) c-d^3 e g (f g+e h)\right )\right ) a^3-3 b^2 \left (5 A d \left (-\left (\left (f^2 g^2+3 e f h g+e^2 h^2\right ) d^2\right )-c f h (f g+e h) d+c^2 f^2 h^2\right )-C \left (2 f h (f g+e h) c^3+d \left (3 f^2 g^2+17 e f h g+3 e^2 h^2\right ) c^2+d^2 e g (f g+e h) c+d^3 e^2 g^2\right )\right ) a^2-b^3 \left (2 f h (5 C e g-4 A f h) c^3+d (f g+e h) (20 C e g-3 A f h) c^2+d^2 \left (10 C e^2 g^2+11 A f^2 g^2+14 A e f h g+11 A e^2 h^2\right ) c+19 A d^3 e g (f g+e h)\right ) a+b^4 \left (-4 A f h (f g+e h) c^3+d \left (15 C e^2 g^2+4 A f^2 g^2+A e f h g+4 A e^2 h^2\right ) c^2+3 A d^2 e g (f g+e h) c+8 A d^3 e^2 g^2\right )\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \int \frac {1}{\sqrt {\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}+1} \sqrt {1-\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}}}d\frac {\sqrt {e+f x}}{\sqrt {a+b x}}}{(f g-e h) \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}}{(b c-a d) (b e-a f) (b g-a h)}}{3 (b c-a d) (b e-a f) (b g-a h)}}{5 (b c-a d) (b e-a f) (b g-a h)}\)

