\(\int \frac {A+B x+C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [42]

Optimal result

Integrand size = 47, antiderivative size = 827 \[ \int \frac {A+B x+C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {2 (d g-c h) \left (b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a^2 b (B d f g+B d e h+B c f h+6 A d f h-4 C (d e g+c f g+c e h))-a b^2 (B d e g-4 A d (f g+e h)+c (6 C e g+B f g+B e h-4 A f h))+a^3 (3 B d f h-2 C (d f g+d e h+c f h))\right ) \sqrt {e+f x} \sqrt {-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}} E\left (\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {d e-c f} \sqrt {a+b x}}\right )|\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{3 (b c-a d)^2 (b e-a f)^{3/2} \sqrt {d e-c f} (b g-a h)^2 \sqrt {-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}}-\frac {2 \left (a^2 (3 d f h (B g-A h)-c C h (f g-e h)-C d g (2 f g+e h))+a b \left (3 C (d e+c f) g^2+3 A (d e+c f) h^2-B c h (2 f g+e h)-B d g (f g+2 e h)\right )-b^2 \left (3 c C e g^2-A d g (f g-e h)-c h (3 B e g-A f g-2 A e h)\right )\right ) \sqrt {e+f x} \sqrt {-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {d e-c f} \sqrt {a+b x}}\right ),\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{3 (b c-a d) (b e-a f)^{3/2} \sqrt {d e-c f} (b g-a h)^2 \sqrt {-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}} \] Output:

-2/3*(A*b^2-a*(B*b-C*a))*(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)-2/3*(-c*h+d*g)*(b^3*(3*B*c*e*g-2*A 
*(c*e*h+c*f*g+d*e*g))-a^2*b*(B*d*f*g+B*d*e*h+B*c*f*h+6*A*d*f*h-4*C*(c*e*h+ 
c*f*g+d*e*g))-a*b^2*(B*d*e*g-4*A*d*(e*h+f*g)+c*(-4*A*f*h+B*e*h+B*f*g+6*C*e 
*g))+a^3*(3*B*d*f*h-2*C*(c*f*h+d*e*h+d*f*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*(a^2*(3*d*f*h 
*(-A*h+B*g)-c*C*h*(-e*h+f*g)-C*d*g*(e*h+2*f*g))+a*b*(3*C*(c*f+d*e)*g^2+3*A 
*(c*f+d*e)*h^2-B*c*h*(e*h+2*f*g)-B*d*g*(2*e*h+f*g))-b^2*(3*c*C*e*g^2-A*d*g 
*(-e*h+f*g)-c*h*(-2*A*e*h-A*f*g+3*B*e*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. \(16787\) vs. \(2(827)=1654\).

Time = 43.42 (sec) , antiderivative size = 16787, normalized size of antiderivative = 20.30 \[ \int \frac {A+B x+C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Result too large to show} \] Input:

Integrate[(A + B*x + C*x^2)/((a + b*x)^(5/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*S 
qrt[g + h*x]),x]
 

Output:

Result too large to show
 

Rubi [A] (warning: unable to verify)

Time = 6.53 (sec) , antiderivative size = 1243, normalized size of antiderivative = 1.50, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.170, Rules used = {2107, 2102, 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.

