\(\int \frac {d e+c f+2 d f x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [13]

Optimal result

Integrand size = 49, antiderivative size = 456 \[ \int \frac {d e+c f+2 d f x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 \sqrt {d e-c f} (b d e+b c f-2 a d f) \sqrt {-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {d e-c f} \sqrt {a+b x}}\right ),\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{b \sqrt {b e-a f} (d g-c h) \sqrt {e+f x} \sqrt {-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}}}+\frac {4 f \sqrt {d e-c f} (a+b x) \sqrt {-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}} \sqrt {-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}} \operatorname {EllipticPi}\left (\frac {b (d e-c f)}{d (b e-a f)},\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {d e-c f} \sqrt {a+b x}}\right ),\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{b \sqrt {b e-a f} \sqrt {e+f x} \sqrt {g+h x}} \] Output:

2*(-c*f+d*e)^(1/2)*(-2*a*d*f+b*c*f+b*d*e)*(-(-a*d+b*c)*(f*x+e)/(-c*f+d*e)/ 
(b*x+a))^(1/2)*(h*x+g)^(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/(-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)+4*f*(-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)*El 
lipticPi((-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/(-a*f+b*e)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 25.29 (sec) , antiderivative size = 723, normalized size of antiderivative = 1.59 \[ \int \frac {d e+c f+2 d f x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 \sqrt {a+b x} \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \left (-b d e (b e-a f) h \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} (g+h x) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right ),\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )+2 a d f (b e-a f) h \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} (g+h x) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right ),\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )+b c f (-b e+a f) h \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} (g+h x) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right ),\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )-2 d f (b g-a h) (f g-e h) (a+b x) \sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}} \sqrt {\frac {(-b e+a f) (b g-a h) (e+f x) (g+h x)}{(f g-e h)^2 (a+b x)^2}} \operatorname {EllipticPi}\left (\frac {b (-f g+e h)}{(b e-a f) h},\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right ),\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )\right )}{b (b e-a f) h (b g-a h) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}} \] Input:

Integrate[(d*e + c*f + 2*d*f*x)/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x] 
*Sqrt[g + h*x]),x]
 

Output:

(2*Sqrt[a + b*x]*Sqrt[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]*(-( 
b*d*e*(b*e - a*f)*h*Sqrt[((b*g - a*h)*(e + f*x))/((f*g - e*h)*(a + b*x))]* 
(g + h*x)*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))]) + 
2*a*d*f*(b*e - a*f)*h*Sqrt[((b*g - a*h)*(e + f*x))/((f*g - e*h)*(a + b*x)) 
]*(g + h*x)*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))] + 
 b*c*f*(-(b*e) + a*f)*h*Sqrt[((b*g - a*h)*(e + f*x))/((f*g - e*h)*(a + b*x 
))]*(g + h*x)*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))] 
 - 2*d*f*(b*g - a*h)*(f*g - e*h)*(a + b*x)*Sqrt[((-(b*e) + a*f)*(g + h*x)) 
/((f*g - e*h)*(a + b*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)]*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*(b*e - a*f 
)*h*(b*g - a*h)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]*Sqrt[((-(b*e) + 
a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))])
 

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 449, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.102, Rules used = {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.

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

\(\Big \downarrow \) 2101

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

\(\Big \downarrow \) 183

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

\(\Big \downarrow \) 188

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 412

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

Input:

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

Output:

(2*(b*d*e + b*c*f - 2*a*d*f)*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])/(S 
qrt[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)))]) + (4*d*f*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))]*EllipticPi[-((b*(d*g - c*h)) 
/((b*c - a*d)*h)), ArcSin[(Sqrt[b*c - a*d]*Sqrt[g + h*x])/(Sqrt[-(d*g) + c 
*h]*Sqrt[a + b*x])], ((b*e - a*f)*(d*g - c*h))/((b*c - a*d)*(f*g - e*h))]) 
/(b*Sqrt[b*c - a*d]*h*Sqrt[c + d*x]*Sqrt[e + f*x])
 

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 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 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(854\) vs. \(2(418)=836\).

