\(\int \frac {A+B x+C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [41]

Optimal result

Integrand size = 47, antiderivative size = 718 \[ \int \frac {A+B x+C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 \left (A b^2-a (b B-a C)\right ) (d g-c h) \sqrt {e+f x} \sqrt {-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}} E\left (\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {d e-c f} \sqrt {a+b x}}\right )|\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{b (b c-a d) \sqrt {b e-a f} \sqrt {d e-c f} (b g-a h) \sqrt {-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}}-\frac {2 \left (2 a b C g-a^2 C h-b^2 (B g-A h)\right ) \sqrt {e+f x} \sqrt {-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {d e-c f} \sqrt {a+b x}}\right ),\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{b^2 \sqrt {b e-a f} \sqrt {d e-c f} (b g-a h) \sqrt {-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}}+\frac {2 C \sqrt {d e-c f} (a+b x) \sqrt {-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}} \sqrt {-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}} \operatorname {EllipticPi}\left (\frac {b (d e-c f)}{d (b e-a f)},\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {d e-c f} \sqrt {a+b x}}\right ),\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{b^2 d \sqrt {b e-a f} \sqrt {e+f x} \sqrt {g+h x}} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 35.33 (sec) , antiderivative size = 487, normalized size of antiderivative = 0.68 \[ \int \frac {A+B x+C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {e+f x} \sqrt {g+h x} \left (b \left (A b^2+a (-b B+a C)\right ) h (-d g+c h) E\left (\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right )|\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )+(b g-a h) \left (-\left (\left (-2 a b c C+a^2 C d+b^2 (B c-A d)\right ) h \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right ),\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )\right )+C (b c-a d) (b g-a h) \operatorname {EllipticPi}\left (\frac {b (-f g+e h)}{(b e-a f) h},\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right ),\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )\right )\right )}{b^2 (b c-a d) h (b g-a h) (-f g+e h) \sqrt {a+b x} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (b g-a h) (e+f x) (g+h x)}{(f g-e h)^2 (a+b x)^2}}} \] Input:

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

Output:

(-2*Sqrt[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]*Sqrt[e + f*x]*Sq 
rt[g + h*x]*(b*(A*b^2 + a*(-(b*B) + a*C))*h*(-(d*g) + c*h)*EllipticE[ArcSi 
n[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((-(b*c) + a* 
d)*(-(f*g) + e*h))/((b*e - a*f)*(d*g - c*h))] + (b*g - a*h)*(-((-2*a*b*c*C 
 + a^2*C*d + b^2*(B*c - A*d))*h*EllipticF[ArcSin[Sqrt[((-(b*e) + a*f)*(g + 
 h*x))/((f*g - e*h)*(a + b*x))]], ((-(b*c) + a*d)*(-(f*g) + e*h))/((b*e - 
a*f)*(d*g - c*h))]) + C*(b*c - a*d)*(b*g - a*h)*EllipticPi[(b*(-(f*g) + e* 
h))/((b*e - a*f)*h), ArcSin[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*( 
a + b*x))]], ((-(b*c) + a*d)*(-(f*g) + e*h))/((b*e - a*f)*(d*g - c*h))]))) 
/(b^2*(b*c - a*d)*h*(b*g - a*h)*(-(f*g) + e*h)*Sqrt[a + b*x]*Sqrt[c + d*x] 
*Sqrt[-(((b*e - a*f)*(b*g - a*h)*(e + f*x)*(g + h*x))/((f*g - e*h)^2*(a + 
b*x)^2))])
 

Rubi [A] (warning: unable to verify)

Time = 2.24 (sec) , antiderivative size = 875, normalized size of antiderivative = 1.22, 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.

