\(\int \frac {A+C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [36]

Optimal result

Integrand size = 44, antiderivative size = 727 \[ \int \frac {A+C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 \left (A b^2+a^2 C\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}+\frac {4 (d g-c h) \left (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\right ) \sqrt {e+f x} \sqrt {-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}} E\left (\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {d e-c f} \sqrt {a+b x}}\right )|\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{3 (b c-a d)^2 (b e-a f)^{3/2} \sqrt {d e-c f} (b g-a h)^2 \sqrt {-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}}-\frac {2 \left (3 a b (d e+c f) \left (C g^2+A h^2\right )-a^2 \left (3 A d f h^2+c C h (f g-e h)+C d g (2 f g+e h)\right )-b^2 \left (3 c C e g^2-A d g (f g-e h)+A c h (f g+2 e h)\right )\right ) \sqrt {e+f x} \sqrt {-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {d e-c f} \sqrt {a+b x}}\right ),\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{3 (b c-a d) (b e-a f)^{3/2} \sqrt {d e-c f} (b g-a h)^2 \sqrt {-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}} \] Output:

-2/3*(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)^(3/2)+4/3*(-c*h+d*g)*(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*(3*A*d*f*h-2*C*(c*e*h+c*f*g+d*e*g))-a*b^ 
2*(2*A*d*(e*h+f*g)-c*(-2*A*f*h+3*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)^2/(-a*f+b*e)^(3/2)/(-c*f+d*e)^(1/2)/(-a*h+b*g)^2/(-(-a*d+ 
b*c)*(f*x+e)/(-c*f+d*e)/(b*x+a))^(1/2)/(h*x+g)^(1/2)-2/3*(3*a*b*(c*f+d*e)* 
(A*h^2+C*g^2)-a^2*(3*A*d*f*h^2+c*C*h*(-e*h+f*g)+C*d*g*(e*h+2*f*g))-b^2*(3* 
c*C*e*g^2-A*d*g*(-e*h+f*g)+A*c*h*(2*e*h+f*g)))*(f*x+e)^(1/2)*(-(-a*d+b*c)* 
(h*x+g)/(-c*h+d*g)/(b*x+a))^(1/2)*EllipticF((-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)/(-a*f+b*e)^(3/2)/(-c*f+d*e)^(1/2)/(-a*h+b*g)^2/(-(-a 
*d+b*c)*(f*x+e)/(-c*f+d*e)/(b*x+a))^(1/2)/(h*x+g)^(1/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(11363\) vs. \(2(727)=1454\).

Time = 40.46 (sec) , antiderivative size = 11363, normalized size of antiderivative = 15.63 \[ \int \frac {A+C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Result too large to show} \] Input:

Integrate[(A + C*x^2)/((a + b*x)^(5/2)*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 = 3.93 (sec) , antiderivative size = 1057, normalized size of antiderivative = 1.45, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2108, 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.

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

\(\Big \downarrow \) 2108

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2102

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2105

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 188

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

\(\Big \downarrow \) 194

\(\displaystyle -\frac {2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (C a^2+A b^2\right )}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {-\frac {4 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (C (d f g+d e h+c f h) a^3+b (3 A d f h-2 C (d e g+c f g+c e h)) a^2-b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h)) a+A b^3 (d e g+c f g+c e h)\right )}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}-\frac {\frac {4 (d g-c h) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} \int \frac {\sqrt {1-\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}}}{\sqrt {1-\frac {(d g-c h) (e+f x)}{(f g-e h) (c+d x)}}}d\frac {\sqrt {e+f x}}{\sqrt {c+d x}} \left (C (d f g+d e h+c f h) a^3+b (3 A d f h-2 C (d e g+c f g+c e h)) a^2-b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h)) a+A b^3 (d e g+c f g+c e h)\right )}{\sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {4 d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (C (d f g+d e h+c f h) a^3+b (3 A d f h-2 C (d e g+c f g+c e h)) a^2-b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h)) a+A b^3 (d e g+c f g+c e h)\right )}{\sqrt {c+d x}}-\frac {2 (b e-a f) (b g-a h) \left (-\left (\left (3 A d^2 f h-C \left (-2 f h c^2-d (f g+e h) c+d^2 e g\right )\right ) a^2\right )+3 b \left (C c^2+A d^2\right ) (f g+e h) a-b^2 \left ((3 C e g-A f h) c^2+A d (f g+e h) c+2 A d^2 e g\right )\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \int \frac {1}{\sqrt {\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}+1} \sqrt {1-\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}}}d\frac {\sqrt {e+f x}}{\sqrt {a+b x}}}{(f g-e h) \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}}{(b c-a d) (b e-a f) (b g-a h)}}{3 (b c-a d) (b e-a f) (b g-a h)}\)