\(\Big \downarrow \) 321

\(\displaystyle -\frac {2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (C a^2+A b^2\right )}{5 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{5/2}}-\frac {-\frac {4 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (C d f h a^4+b C (d f g+d e h+c f h) a^3+3 b^2 (2 A d f h-C (d e g+c f g+c e h)) a^2-b^3 (4 A d (f g+e h)-c (5 C e g-4 A f h)) a+2 A b^4 (d e g+c f g+c e h)\right )}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {-\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (4 a \left (C a^2+A b^2\right ) d f h \left ((d f g+d e h+c f h) a^2-\frac {3}{2} b (d e g+c f g+c e h) a+\frac {3}{2} b^2 c e g\right )+b \left (3 d f h a^2-3 b (d f g+d e h+c f h) a+2 b^2 (d e g+c f g+c e h)\right ) \left ((5 A d f h-C (d e g+c f g+c e h)) a^2+5 b (c C e g-A d f g-A d e h-A c f h) a+4 A b^2 (d e g+c f g+c e h)\right )+2 a (3 a d f h-b (d f g+d e h+c f h)) \left (C (d f g+d e h+c f h) a^3+b (5 A d f h-3 C (d e g+c f g+c e h)) a^2-b^2 (4 A d (f g+e h)-c (5 C e g-4 A f h)) a+2 A b^3 (d e g+c f g+c e h)\right )+b (3 b c e g-a (d e g+c f g+c e h)) \left (2 C d f h a^3+2 b C (d f g+d e h+c f h) a^2+b^2 (7 A d f h-5 C (d e g+c f g+c e h)) a-b^3 (3 A d (f g+e h)-c (5 C e g-3 A f h))\right )\right )}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}-\frac {\frac {2 (d g-c h) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} \int \frac {\sqrt {1-\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}}}{\sqrt {1-\frac {(d g-c h) (e+f x)}{(f g-e h) (c+d x)}}}d\frac {\sqrt {e+f x}}{\sqrt {c+d x}} \left (4 a \left (C a^2+A b^2\right ) d f h \left ((d f g+d e h+c f h) a^2-\frac {3}{2} b (d e g+c f g+c e h) a+\frac {3}{2} b^2 c e g\right )+b \left (3 d f h a^2-3 b (d f g+d e h+c f h) a+2 b^2 (d e g+c f g+c e h)\right ) \left ((5 A d f h-C (d e g+c f g+c e h)) a^2+5 b (c C e g-A d f g-A d e h-A c f h) a+4 A b^2 (d e g+c f g+c e h)\right )+2 a (3 a d f h-b (d f g+d e h+c f h)) \left (C (d f g+d e h+c f h) a^3+b (5 A d f h-3 C (d e g+c f g+c e h)) a^2-b^2 (4 A d (f g+e h)-c (5 C e g-4 A f h)) a+2 A b^3 (d e g+c f g+c e h)\right )+b (3 b c e g-a (d e g+c f g+c e h)) \left (2 C d f h a^3+2 b C (d f g+d e h+c f h) a^2+b^2 (7 A d f h-5 C (d e g+c f g+c e h)) a-b^3 (3 A d (f g+e h)-c (5 C e g-3 A f h))\right )\right )}{\sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {2 d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (4 a \left (C a^2+A b^2\right ) d f h \left ((d f g+d e h+c f h) a^2-\frac {3}{2} b (d e g+c f g+c e h) a+\frac {3}{2} b^2 c e g\right )+b \left (3 d f h a^2-3 b (d f g+d e h+c f h) a+2 b^2 (d e g+c f g+c e h)\right ) \left ((5 A d f h-C (d e g+c f g+c e h)) a^2+5 b (c C e g-A d f g-A d e h-A c f h) a+4 A b^2 (d e g+c f g+c e h)\right )+2 a (3 a d f h-b (d f g+d e h+c f h)) \left (C (d f g+d e h+c f h) a^3+b (5 A d f h-3 C (d e g+c f g+c e h)) a^2-b^2 (4 A d (f g+e h)-c (5 C e g-4 A f h)) a+2 A b^3 (d e g+c f g+c e h)\right )+b (3 b c e g-a (d e g+c f g+c e h)) \left (2 C d f h a^3+2 b C (d f g+d e h+c f h) a^2+b^2 (7 A d f h-5 C (d e g+c f g+c e h)) a-b^3 (3 A d (f g+e h)-c (5 C e g-3 A f h))\right )\right )}{\sqrt {c+d x}}+\frac {2 (b e-a f) \sqrt {b g-a h} \left (5 d f h \left (3 A d^2 f h-C \left (-2 f h c^2-d f g c-d e h c+d^2 e g\right )\right ) a^4-b \left (30 A f h (f g+e h) d^3+C \left (2 f^2 h^2 c^3+27 d f h (f g+e h) c^2+d^2 \left (f^2 g^2+4 e f h g+e^2 h^2\right ) c-d^3 e g (f g+e h)\right )\right ) a^3-3 b^2 \left (5 A d \left (-\left (\left (f^2 g^2+3 e f h g+e^2 h^2\right ) d^2\right )-c f h (f g+e h) d+c^2 f^2 h^2\right )-C \left (2 f h (f g+e h) c^3+d \left (3 f^2 g^2+17 e f h g+3 e^2 h^2\right ) c^2+d^2 e g (f g+e h) c+d^3 e^2 g^2\right )\right ) a^2-b^3 \left (2 f h (5 C e g-4 A f h) c^3+d (f g+e h) (20 C e g-3 A f h) c^2+d^2 \left (10 C e^2 g^2+11 A f^2 g^2+14 A e f h g+11 A e^2 h^2\right ) c+19 A d^3 e g (f g+e h)\right ) a+b^4 \left (-4 A f h (f g+e h) c^3+d \left (15 C e^2 g^2+4 A f^2 g^2+A e f h g+4 A e^2 h^2\right ) c^2+3 A d^2 e g (f g+e h) c+8 A d^3 e^2 g^2\right )\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}}{(b c-a d) (b e-a f) (b g-a h)}}{3 (b c-a d) (b e-a f) (b g-a h)}}{5 (b c-a d) (b e-a f) (b g-a h)}\)