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

\(\Big \downarrow \) 2107

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

\(\Big \downarrow \) 2105

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 194

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 2101

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

\(\Big \downarrow \) 183

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

\(\Big \downarrow \) 188

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 412

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

Input:

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

Output:

(-2*(A*b^2 - a*(b*B - a*C))*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/((b 
*c - a*d)*(b*e - a*f)*(b*g - a*h)*Sqrt[a + b*x]) + ((2*(A*b^2 - a*(b*B - a 
*C))*d*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(b*Sqrt[c + d*x]) - (2*( 
A*b^2 - a*(b*B - a*C))*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))])/(b*Sqrt[((d*e - c*f)*(a + b*x))/( 
(b*e - a*f)*(c + d*x))]*Sqrt[g + h*x]) + ((b*e - a*f)*(b*g - a*h)*((-2*(2* 
a*b*c*C - a^2*C*d - b^2*(B*c - A*d))*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]*Sq 
rt[-(((b*e - a*f)*(g + h*x))/((f*g - e*h)*(a + b*x)))]) + (2*C*Sqrt[b*c - 
a*d]*Sqrt[-(d*g) + c*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))]*Ellipt 
icPi[-((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*h*Sqrt[c + d*x]*Sqrt[e + f*x])))/b)/((b*c - a*d) 
*(b*e - a*f)*(b*g - a*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. \(2320\) vs. \(2(661)=1322\).

Time = 24.10 (sec) , antiderivative size = 2321, normalized size of antiderivative = 3.23

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(2321\)
default	\(\text {Expression too large to display}\)	\(39149\)

Input:

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

Output:

((h*x+g)*(d*x+c)*(b*x+a)*(f*x+e))^(1/2)/(h*x+g)^(1/2)/(d*x+c)^(1/2)/(b*x+a 
)^(1/2)/(f*x+e)^(1/2)*(2*(b*d*f*h*x^3+b*c*f*h*x^2+b*d*e*h*x^2+b*d*f*g*x^2+ 
b*c*e*h*x+b*c*f*g*x+b*d*e*g*x+b*c*e*g)/(a^3*d*f*h-a^2*b*c*f*h-a^2*b*d*e*h- 
a^2*b*d*f*g+a*b^2*c*e*h+a*b^2*c*f*g+a*b^2*d*e*g-b^3*c*e*g)/b*(A*b^2-B*a*b+ 
C*a^2)/((x+a/b)*(b*d*f*h*x^3+b*c*f*h*x^2+b*d*e*h*x^2+b*d*f*g*x^2+b*c*e*h*x 
+b*c*f*g*x+b*d*e*g*x+b*c*e*g))^(1/2)+2*((B*b-C*a)/b^2+1/b^2*(a^2*d*f*h-a*b 
*c*f*h-a*b*d*e*h-a*b*d*f*g+b^2*c*e*h+b^2*c*f*g+b^2*d*e*g)*(A*b^2-B*a*b+C*a 
^2)/(a^3*d*f*h-a^2*b*c*f*h-a^2*b*d*e*h-a^2*b*d*f*g+a*b^2*c*e*h+a*b^2*c*f*g 
+a*b^2*d*e*g-b^3*c*e*g)-(b*c*e*h+b*c*f*g+b*d*e*g)/(a^3*d*f*h-a^2*b*c*f*h-a 
^2*b*d*e*h-a^2*b*d*f*g+a*b^2*c*e*h+a*b^2*c*f*g+a*b^2*d*e*g-b^3*c*e*g)/b*(A 
*b^2-B*a*b+C*a^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/b-1/b*(a*d*f*h 
-b*c*f*h-b*d*e*h-b*d*f*g)*(A*b^2-B*a*b+C*a^2)/(a^3*d*f*h-a^2*b*c*f*h-a^2*b 
*d*e*h-a^2*b*d*f*g+a*b^2*c*e*h+a*b^2*c*f*g+a*b^2*d*e*g-b^3*c*e*g)-(2*b*c*f 
*h+2*b*d*e*h+2*b*d*f*g)/(a^3*d*f*h-a^2*b*c*f*h-a^2*b*d*e*h-a^2*b*d*f*g+a*b 
^2*c*e*h+a*b^2*c*f*g+a*b^2*d*e*g-b^3*c*e*g)/b*(A*b^2-B*a*b+C*a^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. \[ \int \frac {A+B x+C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m operator + Error: 
Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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