\(\int \frac {\sqrt {a+b x} (a b B-a^2 C+b^2 B x+b^2 C x^2)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [22]

Optimal result

Integrand size = 62, antiderivative size = 988 \[ \int \frac {\sqrt {a+b x} \left (a b B-a^2 C+b^2 B x+b^2 C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {b^2 (4 b B d f h+a C d f h-3 b C (d f g+d e h+c f h)) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{4 d^2 f^2 h^2 \sqrt {a+b x}}+\frac {b^2 C \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{2 d f h}-\frac {b \sqrt {b e-a f} (d g-c h) (4 b B d f h+a C d f h-3 b C (d f g+d e h+c f h)) \sqrt {e+f x} \sqrt {-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}} E\left (\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {d e-c f} \sqrt {a+b x}}\right )|\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{4 d^2 f^2 \sqrt {d e-c f} h^2 \sqrt {-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}}-\frac {(b c-a d) \sqrt {b e-a f} (4 b B d f h-a C d f h-b C (d f g+3 d e h+3 c f h)) \sqrt {e+f x} \sqrt {-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {d e-c f} \sqrt {a+b x}}\right ),\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{4 d^2 f^2 \sqrt {d e-c f} h \sqrt {-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}}-\frac {\sqrt {d e-c f} \left ((a d f h+b (d f g+d e h+c f h)) (4 b B d f h+a C d f h-3 b C (d f g+d e h+c f h))+4 d f h \left (2 a^2 C d f h+b^2 C (d e g+c f g+c e h)-a b (4 B d f h-C (d f g+d e h+c f h))\right )\right ) (a+b x) \sqrt {-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}} \sqrt {-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}} \operatorname {EllipticPi}\left (\frac {b (d e-c f)}{d (b e-a f)},\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {d e-c f} \sqrt {a+b x}}\right ),\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{4 d^3 f^2 \sqrt {b e-a f} h^2 \sqrt {e+f x} \sqrt {g+h x}} \] Output:

1/4*b^2*(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*b^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*b*(-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))/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))/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*d*f*h*(2*a^2*C 
*d*f*h+b^2*C*(c*e*h+c*f*g+d*e*g)-a*b*(4*B*d*f*h-C*(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))/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. \(21961\) vs. \(2(988)=1976\).

Time = 37.00 (sec) , antiderivative size = 21961, normalized size of antiderivative = 22.23 \[ \int \frac {\sqrt {a+b x} \left (a b B-a^2 C+b^2 B x+b^2 C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Result too large to show} \] Input:

Integrate[(Sqrt[a + b*x]*(a*b*B - a^2*C + b^2*B*x + b^2*C*x^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 = 3.53 (sec) , antiderivative size = 978, normalized size of antiderivative = 0.99, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2004, 2100, 2105, 25, 27, 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.

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

\(\Big \downarrow \) 2004

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

\(\Big \downarrow \) 2100

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

\(\Big \downarrow \) 2105

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 194

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 2101

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

\(\Big \downarrow \) 183

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

\(\Big \downarrow \) 188

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 412

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

Input:

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

Output:

(b^2*C*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(2*d*f*h) 
+ ((b*(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[e + f*x]*Sqrt[g + h*x])/(f*h*Sqrt[c + d*x]) - (b*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)))]*Ell 
ipticE[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))])/(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)))])/(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*Sqrt[-(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*d*f*h*(2*a^2*C 
*d*f*h + b^2*C*(d*e*g + c*f*g + c*e*h) - a*b*(4*B*d*f*h - C*(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)*...
 

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 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 2004

Int[(u_)*((d_) + (e_.)*(x_))^(q_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.) 
, x_Symbol] :> Int[u*(d + e*x)^(p + q)*(a/d + (c/e)*x)^p, x] /; FreeQ[{a, b 
, c, d, e, q}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]
 

rule 2100

Int[(((a_.) + (b_.)*(x_))^(m_)*((A_.) + (B_.)*(x_)))/(Sqrt[(c_.) + (d_.)*(x 
_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbol] :> Simp[2 
*b*B*(a + b*x)^(m - 1)*Sqrt[c + d*x]*Sqrt[e + f*x]*(Sqrt[g + h*x]/(d*f*h*(2 
*m + 1))), x] + Simp[1/(d*f*h*(2*m + 1))   Int[((a + b*x)^(m - 2)/(Sqrt[c + 
 d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[(-b)*B*(a*(d*e*g + c*f*g + c*e*h) 
+ 2*b*c*e*g*(m - 1)) + a^2*A*d*f*h*(2*m + 1) + (2*a*A*b*d*f*h*(2*m + 1) - B 
*(2*a*b*(d*f*g + d*e*h + c*f*h) + b^2*(d*e*g + c*f*g + c*e*h)*(2*m - 1) - a 
^2*d*f*h*(2*m + 1)))*x + b*(A*b*d*f*h*(2*m + 1) - B*(2*b*(d*f*g + d*e*h + c 
*f*h)*m - a*d*f*h*(4*m - 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g 
, h, A, B}, x] && IntegerQ[2*m] && GtQ[m, 1]
 

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 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1833\) vs. \(2(905)=1810\).

Time = 19.40 (sec) , antiderivative size = 1834, normalized size of antiderivative = 1.86

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1834\)
default	\(\text {Expression too large to display}\)	\(56119\)

Input:

int((b*x+a)^(1/2)*(C*b^2*x^2+B*b^2*x+B*a*b-C*a^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)*(1/2*C*b^2/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^2*b*B-C*a^3-1/2*C*b^2/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*(2*a*b^2*B-a^2*b*C-1/ 
2*C*b^2/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^3+a*C*b^2-1/2*C*b^2/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...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x} \left (a b B-a^2 C+b^2 B x+b^2 C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {\left (a + b x\right )^{\frac {3}{2}} \left (B b - C a + C b x\right )}{\sqrt {c + d x} \sqrt {e + f x} \sqrt {g + h x}}\, dx \] Input:

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

Output:

Integral((a + b*x)**(3/2)*(B*b - C*a + C*b*x)/(sqrt(c + d*x)*sqrt(e + f*x) 
*sqrt(g + h*x)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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