Time = 21.12 (sec) , antiderivative size = 855, normalized size of antiderivative = 1.88

method	result	size
elliptic	\(\frac {\sqrt {\left (h x +g \right ) \left (x d +c \right ) \left (b x +a \right ) \left (f x +e \right )}\, \left (\frac {2 \left (c f +d e \right ) \left (\frac {e}{f}-\frac {g}{h}\right ) \sqrt {\frac {\left (\frac {c}{d}-\frac {e}{f}\right ) \left (x +\frac {g}{h}\right )}{\left (-\frac {e}{f}+\frac {g}{h}\right ) \left (x +\frac {c}{d}\right )}}\, \left (x +\frac {c}{d}\right )^{2} \sqrt {\frac {\left (-\frac {c}{d}+\frac {g}{h}\right ) \left (x +\frac {a}{b}\right )}{\left (-\frac {a}{b}+\frac {g}{h}\right ) \left (x +\frac {c}{d}\right )}}\, \sqrt {\frac {\left (-\frac {c}{d}+\frac {g}{h}\right ) \left (x +\frac {e}{f}\right )}{\left (-\frac {e}{f}+\frac {g}{h}\right ) \left (x +\frac {c}{d}\right )}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (\frac {c}{d}-\frac {e}{f}\right ) \left (x +\frac {g}{h}\right )}{\left (-\frac {e}{f}+\frac {g}{h}\right ) \left (x +\frac {c}{d}\right )}}, \sqrt {\frac {\left (-\frac {c}{d}+\frac {a}{b}\right ) \left (\frac {e}{f}-\frac {g}{h}\right )}{\left (\frac {a}{b}-\frac {g}{h}\right ) \left (-\frac {c}{d}+\frac {e}{f}\right )}}\right )}{\left (\frac {c}{d}-\frac {e}{f}\right ) \left (-\frac {c}{d}+\frac {g}{h}\right ) \sqrt {h d b f \left (x +\frac {g}{h}\right ) \left (x +\frac {c}{d}\right ) \left (x +\frac {a}{b}\right ) \left (x +\frac {e}{f}\right )}}+\frac {4 d f \left (\frac {e}{f}-\frac {g}{h}\right ) \sqrt {\frac {\left (\frac {c}{d}-\frac {e}{f}\right ) \left (x +\frac {g}{h}\right )}{\left (-\frac {e}{f}+\frac {g}{h}\right ) \left (x +\frac {c}{d}\right )}}\, \left (x +\frac {c}{d}\right )^{2} \sqrt {\frac {\left (-\frac {c}{d}+\frac {g}{h}\right ) \left (x +\frac {a}{b}\right )}{\left (-\frac {a}{b}+\frac {g}{h}\right ) \left (x +\frac {c}{d}\right )}}\, \sqrt {\frac {\left (-\frac {c}{d}+\frac {g}{h}\right ) \left (x +\frac {e}{f}\right )}{\left (-\frac {e}{f}+\frac {g}{h}\right ) \left (x +\frac {c}{d}\right )}}\, \left (-\frac {c \operatorname {EllipticF}\left (\sqrt {\frac {\left (\frac {c}{d}-\frac {e}{f}\right ) \left (x +\frac {g}{h}\right )}{\left (-\frac {e}{f}+\frac {g}{h}\right ) \left (x +\frac {c}{d}\right )}}, \sqrt {\frac {\left (-\frac {c}{d}+\frac {a}{b}\right ) \left (\frac {e}{f}-\frac {g}{h}\right )}{\left (\frac {a}{b}-\frac {g}{h}\right ) \left (-\frac {c}{d}+\frac {e}{f}\right )}}\right )}{d}+\left (\frac {c}{d}-\frac {g}{h}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {\left (\frac {c}{d}-\frac {e}{f}\right ) \left (x +\frac {g}{h}\right )}{\left (-\frac {e}{f}+\frac {g}{h}\right ) \left (x +\frac {c}{d}\right )}}, \frac {-\frac {e}{f}+\frac {g}{h}}{\frac {c}{d}-\frac {e}{f}}, \sqrt {\frac {\left (-\frac {c}{d}+\frac {a}{b}\right ) \left (\frac {e}{f}-\frac {g}{h}\right )}{\left (\frac {a}{b}-\frac {g}{h}\right ) \left (-\frac {c}{d}+\frac {e}{f}\right )}}\right )\right )}{\left (\frac {c}{d}-\frac {e}{f}\right ) \left (-\frac {c}{d}+\frac {g}{h}\right ) \sqrt {h d b f \left (x +\frac {g}{h}\right ) \left (x +\frac {c}{d}\right ) \left (x +\frac {a}{b}\right ) \left (x +\frac {e}{f}\right )}}\right )}{\sqrt {h x +g}\, \sqrt {x d +c}\, \sqrt {b x +a}\, \sqrt {f x +e}}\)	\(855\)
default	\(\text {Expression too large to display}\)	\(2527\)

Input:

int((2*d*f*x+c*f+d*e)/(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*(c*f+d*e)*(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))+4*d 
*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-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))))
 

Fricas [F(-1)]

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

integrate((2*d*f*x+c*f+d*e)/(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]

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

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

Output:

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

Maxima [F]

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

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

Giac [F]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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