\(\Big \downarrow \) 327

\(\displaystyle -\frac {2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (C a^2+A b^2\right )}{5 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{5/2}}-\frac {-\frac {4 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (C d f h a^4+b C (d f g+d e h+c f h) a^3+3 b^2 (2 A d f h-C (d e g+c f g+c e h)) a^2-b^3 (4 A d (f g+e h)-c (5 C e g-4 A f h)) a+2 A b^4 (d e g+c f g+c e h)\right )}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {-\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (4 a \left (C a^2+A b^2\right ) d f h \left ((d f g+d e h+c f h) a^2-\frac {3}{2} b (d e g+c f g+c e h) a+\frac {3}{2} b^2 c e g\right )+b \left (3 d f h a^2-3 b (d f g+d e h+c f h) a+2 b^2 (d e g+c f g+c e h)\right ) \left ((5 A d f h-C (d e g+c f g+c e h)) a^2+5 b (c C e g-A d f g-A d e h-A c f h) a+4 A b^2 (d e g+c f g+c e h)\right )+2 a (3 a d f h-b (d f g+d e h+c f h)) \left (C (d f g+d e h+c f h) a^3+b (5 A d f h-3 C (d e g+c f g+c e h)) a^2-b^2 (4 A d (f g+e h)-c (5 C e g-4 A f h)) a+2 A b^3 (d e g+c f g+c e h)\right )+b (3 b c e g-a (d e g+c f g+c e h)) \left (2 C d f h a^3+2 b C (d f g+d e h+c f h) a^2+b^2 (7 A d f h-5 C (d e g+c f g+c e h)) a-b^3 (3 A d (f g+e h)-c (5 C e g-3 A f h))\right )\right )}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}-\frac {\frac {2 \sqrt {d g-c h} \sqrt {f g-e h} \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\arcsin \left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right ) \left (4 a \left (C a^2+A b^2\right ) d f h \left ((d f g+d e h+c f h) a^2-\frac {3}{2} b (d e g+c f g+c e h) a+\frac {3}{2} b^2 c e g\right )+b \left (3 d f h a^2-3 b (d f g+d e h+c f h) a+2 b^2 (d e g+c f g+c e h)\right ) \left ((5 A d f h-C (d e g+c f g+c e h)) a^2+5 b (c C e g-A d f g-A d e h-A c f h) a+4 A b^2 (d e g+c f g+c e h)\right )+2 a (3 a d f h-b (d f g+d e h+c f h)) \left (C (d f g+d e h+c f h) a^3+b (5 A d f h-3 C (d e g+c f g+c e h)) a^2-b^2 (4 A d (f g+e h)-c (5 C e g-4 A f h)) a+2 A b^3 (d e g+c f g+c e h)\right )+b (3 b c e g-a (d e g+c f g+c e h)) \left (2 C d f h a^3+2 b C (d f g+d e h+c f h) a^2+b^2 (7 A d f h-5 C (d e g+c f g+c e h)) a-b^3 (3 A d (f g+e h)-c (5 C e g-3 A f h))\right )\right )}{\sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {2 d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (4 a \left (C a^2+A b^2\right ) d f h \left ((d f g+d e h+c f h) a^2-\frac {3}{2} b (d e g+c f g+c e h) a+\frac {3}{2} b^2 c e g\right )+b \left (3 d f h a^2-3 b (d f g+d e h+c f h) a+2 b^2 (d e g+c f g+c e h)\right ) \left ((5 A d f h-C (d e g+c f g+c e h)) a^2+5 b (c C e g-A d f g-A d e h-A c f h) a+4 A b^2 (d e g+c f g+c e h)\right )+2 a (3 a d f h-b (d f g+d e h+c f h)) \left (C (d f g+d e h+c f h) a^3+b (5 A d f h-3 C (d e g+c f g+c e h)) a^2-b^2 (4 A d (f g+e h)-c (5 C e g-4 A f h)) a+2 A b^3 (d e g+c f g+c e h)\right )+b (3 b c e g-a (d e g+c f g+c e h)) \left (2 C d f h a^3+2 b C (d f g+d e h+c f h) a^2+b^2 (7 A d f h-5 C (d e g+c f g+c e h)) a-b^3 (3 A d (f g+e h)-c (5 C e g-3 A f h))\right )\right )}{\sqrt {c+d x}}+\frac {2 (b e-a f) \sqrt {b g-a h} \left (5 d f h \left (3 A d^2 f h-C \left (-2 f h c^2-d f g c-d e h c+d^2 e g\right )\right ) a^4-b \left (30 A f h (f g+e h) d^3+C \left (2 f^2 h^2 c^3+27 d f h (f g+e h) c^2+d^2 \left (f^2 g^2+4 e f h g+e^2 h^2\right ) c-d^3 e g (f g+e h)\right )\right ) a^3-3 b^2 \left (5 A d \left (-\left (\left (f^2 g^2+3 e f h g+e^2 h^2\right ) d^2\right )-c f h (f g+e h) d+c^2 f^2 h^2\right )-C \left (2 f h (f g+e h) c^3+d \left (3 f^2 g^2+17 e f h g+3 e^2 h^2\right ) c^2+d^2 e g (f g+e h) c+d^3 e^2 g^2\right )\right ) a^2-b^3 \left (2 f h (5 C e g-4 A f h) c^3+d (f g+e h) (20 C e g-3 A f h) c^2+d^2 \left (10 C e^2 g^2+11 A f^2 g^2+14 A e f h g+11 A e^2 h^2\right ) c+19 A d^3 e g (f g+e h)\right ) a+b^4 \left (-4 A f h (f g+e h) c^3+d \left (15 C e^2 g^2+4 A f^2 g^2+A e f h g+4 A e^2 h^2\right ) c^2+3 A d^2 e g (f g+e h) c+8 A d^3 e^2 g^2\right )\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}}{(b c-a d) (b e-a f) (b g-a h)}}{3 (b c-a d) (b e-a f) (b g-a h)}}{5 (b c-a d) (b e-a f) (b g-a h)}\)

