\(\int \frac {\sqrt {a+b x} (A+B x+C x^2)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [39]

Optimal result

Integrand size = 47, antiderivative size = 983 \[ \int \frac {\sqrt {a+b x} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {(4 b B d f h+a C d f h-3 b C (d f g+d e h+c f h)) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{4 d^2 f^2 h^2 \sqrt {a+b x}}+\frac {C \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{2 d f h}-\frac {\sqrt {b e-a f} (d g-c h) (4 b B d f h+a C d f h-3 b C (d f g+d e h+c f h)) \sqrt {e+f x} \sqrt {-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}} E\left (\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {d e-c f} \sqrt {a+b x}}\right )|\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{4 b d^2 f^2 \sqrt {d e-c f} h^2 \sqrt {-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}}-\frac {(b c-a d) \sqrt {b e-a f} (4 b B d f h-a C d f h-b C (d f g+3 d e h+3 c f h)) \sqrt {e+f x} \sqrt {-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {d e-c f} \sqrt {a+b x}}\right ),\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{4 b^2 d^2 f^2 \sqrt {d e-c f} h \sqrt {-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}}-\frac {\sqrt {d e-c f} ((a d f h+b (d f g+d e h+c f h)) (4 b B d f h+a C d f h-3 b C (d f g+d e h+c f h))-4 b d f h (2 (A b+a B) d f h-C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) (a+b x) \sqrt {-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}} \sqrt {-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}} \operatorname {EllipticPi}\left (\frac {b (d e-c f)}{d (b e-a f)},\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {d e-c f} \sqrt {a+b x}}\right ),\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{4 b^2 d^3 f^2 \sqrt {b e-a f} h^2 \sqrt {e+f x} \sqrt {g+h x}} \] Output:

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(22316\) vs. \(2(983)=1966\).

Time = 36.78 (sec) , antiderivative size = 22316, normalized size of antiderivative = 22.70 \[ \int \frac {\sqrt {a+b x} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Result too large to show} \] Input:

Integrate[(Sqrt[a + b*x]*(A + B*x + C*x^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 = 2.89 (sec) , antiderivative size = 973, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.213, Rules used = {2103, 2105, 25, 194, 327, 2101, 183, 188, 321, 412}

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.

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

\(\Big \downarrow \) 2103

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

\(\Big \downarrow \) 2105

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 194

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 2101

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

\(\Big \downarrow \) 183

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

\(\Big \downarrow \) 188

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 412

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

Input:

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

Output:

(C*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(2*d*f*h) + (- 
(((3*c*C - 4*B*d - C*d*(a/b - (3*e)/f - (3*g)/h))*Sqrt[a + b*x]*Sqrt[e + f 
*x]*Sqrt[g + h*x])/Sqrt[c + d*x]) - (Sqrt[d*g - c*h]*Sqrt[f*g - e*h]*(4*b* 
B*d*f*h + a*C*d*f*h - 3*b*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))])/(b*d*f*h*Sqrt[((d*e - c*f)*(a + b*x 
))/((b*e - a*f)*(c + d*x))]*Sqrt[g + h*x]) - ((2*d*(b*e - a*f)*Sqrt[b*g - 
a*h]*(4*b*B*d*f*h - a*C*d*f*h - b*C*(c*f*h + 3*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))/((d*e - c*f)*(b*g - a*h)))])/(b*Sqrt[f*g - e*h]*Sqrt[c 
 + d*x]*Sqrt[-(((b*e - a*f)*(g + h*x))/((f*g - e*h)*(a + b*x)))]) + (2*Sqr 
t[-(d*g) + c*h]*((a*d*f*h + b*(d*f*g + d*e*h + c*f*h))*(4*b*B*d*f*h + a*C* 
d*f*h - 3*b*C*(d*f*g + d*e*h + c*f*h)) - 4*b*d*f*h*(2*(A*b + a*B)*d*f*h - 
C*(b*(d*e*g + c*f*g + c*e*h) + a*(d*f*g + d*e*h + c*f*h))))*(a + b*x)*Sqrt 
[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]*Sqrt[((b*g - a*h)*(e + f 
*x))/((f*g - e*h)*(a + b*x))]*EllipticPi[-((b*(d*g - c*h))/((b*c - a*d)*h) 
), ArcSin[(Sqrt[b*c - a*d]*Sqrt[g + h*x])/(Sqrt[-(d*g) + c*h]*Sqrt[a + b*x 
])], ((b*e - a*f)*(d*g - c*h))/((b*c - a*d)*(f*g - e*h))])/(b*Sqrt[b*c ...
 

