\(\int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [40]

Optimal result

Integrand size = 47, antiderivative size = 782 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {C \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{d f h \sqrt {a+b x}}-\frac {C \sqrt {b e-a f} (d g-c h) \sqrt {e+f x} \sqrt {-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}} E\left (\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {d e-c f} \sqrt {a+b x}}\right )|\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{b d f \sqrt {d e-c f} h \sqrt {-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}}+\frac {\left (a^2 C d f-b^2 (c C e-2 A d f)+a b (C d e+c C f-2 B d f)\right ) \sqrt {e+f x} \sqrt {-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {d e-c f} \sqrt {a+b x}}\right ),\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{b^2 d f \sqrt {b e-a f} \sqrt {d e-c f} \sqrt {-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}}+\frac {\sqrt {d e-c f} (2 b B d f h-C (a d f h+b (d f g+d e h+c f h))) (a+b x) \sqrt {-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}} \sqrt {-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}} \operatorname {EllipticPi}\left (\frac {b (d e-c f)}{d (b e-a f)},\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {d e-c f} \sqrt {a+b x}}\right ),\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{b^2 d^2 f \sqrt {b e-a f} h \sqrt {e+f x} \sqrt {g+h x}} \] Output:

C*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/d/f/h/(b*x+a)^(1/2)-C*(-a*f+b* 
e)^(1/2)*(-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/d/f/(-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)+(a^2 
*C*d*f-b^2*(-2*A*d*f+C*c*e)+a*b*(-2*B*d*f+C*c*f+C*d*e))*(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/f/(-a*f+b*e)^(1/2)/(-c*f+d*e)^(1/2)/(-(-a*d+b*c) 
*(f*x+e)/(-c*f+d*e)/(b*x+a))^(1/2)/(h*x+g)^(1/2)+(-c*f+d*e)^(1/2)*(2*b*B*d 
*f*h-C*(a*d*f*h+b*(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)*Ellipt 
icPi((-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^2/f/(-a*f+b*e)^(1/2)/h/(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. \(7019\) vs. \(2(782)=1564\).

Time = 34.82 (sec) , antiderivative size = 7019, normalized size of antiderivative = 8.98 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 1.55 (sec) , antiderivative size = 780, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.170, Rules used = {2105, 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.

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

\(\Big \downarrow \) 2105

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

\(\Big \downarrow \) 194

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 2101

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

\(\Big \downarrow \) 183

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

\(\Big \downarrow \) 188

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 412

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

Input:

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

Output:

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

Defintions of rubi rules used

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1535\) vs. \(2(717)=1434\).

Time = 21.58 (sec) , antiderivative size = 1536, normalized size of antiderivative = 1.96

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1536\)
default	\(\text {Expression too large to display}\)	\(19862\)

Input:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Giac [F]

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

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

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

Reduce [F]

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

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