Input:

Int[(A + C*x^2)/((a + b*x)^(7/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x] 
),x]
 

Output:

(-2*(A*b^2 + a^2*C)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(5*(b*c - a 
*d)*(b*e - a*f)*(b*g - a*h)*(a + b*x)^(5/2)) - ((-4*(a^4*C*d*f*h + 2*A*b^4 
*(d*e*g + c*f*g + c*e*h) + a^3*b*C*(d*f*g + d*e*h + c*f*h) + 3*a^2*b^2*(2* 
A*d*f*h - C*(d*e*g + c*f*g + c*e*h)) - a*b^3*(4*A*d*(f*g + e*h) - c*(5*C*e 
*g - 4*A*f*h)))*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*(b*c - a*d)* 
(b*e - a*f)*(b*g - a*h)*(a + b*x)^(3/2)) - ((-2*b*(4*a*(A*b^2 + a^2*C)*d*f 
*h*((3*b^2*c*e*g)/2 - (3*a*b*(d*e*g + c*f*g + c*e*h))/2 + a^2*(d*f*g + d*e 
*h + c*f*h)) + b*(3*a^2*d*f*h + 2*b^2*(d*e*g + c*f*g + c*e*h) - 3*a*b*(d*f 
*g + d*e*h + c*f*h))*(4*A*b^2*(d*e*g + c*f*g + c*e*h) + 5*a*b*(c*C*e*g - A 
*d*f*g - A*d*e*h - A*c*f*h) + a^2*(5*A*d*f*h - C*(d*e*g + c*f*g + c*e*h))) 
 + 2*a*(3*a*d*f*h - b*(d*f*g + d*e*h + c*f*h))*(2*A*b^3*(d*e*g + c*f*g + c 
*e*h) + a^3*C*(d*f*g + d*e*h + c*f*h) + a^2*b*(5*A*d*f*h - 3*C*(d*e*g + c* 
f*g + c*e*h)) - a*b^2*(4*A*d*(f*g + e*h) - c*(5*C*e*g - 4*A*f*h))) + b*(3* 
b*c*e*g - a*(d*e*g + c*f*g + c*e*h))*(2*a^3*C*d*f*h + 2*a^2*b*C*(d*f*g + d 
*e*h + c*f*h) + a*b^2*(7*A*d*f*h - 5*C*(d*e*g + c*f*g + c*e*h)) - b^3*(3*A 
*d*(f*g + e*h) - c*(5*C*e*g - 3*A*f*h))))*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt 
[g + h*x])/((b*c - a*d)*(b*e - a*f)*(b*g - a*h)*Sqrt[a + b*x]) - ((-2*d*(4 
*a*(A*b^2 + a^2*C)*d*f*h*((3*b^2*c*e*g)/2 - (3*a*b*(d*e*g + c*f*g + c*e*h) 
)/2 + a^2*(d*f*g + d*e*h + c*f*h)) + b*(3*a^2*d*f*h + 2*b^2*(d*e*g + c*f*g 
 + c*e*h) - 3*a*b*(d*f*g + d*e*h + c*f*h))*(4*A*b^2*(d*e*g + c*f*g + c*...
 

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 188

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

rule 194

Int[Sqrt[(c_.) + (d_.)*(x_)]/(((a_.) + (b_.)*(x_))^(3/2)*Sqrt[(e_.) + (f_.) 
*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2*Sqrt[c + d*x]*(Sqrt[(-(b*e 
 - a*f))*((g + h*x)/((f*g - e*h)*(a + b*x)))]/((b*e - a*f)*Sqrt[g + h*x]*Sq 
rt[(b*e - a*f)*((c + d*x)/((d*e - c*f)*(a + b*x)))]))   Subst[Int[Sqrt[1 + 
(b*c - a*d)*(x^2/(d*e - c*f))]/Sqrt[1 - (b*g - a*h)*(x^2/(f*g - e*h))], x], 
 x, Sqrt[e + f*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 2102

