\(\int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2} \sqrt {g+h x}} \, dx\) [43]

Optimal result

Integrand size = 47, antiderivative size = 713 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2} \sqrt {g+h x}} \, dx=-\frac {2 \left (C e^2-B e f+A f^2\right ) \sqrt {b g-a h} \sqrt {c+d x} \sqrt {\frac {(b e-a f) (g+h x)}{(b g-a h) (e+f x)}} E\left (\arcsin \left (\frac {\sqrt {f g-e h} \sqrt {a+b x}}{\sqrt {b g-a h} \sqrt {e+f x}}\right )|-\frac {(d e-c f) (b g-a h)}{(b c-a d) (f g-e h)}\right )}{f (b e-a f) (d e-c f) \sqrt {f g-e h} \sqrt {\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}} \sqrt {g+h x}}-\frac {2 \left ((B c-A d) f^2+C e (d e-2 c f)\right ) \sqrt {b g-a h} \sqrt {c+d x} \sqrt {\frac {(b e-a f) (g+h x)}{(b g-a h) (e+f x)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f g-e h} \sqrt {a+b x}}{\sqrt {b g-a h} \sqrt {e+f x}}\right ),-\frac {(d e-c f) (b g-a h)}{(b c-a d) (f g-e h)}\right )}{(b c-a d) f^2 (d e-c f) \sqrt {f g-e h} \sqrt {\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}} \sqrt {g+h x}}+\frac {2 C \sqrt {b g-a h} \sqrt {\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}} (e+f x) \sqrt {\frac {(b e-a f) (g+h x)}{(b g-a h) (e+f x)}} \operatorname {EllipticPi}\left (\frac {f (b g-a h)}{b (f g-e h)},\arcsin \left (\frac {\sqrt {f g-e h} \sqrt {a+b x}}{\sqrt {b g-a h} \sqrt {e+f x}}\right ),-\frac {(d e-c f) (b g-a h)}{(b c-a d) (f g-e h)}\right )}{b f^2 \sqrt {f g-e h} \sqrt {c+d x} \sqrt {g+h x}} \] Output:

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(11158\) vs. \(2(713)=1426\).

Time = 35.94 (sec) , antiderivative size = 11158, normalized size of antiderivative = 15.65 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2} \sqrt {g+h x}} \, dx=\text {Result too large to show} \] Input:

Integrate[(A + B*x + C*x^2)/(Sqrt[a + b*x]*Sqrt[c + d*x]*(e + f*x)^(3/2)*S 
qrt[g + h*x]),x]
 

Output:

Result too large to show
 

Rubi [A] (warning: unable to verify)

Time = 2.28 (sec) , antiderivative size = 882, normalized size of antiderivative = 1.24, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.213, Rules used = {2107, 2105, 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.

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

\(\Big \downarrow \) 2107

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

\(\Big \downarrow \) 2105

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 194

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 2101

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

\(\Big \downarrow \) 183

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

\(\Big \downarrow \) 188

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 412

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

Input:

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

Output:

(-2*(C*e^2 - B*e*f + A*f^2)*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[g + h*x])/((b 
*e - a*f)*(d*e - c*f)*(f*g - e*h)*Sqrt[e + f*x]) + ((2*d*(C*e^2 - B*e*f + 
A*f^2)*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(f*Sqrt[c + d*x]) - (2*( 
C*e^2 - B*e*f + A*f^2)*Sqrt[d*g - c*h]*Sqrt[f*g - e*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))])/(f*Sqrt[((d*e - c*f)*(a + b*x))/( 
(b*e - a*f)*(c + d*x))]*Sqrt[g + h*x]) - ((b*e - a*f)*(d*e - c*f)*((2*(a*C 
*(f*g - e*h) + b*(C*e*g - B*f*g + A*f*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[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*C*Sqrt[- 
(d*g) + c*h]*(f*g - e*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))]*Ellip 
ticPi[-((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 - a*d]*h*Sqrt[c + d*x]*Sqrt[e + f*x])) 
)/f)/((b*e - a*f)*(d*e - c*f)*(f*g - e*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 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 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. \(2317\) vs. \(2(656)=1312\).

Time = 23.81 (sec) , antiderivative size = 2318, normalized size of antiderivative = 3.25

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(2318\)
default	\(\text {Expression too large to display}\)	\(25793\)

Input:

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

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2} \sqrt {g+h x}} \, dx=\text {Timed out} \] Input:

integrate((C*x^2+B*x+A)/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(3/2)/(h*x+g)^ 
(1/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((C*x^2+B*x+A)/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(3/2)/(h*x+g)^ 
(1/2),x, algorithm="maxima")
 

Output:

integrate((C*x^2 + B*x + A)/(sqrt(b*x + a)*sqrt(d*x + c)*(f*x + e)^(3/2)*s 
qrt(h*x + g)), x)
 

Giac [F]

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

integrate((C*x^2+B*x+A)/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(3/2)/(h*x+g)^ 
(1/2),x, algorithm="giac")
 

Output:

integrate((C*x^2 + B*x + A)/(sqrt(b*x + a)*sqrt(d*x + c)*(f*x + e)^(3/2)*s 
qrt(h*x + g)), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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