\(\int \frac {\sqrt {a+b x} (d e+c f+2 d f x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [12]

Optimal result

Integrand size = 49, antiderivative size = 702 \[ \int \frac {\sqrt {a+b x} (d e+c f+2 d f x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{h \sqrt {a+b x}}-\frac {2 \sqrt {b e-a f} (d g-c h) \sqrt {e+f x} \sqrt {-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}} E\left (\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {d e-c f} \sqrt {a+b x}}\right )|\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{\sqrt {d e-c f} h \sqrt {-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}}-\frac {2 (b c-a d) \sqrt {b e-a f} \sqrt {e+f x} \sqrt {-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {d e-c f} \sqrt {a+b x}}\right ),\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{b \sqrt {d e-c f} \sqrt {-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}}-\frac {2 f \sqrt {d e-c f} (b g-a h) (a+b x) \sqrt {-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}} \sqrt {-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}} \operatorname {EllipticPi}\left (\frac {b (d e-c f)}{d (b e-a f)},\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {d e-c f} \sqrt {a+b x}}\right ),\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{b \sqrt {b e-a f} h \sqrt {e+f x} \sqrt {g+h x}} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 36.19 (sec) , antiderivative size = 443, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {a+b x} (d e+c f+2 d f x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \left (-\frac {d h (e+f x) (g+h x)}{c+d x}-\frac {(f g-e h) \sqrt {\frac {(-d e+c f) (d g-c h) (e+f x) (g+h x)}{(f g-e h)^2 (c+d x)^2}} \left ((d e-c f) h E\left (\arcsin \left (\sqrt {\frac {(-d e+c f) (g+h x)}{(f g-e h) (c+d x)}}\right )|\frac {(b c-a d) (-f g+e h)}{(d e-c f) (b g-a h)}\right )+(-d e h+c f h) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {(-d e+c f) (g+h x)}{(f g-e h) (c+d x)}}\right ),\frac {(b c-a d) (-f g+e h)}{(d e-c f) (b g-a h)}\right )+f (d g-c h) \operatorname {EllipticPi}\left (\frac {d (-f g+e h)}{(d e-c f) h},\arcsin \left (\sqrt {\frac {(-d e+c f) (g+h x)}{(f g-e h) (c+d x)}}\right ),\frac {(b c-a d) (-f g+e h)}{(d e-c f) (b g-a h)}\right )\right )}{(d e-c f) \sqrt {\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}}}\right )}{h^2 \sqrt {e+f x} \sqrt {g+h x}} \] Input:

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

Output:

(-2*Sqrt[a + b*x]*Sqrt[c + d*x]*(-((d*h*(e + f*x)*(g + h*x))/(c + d*x)) - 
((f*g - e*h)*Sqrt[((-(d*e) + c*f)*(d*g - c*h)*(e + f*x)*(g + h*x))/((f*g - 
 e*h)^2*(c + d*x)^2)]*((d*e - c*f)*h*EllipticE[ArcSin[Sqrt[((-(d*e) + c*f) 
*(g + h*x))/((f*g - e*h)*(c + d*x))]], ((b*c - a*d)*(-(f*g) + e*h))/((d*e 
- c*f)*(b*g - a*h))] + (-(d*e*h) + c*f*h)*EllipticF[ArcSin[Sqrt[((-(d*e) + 
 c*f)*(g + h*x))/((f*g - e*h)*(c + d*x))]], ((b*c - a*d)*(-(f*g) + e*h))/( 
(d*e - c*f)*(b*g - a*h))] + f*(d*g - c*h)*EllipticPi[(d*(-(f*g) + e*h))/(( 
d*e - c*f)*h), ArcSin[Sqrt[((-(d*e) + c*f)*(g + h*x))/((f*g - e*h)*(c + d* 
x))]], ((b*c - a*d)*(-(f*g) + e*h))/((d*e - c*f)*(b*g - a*h))]))/((d*e - c 
*f)*Sqrt[((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])))/(h^2*Sqrt[e + 
 f*x]*Sqrt[g + h*x])
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 472, normalized size of antiderivative = 0.67, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.102, Rules used = {2098, 183, 194, 327, 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.

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

\(\Big \downarrow \) 2098

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

\(\Big \downarrow \) 183

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

\(\Big \downarrow \) 194

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 412

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

Input:

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

Output:

(2*b*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(h*Sqrt[a + b*x]) - (2*Sqr 
t[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)))]*EllipticE[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)))])/(h*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))] 
*Sqrt[g + h*x]) - (2*d*(b*g - a*h)^(3/2)*Sqrt[((f*g - e*h)*(a + b*x))/((b* 
g - a*h)*(e + f*x))]*Sqrt[((f*g - e*h)*(c + d*x))/((d*g - c*h)*(e + f*x))] 
*(e + f*x)*EllipticPi[(f*(b*g - a*h))/((b*e - a*f)*h), ArcSin[(Sqrt[b*e - 
a*f]*Sqrt[g + h*x])/(Sqrt[b*g - a*h]*Sqrt[e + f*x])], ((d*e - c*f)*(b*g - 
a*h))/((b*e - a*f)*(d*g - c*h))])/(Sqrt[b*e - a*f]*h^2*Sqrt[a + b*x]*Sqrt[ 
c + d*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 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 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 2098

Int[(Sqrt[(a_.) + (b_.)*(x_)]*((A_.) + (B_.)*(x_)))/(Sqrt[(c_.) + (d_.)*(x_ 
)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbol] :> Simp[b* 
B*Sqrt[c + d*x]*Sqrt[e + f*x]*(Sqrt[g + h*x]/(d*f*h*Sqrt[a + b*x])), x] + ( 
-Simp[B*((b*g - a*h)/(2*f*h))   Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d 
*x]*Sqrt[g + h*x]), x], x] + Simp[B*(b*e - a*f)*((b*g - a*h)/(2*d*f*h))   I 
nt[Sqrt[c + d*x]/((a + b*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x]) /; 
FreeQ[{a, b, c, d, e, f, g, h, A, B}, x] && EqQ[2*A*d*f - B*(d*e + c*f), 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1559\) vs. \(2(637)=1274\).

Time = 7.44 (sec) , antiderivative size = 1560, normalized size of antiderivative = 2.22

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1560\)
default	\(\text {Expression too large to display}\)	\(14663\)

Input:

int((b*x+a)^(1/2)*(2*d*f*x+c*f+d*e)/(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*c*f+a*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)) 
+2*(2*a*d*f+b*c*f+b*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)*(-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)))+2*b*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/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)*...
 

Fricas [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((b*x+a)^(1/2)*(2*d*f*x+c*f+d*e)/(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]

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

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

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

Reduce [F]

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

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

Output:

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