Int[(((a_.) + (b_.)*(x_))^(m_)*((A_.) + (B_.)*(x_)))/(Sqrt[(c_.) + (d_.)*(x 
_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbol] :> Simp[( 
A*b^2 - a*b*B)*(a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[e + f*x]*(Sqrt[g + h*x] 
/((m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h))), x] - Simp[1/(2*(m + 1)*(b* 
c - a*d)*(b*e - a*f)*(b*g - a*h))   Int[((a + b*x)^(m + 1)/(Sqrt[c + d*x]*S 
qrt[e + f*x]*Sqrt[g + h*x]))*Simp[A*(2*a^2*d*f*h*(m + 1) - 2*a*b*(m + 1)*(d 
*f*g + d*e*h + c*f*h) + b^2*(2*m + 3)*(d*e*g + c*f*g + c*e*h)) - b*B*(a*(d* 
e*g + c*f*g + c*e*h) + 2*b*c*e*g*(m + 1)) - 2*((A*b - a*B)*(a*d*f*h*(m + 1) 
 - b*(m + 2)*(d*f*g + d*e*h + c*f*h)))*x + d*f*h*(2*m + 5)*(A*b^2 - a*b*B)* 
x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B}, x] && IntegerQ[2*m 
] && LtQ[m, -1]
 

rule 2105

Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_. 
) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbo 
l] :> Simp[C*Sqrt[a + b*x]*Sqrt[e + f*x]*(Sqrt[g + h*x]/(b*f*h*Sqrt[c + d*x 
])), x] + (Simp[1/(2*b*d*f*h)   Int[(1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e 
+ f*x]*Sqrt[g + h*x]))*Simp[2*A*b*d*f*h - C*(b*d*e*g + a*c*f*h) + (2*b*B*d* 
f*h - C*(a*d*f*h + b*(d*f*g + d*e*h + c*f*h)))*x, x], x], x] + Simp[C*(d*e 
- c*f)*((d*g - c*h)/(2*b*d*f*h))   Int[Sqrt[a + b*x]/((c + d*x)^(3/2)*Sqrt[ 
e + f*x]*Sqrt[g + h*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, A, B, C} 
, x]
 

rule 2107

Int[(((a_.) + (b_.)*(x_))^(m_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2))/(Sqrt[( 
c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sy 
mbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*(a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[ 
e + f*x]*(Sqrt[g + h*x]/((m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h))), x] 
- Simp[1/(2*(m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h))   Int[((a + b*x)^( 
m + 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[A*(2*a^2*d*f*h*(m 
+ 1) - 2*a*b*(m + 1)*(d*f*g + d*e*h + c*f*h) + b^2*(2*m + 3)*(d*e*g + c*f*g 
 + c*e*h)) - (b*B - a*C)*(a*(d*e*g + c*f*g + c*e*h) + 2*b*c*e*g*(m + 1)) - 
2*((A*b - a*B)*(a*d*f*h*(m + 1) - b*(m + 2)*(d*f*g + d*e*h + c*f*h)) - C*(a 
^2*(d*f*g + d*e*h + c*f*h) - b^2*c*e*g*(m + 1) + a*b*(m + 1)*(d*e*g + c*f*g 
 + c*e*h)))*x + d*f*h*(2*m + 5)*(A*b^2 - a*b*B + a^2*C)*x^2, x], x], x] /; 
FreeQ[{a, b, c, d, e, f, g, h, A, B, C}, x] && IntegerQ[2*m] && LtQ[m, -1]
 

rule 2108

Int[(((a_.) + (b_.)*(x_))^(m_)*((A_.) + (C_.)*(x_)^2))/(Sqrt[(c_.) + (d_.)* 
(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbol] :> Simp 
[(A*b^2 + a^2*C)*(a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[e + f*x]*(Sqrt[g + h* 
x]/((m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h))), x] - Simp[1/(2*(m + 1)*( 
b*c - a*d)*(b*e - a*f)*(b*g - a*h))   Int[((a + b*x)^(m + 1)/(Sqrt[c + d*x] 
*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[A*(2*a^2*d*f*h*(m + 1) - 2*a*b*(m + 1)* 
(d*f*g + d*e*h + c*f*h) + b^2*(2*m + 3)*(d*e*g + c*f*g + c*e*h)) + a*C*(a*( 
d*e*g + c*f*g + c*e*h) + 2*b*c*e*g*(m + 1)) - 2*(A*b*(a*d*f*h*(m + 1) - b*( 
m + 2)*(d*f*g + d*e*h + c*f*h)) - C*(a^2*(d*f*g + d*e*h + c*f*h) - b^2*c*e* 
g*(m + 1) + a*b*(m + 1)*(d*e*g + c*f*g + c*e*h)))*x + d*f*h*(2*m + 5)*(A*b^ 
2 + a^2*C)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, C}, x] && I 
ntegerQ[2*m] && LtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(7992\) vs. \(2(1551)=3102\).