\(\Big \downarrow \) 321

\(\displaystyle -\frac {2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (C a^2+A b^2\right )}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {-\frac {4 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (C (d f g+d e h+c f h) a^3+b (3 A d f h-2 C (d e g+c f g+c e h)) a^2-b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h)) a+A b^3 (d e g+c f g+c e h)\right )}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}-\frac {\frac {4 (d g-c h) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} \int \frac {\sqrt {1-\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}}}{\sqrt {1-\frac {(d g-c h) (e+f x)}{(f g-e h) (c+d x)}}}d\frac {\sqrt {e+f x}}{\sqrt {c+d x}} \left (C (d f g+d e h+c f h) a^3+b (3 A d f h-2 C (d e g+c f g+c e h)) a^2-b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h)) a+A b^3 (d e g+c f g+c e h)\right )}{\sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {4 d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (C (d f g+d e h+c f h) a^3+b (3 A d f h-2 C (d e g+c f g+c e h)) a^2-b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h)) a+A b^3 (d e g+c f g+c e h)\right )}{\sqrt {c+d x}}-\frac {2 (b e-a f) \sqrt {b g-a h} \left (-\left (\left (3 A d^2 f h-C \left (-2 f h c^2-d (f g+e h) c+d^2 e g\right )\right ) a^2\right )+3 b \left (C c^2+A d^2\right ) (f g+e h) a-b^2 \left ((3 C e g-A f h) c^2+A d (f g+e h) c+2 A d^2 e g\right )\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}}{(b c-a d) (b e-a f) (b g-a h)}}{3 (b c-a d) (b e-a f) (b g-a h)}\)

\(\Big \downarrow \) 327

\(\displaystyle -\frac {2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (C a^2+A b^2\right )}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {-\frac {4 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (C (d f g+d e h+c f h) a^3+b (3 A d f h-2 C (d e g+c f g+c e h)) a^2-b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h)) a+A b^3 (d e g+c f g+c e h)\right )}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}-\frac {\frac {4 \sqrt {d g-c h} \sqrt {f g-e h} \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\arcsin \left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right ) \left (C (d f g+d e h+c f h) a^3+b (3 A d f h-2 C (d e g+c f g+c e h)) a^2-b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h)) a+A b^3 (d e g+c f g+c e h)\right )}{\sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {4 d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (C (d f g+d e h+c f h) a^3+b (3 A d f h-2 C (d e g+c f g+c e h)) a^2-b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h)) a+A b^3 (d e g+c f g+c e h)\right )}{\sqrt {c+d x}}-\frac {2 (b e-a f) \sqrt {b g-a h} \left (-\left (\left (3 A d^2 f h-C \left (-2 f h c^2-d (f g+e h) c+d^2 e g\right )\right ) a^2\right )+3 b \left (C c^2+A d^2\right ) (f g+e h) a-b^2 \left ((3 C e g-A f h) c^2+A d (f g+e h) c+2 A d^2 e g\right )\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}}{(b c-a d) (b e-a f) (b g-a h)}}{3 (b c-a d) (b e-a f) (b g-a h)}\)

Input:

Int[(A + C*x^2)/((a + b*x)^(5/2)*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])/(3*(b*c - a 
*d)*(b*e - a*f)*(b*g - a*h)*(a + b*x)^(3/2)) - ((-4*b*(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*(3*A*d*f*h - 2*C*(d*e*g 
 + c*f*g + c*e*h)) - a*b^2*(2*A*d*(f*g + e*h) - c*(3*C*e*g - 2*A*f*h)))*Sq 
rt[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]) - ((-4*d*(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*(3*A*d*f*h - 2*C*(d*e*g + c*f*g + c*e*h)) - a*b^2 
*(2*A*d*(f*g + e*h) - c*(3*C*e*g - 2*A*f*h)))*Sqrt[a + b*x]*Sqrt[e + f*x]* 
Sqrt[g + h*x])/Sqrt[c + d*x] + (4*Sqrt[d*g - c*h]*Sqrt[f*g - e*h]*(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*(3*A*d*f*h 
- 2*C*(d*e*g + c*f*g + c*e*h)) - a*b^2*(2*A*d*(f*g + e*h) - c*(3*C*e*g - 2 
*A*f*h)))*Sqrt[a + b*x]*Sqrt[-(((d*e - c*f)*(g + h*x))/((f*g - e*h)*(c + d 
*x)))]*EllipticE[ArcSin[(Sqrt[d*g - c*h]*Sqrt[e + f*x])/(Sqrt[f*g - e*h]*S 
qrt[c + d*x])], ((b*c - a*d)*(f*g - e*h))/((b*e - a*f)*(d*g - c*h))])/(Sqr 
t[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))]*Sqrt[g + h*x]) - (2*(b* 
e - a*f)*Sqrt[b*g - a*h]*(3*a*b*(c^2*C + A*d^2)*(f*g + e*h) - b^2*(2*A*d^2 
*e*g + A*c*d*(f*g + e*h) + c^2*(3*C*e*g - A*f*h)) - a^2*(3*A*d^2*f*h - C*( 
d^2*e*g - 2*c^2*f*h - c*d*(f*g + e*h))))*Sqrt[((b*e - a*f)*(c + d*x))/((d* 
e - c*f)*(a + b*x))]*Sqrt[g + h*x]*EllipticF[ArcSin[(Sqrt[b*g - a*h]*Sqrt[ 
e + f*x])/(Sqrt[f*g - e*h]*Sqrt[a + b*x])], -(((b*c - a*d)*(f*g - e*h))...
 

