\(\int \frac {(a+b x) (A+C x^2)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [28]

Optimal result

Integrand size = 40, antiderivative size = 608 \[ \int \frac {(a+b x) \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {4 C (a d f h-2 b (d f g+d e h+c f h)) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{15 d^2 f^2 h^2}+\frac {2 C (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}+\frac {2 \sqrt {-d e+c f} \left (15 A b d^2 f^2 h^2-10 a C d f h (d f g+d e h+c f h)+b C \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{15 d^3 f^{5/2} h^3 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}+\frac {2 \sqrt {-d e+c f} \left (5 a d f h \left (3 A d f h^2+c C h (f g-e h)+C d g (2 f g+e h)\right )-b \left (15 A d^2 f^2 g h^2+C \left (4 c^2 f h^2 (f g-e h)+c d h \left (3 f^2 g^2+e f g h-4 e^2 h^2\right )+d^2 g \left (8 f^2 g^2+3 e f g h+4 e^2 h^2\right )\right )\right )\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{15 d^3 f^{5/2} h^3 \sqrt {e+f x} \sqrt {g+h x}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 26.45 (sec) , antiderivative size = 686, normalized size of antiderivative = 1.13 \[ \int \frac {(a+b x) \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 \left (-d^2 \sqrt {-c+\frac {d e}{f}} \left (15 A b d^2 f^2 h^2-10 a C d f h (d f g+d e h+c f h)+b C \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right ) (e+f x) (g+h x)+C d^2 \sqrt {-c+\frac {d e}{f}} f h (c+d x) (e+f x) (g+h x) (4 b c f h-5 a d f h+b d (4 f g+4 e h-3 f h x))-i (d e-c f) h \left (15 A b d^2 f^2 h^2-10 a C d f h (d f g+d e h+c f h)+b C \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right ) (c+d x)^{3/2} \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right )|\frac {d f g-c f h}{d e h-c f h}\right )-i d h \left (5 a d f h \left (3 A d f^2 h+c C f (-f g+e h)+C d e (f g+2 e h)\right )-b \left (15 A d^2 e f^2 h^2+C \left (4 c^2 f^2 h (-f g+e h)+c d f \left (-4 f^2 g^2+e f g h+3 e^2 h^2\right )+d^2 e \left (4 f^2 g^2+3 e f g h+8 e^2 h^2\right )\right )\right )\right ) (c+d x)^{3/2} \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right ),\frac {d f g-c f h}{d e h-c f h}\right )\right )}{15 d^4 \sqrt {-c+\frac {d e}{f}} f^3 h^3 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \] Input:

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

Output:

(-2*(-(d^2*Sqrt[-c + (d*e)/f]*(15*A*b*d^2*f^2*h^2 - 10*a*C*d*f*h*(d*f*g + 
d*e*h + c*f*h) + b*C*(8*c^2*f^2*h^2 + 7*c*d*f*h*(f*g + e*h) + d^2*(8*f^2*g 
^2 + 7*e*f*g*h + 8*e^2*h^2)))*(e + f*x)*(g + h*x)) + C*d^2*Sqrt[-c + (d*e) 
/f]*f*h*(c + d*x)*(e + f*x)*(g + h*x)*(4*b*c*f*h - 5*a*d*f*h + b*d*(4*f*g 
+ 4*e*h - 3*f*h*x)) - I*(d*e - c*f)*h*(15*A*b*d^2*f^2*h^2 - 10*a*C*d*f*h*( 
d*f*g + d*e*h + c*f*h) + b*C*(8*c^2*f^2*h^2 + 7*c*d*f*h*(f*g + e*h) + d^2* 
(8*f^2*g^2 + 7*e*f*g*h + 8*e^2*h^2)))*(c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/( 
f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticE[I*ArcSinh[Sqrt[- 
c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)] - I*d*h*(5*a 
*d*f*h*(3*A*d*f^2*h + c*C*f*(-(f*g) + e*h) + C*d*e*(f*g + 2*e*h)) - b*(15* 
A*d^2*e*f^2*h^2 + C*(4*c^2*f^2*h*(-(f*g) + e*h) + c*d*f*(-4*f^2*g^2 + e*f* 
g*h + 3*e^2*h^2) + d^2*e*(4*f^2*g^2 + 3*e*f*g*h + 8*e^2*h^2))))*(c + d*x)^ 
(3/2)*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]* 
EllipticF[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d* 
e*h - c*f*h)]))/(15*d^4*Sqrt[-c + (d*e)/f]*f^3*h^3*Sqrt[c + d*x]*Sqrt[e + 
f*x]*Sqrt[g + h*x])
 