Defintions of rubi rules used

rule 25

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

rule 183

Int[Sqrt[(a_.) + (b_.)*(x_)]/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*( 
x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[2*(a + b*x)*Sqrt[(b*g - a*h)*(( 
c + d*x)/((d*g - c*h)*(a + b*x)))]*(Sqrt[(b*g - a*h)*((e + f*x)/((f*g - e*h 
)*(a + b*x)))]/(Sqrt[c + d*x]*Sqrt[e + f*x]))   Subst[Int[1/((h - b*x^2)*Sq 
rt[1 + (b*c - a*d)*(x^2/(d*g - c*h))]*Sqrt[1 + (b*e - a*f)*(x^2/(f*g - e*h) 
)]), x], x, Sqrt[g + h*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d, e, f, g, 
 h}, 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 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* 
(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, 
 f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && S 
implerSqrtQ[-f/e, -d/c])
 

rule 2101

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

rule 2103

Int[(((a_.) + (b_.)*(x_))^(m_.)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2))/(Sqrt[ 
(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_S 
ymbol] :> Simp[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 + 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 + (b 
*B*d*f*h*(2*m + 3) + 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, B, C}, x] && IntegerQ[2 
*m] && GtQ[m, 0]
 

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]
 

Maple [A] (verified)

Time = 19.56 (sec) , antiderivative size = 1800, normalized size of antiderivative = 1.83

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1800\)
default	\(\text {Expression too large to display}\)	\(57266\)

Input:

int((b*x+a)^(1/2)*(C*x^2+B*x+A)/(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)*(1/2*C/h/d/f*(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) 
+2*(A*a-1/2*C/h/d/f*(1/2*a*c*e*h+1/2*a*c*f*g+1/2*a*d*e*g+1/2*b*c*e*g))*(e/ 
f-g/h)*((c/d-e/f)*(x+g/h)/(-e/f+g/h)/(x+c/d))^(1/2)*(x+c/d)^2*((-c/d+g/h)* 
(x+a/b)/(-a/b+g/h)/(x+c/d))^(1/2)*((-c/d+g/h)*(x+e/f)/(-e/f+g/h)/(x+c/d))^ 
(1/2)/(c/d-e/f)/(-c/d+g/h)/(h*d*b*f*(x+g/h)*(x+c/d)*(x+a/b)*(x+e/f))^(1/2) 
*EllipticF(((c/d-e/f)*(x+g/h)/(-e/f+g/h)/(x+c/d))^(1/2),((-c/d+a/b)*(e/f-g 
/h)/(a/b-g/h)/(-c/d+e/f))^(1/2))+2*(A*b+B*a-1/2*C/h/d/f*(a*c*f*h+a*d*e*h+a 
*d*f*g+b*c*e*h+b*c*f*g+b*d*e*g))*(e/f-g/h)*((c/d-e/f)*(x+g/h)/(-e/f+g/h)/( 
x+c/d))^(1/2)*(x+c/d)^2*((-c/d+g/h)*(x+a/b)/(-a/b+g/h)/(x+c/d))^(1/2)*((-c 
/d+g/h)*(x+e/f)/(-e/f+g/h)/(x+c/d))^(1/2)/(c/d-e/f)/(-c/d+g/h)/(h*d*b*f*(x 
+g/h)*(x+c/d)*(x+a/b)*(x+e/f))^(1/2)*(-c/d*EllipticF(((c/d-e/f)*(x+g/h)/(- 
e/f+g/h)/(x+c/d))^(1/2),((-c/d+a/b)*(e/f-g/h)/(a/b-g/h)/(-c/d+e/f))^(1/2)) 
+(c/d-g/h)*EllipticPi(((c/d-e/f)*(x+g/h)/(-e/f+g/h)/(x+c/d))^(1/2),(-e/f+g 
/h)/(c/d-e/f),((-c/d+a/b)*(e/f-g/h)/(a/b-g/h)/(-c/d+e/f))^(1/2)))+(B*b+C*a 
-1/2*C/h/d/f*(3/2*a*d*f*h+3/2*b*c*f*h+3/2*b*d*e*h+3/2*b*g*d*f))*((x+g/h)*( 
x+a/b)*(x+e/f)+(e/f-g/h)*((c/d-e/f)*(x+g/h)/(-e/f+g/h)/(x+c/d))^(1/2)*(x+c 
/d)^2*((-c/d+g/h)*(x+a/b)/(-a/b+g/h)/(x+c/d))^(1/2)*((-c/d+g/h)*(x+e/f)...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Giac [F]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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