Time = 41.69 (sec) , antiderivative size = 7993, normalized size of antiderivative = 4.94

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(7993\)
default	\(\text {Expression too large to display}\)	\(355547\)

Input:

int((C*x^2+A)/(b*x+a)^(7/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x,me 
thod=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [F]

\[ \int \frac {A+C x^2}{(a+b x)^{7/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {C x^{2} + A}{{\left (b x + a\right )}^{\frac {7}{2}} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \] Input:

integrate((C*x^2+A)/(b*x+a)^(7/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2 
),x, algorithm="fricas")
 

Output:

integral((C*x^2 + A)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + 
g)/(b^4*d*f*h*x^7 + a^4*c*e*g + (b^4*d*f*g + (b^4*d*e + (b^4*c + 4*a*b^3*d 
)*f)*h)*x^6 + ((b^4*d*e + (b^4*c + 4*a*b^3*d)*f)*g + ((b^4*c + 4*a*b^3*d)* 
e + 2*(2*a*b^3*c + 3*a^2*b^2*d)*f)*h)*x^5 + (((b^4*c + 4*a*b^3*d)*e + 2*(2 
*a*b^3*c + 3*a^2*b^2*d)*f)*g + 2*((2*a*b^3*c + 3*a^2*b^2*d)*e + (3*a^2*b^2 
*c + 2*a^3*b*d)*f)*h)*x^4 + (2*((2*a*b^3*c + 3*a^2*b^2*d)*e + (3*a^2*b^2*c 
 + 2*a^3*b*d)*f)*g + (2*(3*a^2*b^2*c + 2*a^3*b*d)*e + (4*a^3*b*c + a^4*d)* 
f)*h)*x^3 + ((2*(3*a^2*b^2*c + 2*a^3*b*d)*e + (4*a^3*b*c + a^4*d)*f)*g + ( 
a^4*c*f + (4*a^3*b*c + a^4*d)*e)*h)*x^2 + (a^4*c*e*h + (a^4*c*f + (4*a^3*b 
*c + a^4*d)*e)*g)*x), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+C x^2}{(a+b x)^{7/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \] Input:

integrate((C*x**2+A)/(b*x+a)**(7/2)/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x+g)* 
*(1/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {A+C x^2}{(a+b x)^{7/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {C x^{2} + A}{{\left (b x + a\right )}^{\frac {7}{2}} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \] Input:

integrate((C*x^2+A)/(b*x+a)^(7/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2 
),x, algorithm="maxima")
 

Output:

integrate((C*x^2 + A)/((b*x + a)^(7/2)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h* 
x + g)), x)
 

Giac [F]

\[ \int \frac {A+C x^2}{(a+b x)^{7/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {C x^{2} + A}{{\left (b x + a\right )}^{\frac {7}{2}} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \] Input:

integrate((C*x^2+A)/(b*x+a)^(7/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2 
),x, algorithm="giac")
 

Output:

integrate((C*x^2 + A)/((b*x + a)^(7/2)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h* 
x + g)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+C x^2}{(a+b x)^{7/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {C\,x^2+A}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,{\left (a+b\,x\right )}^{7/2}\,\sqrt {c+d\,x}} \,d x \] Input:

int((A + C*x^2)/((e + f*x)^(1/2)*(g + h*x)^(1/2)*(a + b*x)^(7/2)*(c + d*x) 
^(1/2)),x)
 

Output:

int((A + C*x^2)/((e + f*x)^(1/2)*(g + h*x)^(1/2)*(a + b*x)^(7/2)*(c + d*x) 
^(1/2)), x)
 

Reduce [F]

\[ \int \frac {A+C x^2}{(a+b x)^{7/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {C \,x^{2}+A}{\left (b x +a \right )^{\frac {7}{2}} \sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}d x \] Input:

int((C*x^2+A)/(b*x+a)^(7/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x)
 

Output:

int((C*x^2+A)/(b*x+a)^(7/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x)