Rubi [A] (verified)

Time = 1.53 (sec) , antiderivative size = 632, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2104, 25, 2118, 27, 176, 124, 123, 131, 131, 130}

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.

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

\(\Big \downarrow \) 2104

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2118

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 176

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

\(\Big \downarrow \) 124

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

\(\Big \downarrow \) 123

\(\displaystyle \frac {2 C (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}-\frac {-\frac {\frac {\left (5 a d f h \left (3 A d f h^2+c C h (f g-e h)+C d g (e h+2 f g)\right )-b \left (15 A d^2 f^2 g h^2+C \left (4 c^2 f h^2 (f g-e h)+c d h \left (-4 e^2 h^2+e f g h+3 f^2 g^2\right )+d^2 g \left (4 e^2 h^2+3 e f g h+8 f^2 g^2\right )\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{h}+\frac {2 \sqrt {g+h x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right ) \left (-10 a C d f h (c f h+d e h+d f g)+15 A b d^2 f^2 h^2+b C \left (8 c^2 f^2 h^2+7 c d f h (e h+f g)+d^2 \left (8 e^2 h^2+7 e f g h+8 f^2 g^2\right )\right )\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{3 d f h}-\frac {4 C \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (a d f h-2 b (c f h+d e h+d f g))}{3 d f h}}{5 d f h}\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {2 C (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}-\frac {-\frac {\frac {\sqrt {\frac {d (e+f x)}{d e-c f}} \left (5 a d f h \left (3 A d f h^2+c C h (f g-e h)+C d g (e h+2 f g)\right )-b \left (15 A d^2 f^2 g h^2+C \left (4 c^2 f h^2 (f g-e h)+c d h \left (-4 e^2 h^2+e f g h+3 f^2 g^2\right )+d^2 g \left (4 e^2 h^2+3 e f g h+8 f^2 g^2\right )\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {g+h x}}dx}{h \sqrt {e+f x}}+\frac {2 \sqrt {g+h x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right ) \left (-10 a C d f h (c f h+d e h+d f g)+15 A b d^2 f^2 h^2+b C \left (8 c^2 f^2 h^2+7 c d f h (e h+f g)+d^2 \left (8 e^2 h^2+7 e f g h+8 f^2 g^2\right )\right )\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{3 d f h}-\frac {4 C \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (a d f h-2 b (c f h+d e h+d f g))}{3 d f h}}{5 d f h}\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {2 C (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}-\frac {-\frac {\frac {\sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \left (5 a d f h \left (3 A d f h^2+c C h (f g-e h)+C d g (e h+2 f g)\right )-b \left (15 A d^2 f^2 g h^2+C \left (4 c^2 f h^2 (f g-e h)+c d h \left (-4 e^2 h^2+e f g h+3 f^2 g^2\right )+d^2 g \left (4 e^2 h^2+3 e f g h+8 f^2 g^2\right )\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}}dx}{h \sqrt {e+f x} \sqrt {g+h x}}+\frac {2 \sqrt {g+h x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right ) \left (-10 a C d f h (c f h+d e h+d f g)+15 A b d^2 f^2 h^2+b C \left (8 c^2 f^2 h^2+7 c d f h (e h+f g)+d^2 \left (8 e^2 h^2+7 e f g h+8 f^2 g^2\right )\right )\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{3 d f h}-\frac {4 C \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (a d f h-2 b (c f h+d e h+d f g))}{3 d f h}}{5 d f h}\)

\(\Big \downarrow \) 130

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

Input:

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

Output:

(2*C*(a + b*x)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(5*d*f*h) - ((-4 
*C*(a*d*f*h - 2*b*(d*f*g + d*e*h + c*f*h))*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqr 
t[g + h*x])/(3*d*f*h) - ((2*Sqrt[-(d*e) + c*f]*(15*A*b*d^2*f^2*h^2 - 10*a* 
C*d*f*h*(d*f*g + d*e*h + c*f*h) + b*C*(8*c^2*f^2*h^2 + 7*c*d*f*h*(f*g + e* 
h) + d^2*(8*f^2*g^2 + 7*e*f*g*h + 8*e^2*h^2)))*Sqrt[(d*(e + f*x))/(d*e - c 
*f)]*Sqrt[g + h*x]*EllipticE[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + 
c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/(d*Sqrt[f]*h*Sqrt[e + f*x]*Sqrt[( 
d*(g + h*x))/(d*g - c*h)]) + (2*Sqrt[-(d*e) + c*f]*(5*a*d*f*h*(3*A*d*f*h^2 
 + c*C*h*(f*g - e*h) + C*d*g*(2*f*g + e*h)) - b*(15*A*d^2*f^2*g*h^2 + C*(4 
*c^2*f*h^2*(f*g - e*h) + c*d*h*(3*f^2*g^2 + e*f*g*h - 4*e^2*h^2) + d^2*g*( 
8*f^2*g^2 + 3*e*f*g*h + 4*e^2*h^2))))*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt 
[(d*(g + h*x))/(d*g - c*h)]*EllipticF[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[ 
-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/(d*Sqrt[f]*h*Sqrt[e + f* 
x]*Sqrt[g + h*x]))/(3*d*f*h))/(5*d*f*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 123

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ 
)]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] 
/Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, 
b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !L 
tQ[-(b*c - a*d)/d, 0] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d 
), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)/b, 0])
 

rule 124

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ 
)]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d 
*x]*Sqrt[b*((e + f*x)/(b*e - a*f))]))   Int[Sqrt[b*(e/(b*e - a*f)) + b*f*(x 
/(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))] 
), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !(GtQ[b/(b*c - a*d), 0] && Gt 
Q[b/(b*e - a*f), 0]) &&  !LtQ[-(b*c - a*d)/d, 0]
 

rule 130

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x 
_)]), x_] :> Simp[2*(Rt[-b/d, 2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[ 
Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)/(d*(b*e - 
a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ 
[b/(b*e - a*f), 0] && SimplerQ[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f 
*x] && (PosQ[-(b*c - a*d)/d] || NegQ[-(b*e - a*f)/f])
 

rule 131

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x 
_)]), x_] :> Simp[Sqrt[b*((c + d*x)/(b*c - a*d))]/Sqrt[c + d*x]   Int[1/(Sq 
rt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))]*Sqrt[e + f*x]), x 
], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !GtQ[(b*c - a*d)/b, 0] && Simpler 
Q[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f*x]
 

rule 176

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

rule 2104

Int[(((a_.) + (b_.)*(x_))^(m_.)*((A_.) + (C_.)*(x_)^2))/(Sqrt[(c_.) + (d_.) 
*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbol] :> Sim 
p[2*C*(a + b*x)^m*Sqrt[c + d*x]*Sqrt[e + f*x]*(Sqrt[g + h*x]/(d*f*h*(2*m + 
3))), x] + Simp[1/(d*f*h*(2*m + 3))   Int[((a + b*x)^(m - 1)/(Sqrt[c + d*x] 
*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[a*A*d*f*h*(2*m + 3) - C*(a*(d*e*g + c*f 
*g + c*e*h) + 2*b*c*e*g*m) + (A*b*d*f*h*(2*m + 3) - C*(2*a*(d*f*g + d*e*h + 
 c*f*h) + b*(2*m + 1)*(d*e*g + c*f*g + c*e*h)))*x + 2*C*(a*d*f*h*m - b*(m + 
 1)*(d*f*g + d*e*h + c*f*h))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 h, A, C}, x] && IntegerQ[2*m] && GtQ[m, 0]
 

rule 2118

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f 
_.)*(x_))^(p_.), x_Symbol] :> With[{q = Expon[Px, x], k = Coeff[Px, x, Expo 
n[Px, x]]}, Simp[k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 
1)/(d*f*b^(q - 1)*(m + n + p + q + 1))), x] + Simp[1/(d*f*b^q*(m + n + p + 
q + 1))   Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + 
n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a + b*x)^(q 
- 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + 
 c*f*(p + 1))) + b*(a*d*f*(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m 
 + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; FreeQ[{a, b, c, 
 d, e, f, m, n, p}, x] && PolyQ[Px, x]
 