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 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. \(3341\) vs. \(2(675)=1350\).

Time = 28.05 (sec) , antiderivative size = 3342, normalized size of antiderivative = 4.60

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(3342\)
default	\(\text {Expression too large to display}\)	\(112033\)

Input:

int((C*x^2+A)/(b*x+a)^(5/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x,me 
thod=_RETURNVERBOSE)
 

Output:

((h*x+g)*(d*x+c)*(b*x+a)*(f*x+e))^(1/2)/(h*x+g)^(1/2)/(d*x+c)^(1/2)/(b*x+a 
)^(1/2)/(f*x+e)^(1/2)*(2/3/(a^3*d*f*h-a^2*b*c*f*h-a^2*b*d*e*h-a^2*b*d*f*g+ 
a*b^2*c*e*h+a*b^2*c*f*g+a*b^2*d*e*g-b^3*c*e*g)/b^2*(A*b^2+C*a^2)*(b*d*f*h* 
x^4+a*d*f*h*x^3+b*c*f*h*x^3+b*d*e*h*x^3+b*d*f*g*x^3+a*c*f*h*x^2+a*d*e*h*x^ 
2+a*d*f*g*x^2+b*c*e*h*x^2+b*c*f*g*x^2+b*d*e*g*x^2+a*c*e*h*x+a*c*f*g*x+a*d* 
e*g*x+b*c*e*g*x+a*c*e*g)^(1/2)/(x+a/b)^2+4/3*(b*d*f*h*x^3+b*c*f*h*x^2+b*d* 
e*h*x^2+b*d*f*g*x^2+b*c*e*h*x+b*c*f*g*x+b*d*e*g*x+b*c*e*g)/(a^3*d*f*h-a^2* 
b*c*f*h-a^2*b*d*e*h-a^2*b*d*f*g+a*b^2*c*e*h+a*b^2*c*f*g+a*b^2*d*e*g-b^3*c* 
e*g)^2*(3*A*a^2*b*d*f*h-2*A*a*b^2*c*f*h-2*A*a*b^2*d*e*h-2*A*a*b^2*d*f*g+A* 
b^3*c*e*h+A*b^3*c*f*g+A*b^3*d*e*g+C*a^3*c*f*h+C*a^3*d*e*h+C*a^3*d*f*g-2*C* 
a^2*b*c*e*h-2*C*a^2*b*c*f*g-2*C*a^2*b*d*e*g+3*C*a*b^2*c*e*g)/((x+a/b)*(b*d 
*f*h*x^3+b*c*f*h*x^2+b*d*e*h*x^2+b*d*f*g*x^2+b*c*e*h*x+b*c*f*g*x+b*d*e*g*x 
+b*c*e*g))^(1/2)+2*(C/b^2-1/3/b^2*(3*A*a*b^2*d*f*h-A*b^3*c*f*h-A*b^3*d*e*h 
-A*b^3*d*f*g+3*C*a^3*d*f*h-C*a^2*b*c*f*h-C*a^2*b*d*e*h-C*a^2*b*d*f*g)/(a^3 
*d*f*h-a^2*b*c*f*h-a^2*b*d*e*h-a^2*b*d*f*g+a*b^2*c*e*h+a*b^2*c*f*g+a*b^2*d 
*e*g-b^3*c*e*g)+2/3/b*(a^2*d*f*h-a*b*c*f*h-a*b*d*e*h-a*b*d*f*g+b^2*c*e*h+b 
^2*c*f*g+b^2*d*e*g)*(3*A*a^2*b*d*f*h-2*A*a*b^2*c*f*h-2*A*a*b^2*d*e*h-2*A*a 
*b^2*d*f*g+A*b^3*c*e*h+A*b^3*c*f*g+A*b^3*d*e*g+C*a^3*c*f*h+C*a^3*d*e*h+C*a 
^3*d*f*g-2*C*a^2*b*c*e*h-2*C*a^2*b*c*f*g-2*C*a^2*b*d*e*g+3*C*a*b^2*c*e*g)/ 
(a^3*d*f*h-a^2*b*c*f*h-a^2*b*d*e*h-a^2*b*d*f*g+a*b^2*c*e*h+a*b^2*c*f*g+...
 

Fricas [F]

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

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

Sympy [F(-1)]

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

integrate((C*x**2+A)/(b*x+a)**(5/2)/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x+g)* 
*(1/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((C*x^2+A)/(b*x+a)^(5/2)/(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)^(5/2)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h* 
x + g)), x)
 

Giac [F]

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

integrate((C*x^2+A)/(b*x+a)^(5/2)/(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)^(5/2)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h* 
x + g)), x)
 

Mupad [F(-1)]

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

int((A + C*x^2)/((e + f*x)^(1/2)*(g + h*x)^(1/2)*(a + b*x)^(5/2)*(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)^(5/2)*(c + d*x) 
^(1/2)), x)
 

Reduce [F]

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

int((C*x^2+A)/(b*x+a)^(5/2)/(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)^(5/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x)