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

\(\Big \downarrow \) 2107

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

\(\Big \downarrow \) 2102

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

\(\Big \downarrow \) 2105

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 188

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

\(\Big \downarrow \) 194

\(\displaystyle \frac {\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 ((3 B d f h-2 C (d f g+d e h+c f h)) a^3-b (6 A d f h-4 C (d e g+c f g+c e h)+B (d f g+d e h+c f h)) a^2-b^2 (B d e g-4 A d (f g+e h)+c (6 C e g+B f g+B e h-4 A f h)) a+b^3 (3 B c e g-2 A (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 {2 d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left ((3 B d f h-2 C (d f g+d e h+c f h)) a^3-b (6 A d f h-4 C (d e g+c f g+c e h)+B (d f g+d e h+c f h)) a^2-b^2 (B d e g-4 A d (f g+e h)+c (6 C e g+B f g+B e h-4 A f h)) a+b^3 (3 B c e g-2 A (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 d (B c-A d) f h+C \left (-2 f h c^2-d f g c-d e h c+d^2 e g\right )\right ) a^2\right )-b \left (3 \left (C c^2+A d^2\right ) (f g+e h)-B \left (f h c^2+2 d (f g+e h) c+d^2 e g\right )\right ) a+b^2 \left ((3 C e g-A f h) c^2-d (3 B e g-A f g-A 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)}-\frac {2 b \left ((3 B d f h-2 C (d f g+d e h+c f h)) a^3-b (B d f g+B d e h+B c f h+6 A d f h-4 C (d e g+c f g+c e h)) a^2-b^2 (B d e g-4 A d (f g+e h)+c (6 C e g+B f g+B e h-4 A f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\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)}-\frac {2 \left (A b^2-a (b B-a 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}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\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 ((3 B d f h-2 C (d f g+d e h+c f h)) a^3-b (6 A d f h-4 C (d e g+c f g+c e h)+B (d f g+d e h+c f h)) a^2-b^2 (B d e g-4 A d (f g+e h)+c (6 C e g+B f g+B e h-4 A f h)) a+b^3 (3 B c e g-2 A (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 {2 d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left ((3 B d f h-2 C (d f g+d e h+c f h)) a^3-b (6 A d f h-4 C (d e g+c f g+c e h)+B (d f g+d e h+c f h)) a^2-b^2 (B d e g-4 A d (f g+e h)+c (6 C e g+B f g+B e h-4 A f h)) a+b^3 (3 B c e g-2 A (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 d (B c-A d) f h+C \left (-2 f h c^2-d f g c-d e h c+d^2 e g\right )\right ) a^2\right )-b \left (3 \left (C c^2+A d^2\right ) (f g+e h)-B \left (f h c^2+2 d (f g+e h) c+d^2 e g\right )\right ) a+b^2 \left ((3 C e g-A f h) c^2-d (3 B e g-A f g-A 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)}-\frac {2 b \left ((3 B d f h-2 C (d f g+d e h+c f h)) a^3-b (B d f g+B d e h+B c f h+6 A d f h-4 C (d e g+c f g+c e h)) a^2-b^2 (B d e g-4 A d (f g+e h)+c (6 C e g+B f g+B e h-4 A f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\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)}-\frac {2 \left (A b^2-a (b B-a 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}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\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 ((3 B d f h-2 C (d f g+d e h+c f h)) a^3-b (6 A d f h-4 C (d e g+c f g+c e h)+B (d f g+d e h+c f h)) a^2-b^2 (B d e g-4 A d (f g+e h)+c (6 C e g+B f g+B e h-4 A f h)) a+b^3 (3 B c e g-2 A (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 {2 d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left ((3 B d f h-2 C (d f g+d e h+c f h)) a^3-b (6 A d f h-4 C (d e g+c f g+c e h)+B (d f g+d e h+c f h)) a^2-b^2 (B d e g-4 A d (f g+e h)+c (6 C e g+B f g+B e h-4 A f h)) a+b^3 (3 B c e g-2 A (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 d (B c-A d) f h+C \left (-2 f h c^2-d f g c-d e h c+d^2 e g\right )\right ) a^2\right )-b \left (3 \left (C c^2+A d^2\right ) (f g+e h)-B \left (f h c^2+2 d (f g+e h) c+d^2 e g\right )\right ) a+b^2 \left ((3 C e g-A f h) c^2-d (3 B e g-A f g-A 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)}-\frac {2 b \left ((3 B d f h-2 C (d f g+d e h+c f h)) a^3-b (B d f g+B d e h+B c f h+6 A d f h-4 C (d e g+c f g+c e h)) a^2-b^2 (B d e g-4 A d (f g+e h)+c (6 C e g+B f g+B e h-4 A f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\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)}-\frac {2 \left (A b^2-a (b B-a 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}}\)

Input:

Int[(A + B*x + 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*(b*B - a*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)) + ((-2*b*(b^3*(3*B*c* 
e*g - 2*A*(d*e*g + c*f*g + c*e*h)) - a^2*b*(B*d*f*g + B*d*e*h + B*c*f*h + 
6*A*d*f*h - 4*C*(d*e*g + c*f*g + c*e*h)) - a*b^2*(B*d*e*g - 4*A*d*(f*g + e 
*h) + c*(6*C*e*g + B*f*g + B*e*h - 4*A*f*h)) + a^3*(3*B*d*f*h - 2*C*(d*f*g 
 + d*e*h + c*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*(b^3*(3*B*c*e*g - 2*A*(d* 
e*g + c*f*g + c*e*h)) - a*b^2*(B*d*e*g - 4*A*d*(f*g + e*h) + c*(6*C*e*g + 
B*f*g + B*e*h - 4*A*f*h)) - a^2*b*(6*A*d*f*h - 4*C*(d*e*g + c*f*g + c*e*h) 
 + B*(d*f*g + d*e*h + c*f*h)) + a^3*(3*B*d*f*h - 2*C*(d*f*g + d*e*h + c*f* 
h)))*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/Sqrt[c + d*x] - (2*Sqrt[d* 
g - c*h]*Sqrt[f*g - e*h]*(b^3*(3*B*c*e*g - 2*A*(d*e*g + c*f*g + c*e*h)) - 
a*b^2*(B*d*e*g - 4*A*d*(f*g + e*h) + c*(6*C*e*g + B*f*g + B*e*h - 4*A*f*h) 
) - a^2*b*(6*A*d*f*h - 4*C*(d*e*g + c*f*g + c*e*h) + B*(d*f*g + d*e*h + c* 
f*h)) + a^3*(3*B*d*f*h - 2*C*(d*f*g + d*e*h + c*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]*Sqrt[c + d*x])], ((b*c - a*d)* 
(f*g - e*h))/((b*e - a*f)*(d*g - c*h))])/(Sqrt[((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]*(b^2 
*(2*A*d^2*e*g - c*d*(3*B*e*g - A*f*g - A*e*h) + c^2*(3*C*e*g - A*f*h)) ...
 

Defintions of rubi rules used

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3851\) vs. \(2(775)=1550\).

Time = 38.85 (sec) , antiderivative size = 3852, normalized size of antiderivative = 4.66

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(3852\)
default	\(\text {Expression too large to display}\)	\(168816\)

Input:

int((C*x^2+B*x+A)/(b*x+a)^(5/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2), 
x,method=_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-B*a*b+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+2/3*(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*(6*A*a^2*b*d*f*h-4*A*a*b^2*c*f*h-4*A*a*b^2*d*e*h-4*A*a*b^2*d* 
f*g+2*A*b^3*c*e*h+2*A*b^3*c*f*g+2*A*b^3*d*e*g-3*B*a^3*d*f*h+B*a^2*b*c*f*h+ 
B*a^2*b*d*e*h+B*a^2*b*d*f*g+B*a*b^2*c*e*h+B*a*b^2*c*f*g+B*a*b^2*d*e*g-3*B* 
b^3*c*e*g+2*C*a^3*c*f*h+2*C*a^3*d*e*h+2*C*a^3*d*f*g-4*C*a^2*b*c*e*h-4*C*a^ 
2*b*c*f*g-4*C*a^2*b*d*e*g+6*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*B*a 
^2*b*d*f*h+B*a*b^2*c*f*h+B*a*b^2*d*e*h+B*a*b^2*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)+1/3/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)*(6*A*a^2*b*d*f 
*h-4*A*a*b^2*c*f*h-4*A*a*b^2*d*e*h-4*A*a*b^2*d*f*g+2*A*b^3*c*e*h+2*A*b^...
 

Fricas [F]

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

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

integrate((C*x**2+B*x+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]

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

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

Giac [F]

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

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

Mupad [F(-1)]

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

int((A + B*x + 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 + B*x + 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]

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

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