Maple [A] (verified)

Time = 6.38 (sec) , antiderivative size = 824, normalized size of antiderivative = 1.36

method	result	size
elliptic	\(\frac {\sqrt {\left (h x +g \right ) \left (f x +e \right ) \left (x d +c \right )}\, \left (\frac {2 C b x \sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}{5 d f h}+\frac {2 \left (C a -\frac {2 C b \left (2 c f h +2 d e h +2 d f g \right )}{5 d f h}\right ) \sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}{3 d f h}+\frac {2 \left (A a -\frac {2 C b c e g}{5 d f h}-\frac {2 \left (C a -\frac {2 C b \left (2 c f h +2 d e h +2 d f g \right )}{5 d f h}\right ) \left (\frac {1}{2} c e h +\frac {1}{2} c f g +\frac {1}{2} d e g \right )}{3 d f h}\right ) \left (\frac {c}{d}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {g}{h}}{-\frac {c}{d}+\frac {g}{h}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {e}{f}}}, \sqrt {\frac {-\frac {c}{d}+\frac {e}{f}}{-\frac {c}{d}+\frac {g}{h}}}\right )}{\sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}+\frac {2 \left (A b -\frac {2 C b \left (\frac {3}{2} c e h +\frac {3}{2} c f g +\frac {3}{2} d e g \right )}{5 d f h}-\frac {2 \left (C a -\frac {2 C b \left (2 c f h +2 d e h +2 d f g \right )}{5 d f h}\right ) \left (c f h +d e h +d f g \right )}{3 d f h}\right ) \left (\frac {c}{d}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {g}{h}}{-\frac {c}{d}+\frac {g}{h}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}\, \left (\left (-\frac {c}{d}+\frac {g}{h}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {e}{f}}}, \sqrt {\frac {-\frac {c}{d}+\frac {e}{f}}{-\frac {c}{d}+\frac {g}{h}}}\right )-\frac {g \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {e}{f}}}, \sqrt {\frac {-\frac {c}{d}+\frac {e}{f}}{-\frac {c}{d}+\frac {g}{h}}}\right )}{h}\right )}{\sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}\right )}{\sqrt {h x +g}\, \sqrt {f x +e}\, \sqrt {x d +c}}\)	\(824\)
default	\(\text {Expression too large to display}\)	\(5675\)

Input:

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

Output:

((h*x+g)*(f*x+e)*(d*x+c))^(1/2)/(h*x+g)^(1/2)/(f*x+e)^(1/2)/(d*x+c)^(1/2)* 
(2/5*C*b/d/f/h*x*(d*f*h*x^3+c*f*h*x^2+d*e*h*x^2+d*f*g*x^2+c*e*h*x+c*f*g*x+ 
d*e*g*x+c*e*g)^(1/2)+2/3*(C*a-2/5*C*b/d/f/h*(2*c*f*h+2*d*e*h+2*d*f*g))/d/f 
/h*(d*f*h*x^3+c*f*h*x^2+d*e*h*x^2+d*f*g*x^2+c*e*h*x+c*f*g*x+d*e*g*x+c*e*g) 
^(1/2)+2*(A*a-2/5*C*b/d/f/h*c*e*g-2/3*(C*a-2/5*C*b/d/f/h*(2*c*f*h+2*d*e*h+ 
2*d*f*g))/d/f/h*(1/2*c*e*h+1/2*c*f*g+1/2*d*e*g))*(c/d-e/f)*((x+c/d)/(c/d-e 
/f))^(1/2)*((x+g/h)/(-c/d+g/h))^(1/2)*((x+e/f)/(-c/d+e/f))^(1/2)/(d*f*h*x^ 
3+c*f*h*x^2+d*e*h*x^2+d*f*g*x^2+c*e*h*x+c*f*g*x+d*e*g*x+c*e*g)^(1/2)*Ellip 
ticF(((x+c/d)/(c/d-e/f))^(1/2),((-c/d+e/f)/(-c/d+g/h))^(1/2))+2*(A*b-2/5*C 
*b/d/f/h*(3/2*c*e*h+3/2*c*f*g+3/2*d*e*g)-2/3*(C*a-2/5*C*b/d/f/h*(2*c*f*h+2 
*d*e*h+2*d*f*g))/d/f/h*(c*f*h+d*e*h+d*f*g))*(c/d-e/f)*((x+c/d)/(c/d-e/f))^ 
(1/2)*((x+g/h)/(-c/d+g/h))^(1/2)*((x+e/f)/(-c/d+e/f))^(1/2)/(d*f*h*x^3+c*f 
*h*x^2+d*e*h*x^2+d*f*g*x^2+c*e*h*x+c*f*g*x+d*e*g*x+c*e*g)^(1/2)*((-c/d+g/h 
)*EllipticE(((x+c/d)/(c/d-e/f))^(1/2),((-c/d+e/f)/(-c/d+g/h))^(1/2))-g/h*E 
llipticF(((x+c/d)/(c/d-e/f))^(1/2),((-c/d+e/f)/(-c/d+g/h))^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 1068, normalized size of antiderivative = 1.76 \[ \int \frac {(a+b x) \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx =\text {Too large to display} \] Input:

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

Output:

2/45*(3*(3*C*b*d^3*f^3*h^3*x - 4*C*b*d^3*f^3*g*h^2 - (4*C*b*d^3*e*f^2 + (4 
*C*b*c*d^2 - 5*C*a*d^3)*f^3)*h^3)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g 
) - (8*C*b*d^3*f^3*g^3 + (3*C*b*d^3*e*f^2 + (3*C*b*c*d^2 - 10*C*a*d^3)*f^3 
)*g^2*h + (3*C*b*d^3*e^2*f + (3*C*b*c*d^2 - 5*C*a*d^3)*e*f^2 + (3*C*b*c^2* 
d - 5*C*a*c*d^2 + 15*A*b*d^3)*f^3)*g*h^2 + (8*C*b*d^3*e^3 + (3*C*b*c*d^2 - 
 10*C*a*d^3)*e^2*f + (3*C*b*c^2*d - 5*C*a*c*d^2 + 15*A*b*d^3)*e*f^2 + (8*C 
*b*c^3 - 10*C*a*c^2*d + 15*A*b*c*d^2 - 45*A*a*d^3)*f^3)*h^3)*sqrt(d*f*h)*w 
eierstrassPInverse(4/3*(d^2*f^2*g^2 - (d^2*e*f + c*d*f^2)*g*h + (d^2*e^2 - 
 c*d*e*f + c^2*f^2)*h^2)/(d^2*f^2*h^2), -4/27*(2*d^3*f^3*g^3 - 3*(d^3*e*f^ 
2 + c*d^2*f^3)*g^2*h - 3*(d^3*e^2*f - 4*c*d^2*e*f^2 + c^2*d*f^3)*g*h^2 + ( 
2*d^3*e^3 - 3*c*d^2*e^2*f - 3*c^2*d*e*f^2 + 2*c^3*f^3)*h^3)/(d^3*f^3*h^3), 
 1/3*(3*d*f*h*x + d*f*g + (d*e + c*f)*h)/(d*f*h)) - 3*(8*C*b*d^3*f^3*g^2*h 
 + (7*C*b*d^3*e*f^2 + (7*C*b*c*d^2 - 10*C*a*d^3)*f^3)*g*h^2 + (8*C*b*d^3*e 
^2*f + (7*C*b*c*d^2 - 10*C*a*d^3)*e*f^2 + (8*C*b*c^2*d - 10*C*a*c*d^2 + 15 
*A*b*d^3)*f^3)*h^3)*sqrt(d*f*h)*weierstrassZeta(4/3*(d^2*f^2*g^2 - (d^2*e* 
f + c*d*f^2)*g*h + (d^2*e^2 - c*d*e*f + c^2*f^2)*h^2)/(d^2*f^2*h^2), -4/27 
*(2*d^3*f^3*g^3 - 3*(d^3*e*f^2 + c*d^2*f^3)*g^2*h - 3*(d^3*e^2*f - 4*c*d^2 
*e*f^2 + c^2*d*f^3)*g*h^2 + (2*d^3*e^3 - 3*c*d^2*e^2*f - 3*c^2*d*e*f^2 + 2 
*c^3*f^3)*h^3)/(d^3*f^3*h^3), weierstrassPInverse(4/3*(d^2*f^2*g^2 - (d^2* 
e*f + c*d*f^2)*g*h + (d^2*e^2 - c*d*e*f + c^2*f^2)*h^2)/(d^2*f^2*h^2), ...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(10*sqrt(g + h*x)*sqrt(e + f*x)*sqrt(c + d*x)*a*b*d*f*h - 6*sqrt(g + h*x)* 
sqrt(e + f*x)*sqrt(c + d*x)*b*c**2*e*h - 6*sqrt(g + h*x)*sqrt(e + f*x)*sqr 
t(c + d*x)*b*c**2*f*g + 4*sqrt(g + h*x)*sqrt(e + f*x)*sqrt(c + d*x)*b*c**2 
*f*h*x - 6*sqrt(g + h*x)*sqrt(e + f*x)*sqrt(c + d*x)*b*c*d*e*g + 4*sqrt(g 
+ h*x)*sqrt(e + f*x)*sqrt(c + d*x)*b*c*d*e*h*x + 4*sqrt(g + h*x)*sqrt(e + 
f*x)*sqrt(c + d*x)*b*c*d*f*g*x - 15*int((sqrt(g + h*x)*sqrt(e + f*x)*sqrt( 
c + d*x)*x**2)/(c**2*e*f*g*h + c**2*e*f*h**2*x + c**2*f**2*g*h*x + c**2*f* 
*2*h**2*x**2 + c*d*e**2*g*h + c*d*e**2*h**2*x + c*d*e*f*g**2 + 3*c*d*e*f*g 
*h*x + 2*c*d*e*f*h**2*x**2 + c*d*f**2*g**2*x + 2*c*d*f**2*g*h*x**2 + c*d*f 
**2*h**2*x**3 + d**2*e**2*g*h*x + d**2*e**2*h**2*x**2 + d**2*e*f*g**2*x + 
2*d**2*e*f*g*h*x**2 + d**2*e*f*h**2*x**3 + d**2*f**2*g**2*x**2 + d**2*f**2 
*g*h*x**3),x)*a*b*c*d**2*f**3*h**3 - 15*int((sqrt(g + h*x)*sqrt(e + f*x)*s 
qrt(c + d*x)*x**2)/(c**2*e*f*g*h + c**2*e*f*h**2*x + c**2*f**2*g*h*x + c** 
2*f**2*h**2*x**2 + c*d*e**2*g*h + c*d*e**2*h**2*x + c*d*e*f*g**2 + 3*c*d*e 
*f*g*h*x + 2*c*d*e*f*h**2*x**2 + c*d*f**2*g**2*x + 2*c*d*f**2*g*h*x**2 + c 
*d*f**2*h**2*x**3 + d**2*e**2*g*h*x + d**2*e**2*h**2*x**2 + d**2*e*f*g**2* 
x + 2*d**2*e*f*g*h*x**2 + d**2*e*f*h**2*x**3 + d**2*f**2*g**2*x**2 + d**2* 
f**2*g*h*x**3),x)*a*b*d**3*e*f**2*h**3 - 15*int((sqrt(g + h*x)*sqrt(e + f* 
x)*sqrt(c + d*x)*x**2)/(c**2*e*f*g*h + c**2*e*f*h**2*x + c**2*f**2*g*h*x + 
 c**2*f**2*h**2*x**2 + c*d*e**2*g*h + c*d*e**2*h**2*x + c*d*e*f*g**2 + ...