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

Optimal result

Integrand size = 49, antiderivative size = 449 \[ \int \frac {d e+c f+2 d f x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 (b d e+b c f-2 a d f) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{b \sqrt {b g-a h} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac {4 d f \sqrt {-d g+c h} (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \operatorname {EllipticPi}\left (-\frac {b (d g-c h)}{(b c-a d) h},\arcsin \left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {-d g+c h} \sqrt {a+b x}}\right ),\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{b \sqrt {b c-a d} h \sqrt {c+d x} \sqrt {e+f x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.102, Rules used = {1612, 176, 430, 171, 551} \[ \int \frac {d e+c f+2 d f x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 \sqrt {g+h x} (-2 a d f+b c f+b d e) \sqrt {\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{b \sqrt {c+d x} \sqrt {b g-a h} \sqrt {f g-e h} \sqrt {-\frac {(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}+\frac {4 d f (a+b x) \sqrt {c h-d g} \sqrt {\frac {(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \sqrt {\frac {(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} \operatorname {EllipticPi}\left (-\frac {b (d g-c h)}{(b c-a d) h},\arcsin \left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {c h-d g} \sqrt {a+b x}}\right ),\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{b h \sqrt {c+d x} \sqrt {e+f x} \sqrt {b c-a d}} \]

[In]

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]

[Out]

(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[Arc
Sin[(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)))]) + (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])

Rule 171

Int[Sqrt[(a_.) + (b_.)*(x_)]/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_S
ymbol] :> Dist[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)*Sqrt[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 176

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x
_Symbol] :> Dist[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 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(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] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[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] && SimplerSqrtQ[-f/e, -d/c])

Rule 1612

Int[((A_.) + (B_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.
) + (h_.)*(x_)]), x_Symbol] :> Dist[(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] + Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {(2 d f) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{b}+\frac {(-2 a d f+b (d e+c f)) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{b} \\ & = \frac {\left (4 d f (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)}}\right ) \text {Subst}\left (\int \frac {1}{\left (h-b x^2\right ) \sqrt {1+\frac {(b c-a d) x^2}{d g-c h}} \sqrt {1+\frac {(b e-a f) x^2}{f g-e h}}} \, dx,x,\frac {\sqrt {g+h x}}{\sqrt {a+b x}}\right )}{b \sqrt {c+d x} \sqrt {e+f x}}+\frac {\left (2 (-2 a d f+b (d e+c f)) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {(b c-a d) x^2}{d e-c f}} \sqrt {1-\frac {(b g-a h) x^2}{f g-e h}}} \, dx,x,\frac {\sqrt {e+f x}}{\sqrt {a+b x}}\right )}{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 (b d e+b c f-2 a d f) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} F\left (\sin ^{-1}\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 {4 d f \sqrt {-d g+c 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)}} \Pi \left (-\frac {b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {-d g+c h} \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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 25.23 (sec) , antiderivative size = 723, normalized size of antiderivative = 1.61 \[ \int \frac {d e+c f+2 d f x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 \sqrt {a+b x} \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \left (-b d e (b e-a f) h \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} (g+h x) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right ),\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )+2 a d f (b e-a f) h \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} (g+h x) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right ),\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )+b c f (-b e+a f) h \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} (g+h x) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right ),\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )-2 d f (b g-a h) (f g-e h) (a+b x) \sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}} \sqrt {\frac {(-b e+a f) (b g-a h) (e+f x) (g+h x)}{(f g-e h)^2 (a+b x)^2}} \operatorname {EllipticPi}\left (\frac {b (-f g+e h)}{(b e-a f) h},\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right ),\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )\right )}{b (b e-a f) h (b g-a h) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}} \]

[In]

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]

[Out]

(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))])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(854\) vs. \(2(411)=822\).

Time = 6.28 (sec) , antiderivative size = 855, normalized size of antiderivative = 1.90

method	result	size
elliptic	\(\frac {\sqrt {\left (b x +a \right ) \left (d x +c \right ) \left (f x +e \right ) \left (h x +g \right )}\, \left (\frac {2 \left (c f +d e \right ) \left (\frac {g}{h}-\frac {a}{b}\right ) \sqrt {\frac {\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (x +\frac {a}{b}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}\, \left (x +\frac {c}{d}\right )^{2} \sqrt {\frac {\left (-\frac {c}{d}+\frac {a}{b}\right ) \left (x +\frac {e}{f}\right )}{\left (-\frac {e}{f}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}\, \sqrt {\frac {\left (-\frac {c}{d}+\frac {a}{b}\right ) \left (x +\frac {g}{h}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}\, F\left (\sqrt {\frac {\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (x +\frac {a}{b}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}, \sqrt {\frac {\left (\frac {e}{f}-\frac {c}{d}\right ) \left (\frac {g}{h}-\frac {a}{b}\right )}{\left (-\frac {a}{b}+\frac {e}{f}\right ) \left (-\frac {c}{d}+\frac {g}{h}\right )}}\right )}{\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (-\frac {c}{d}+\frac {a}{b}\right ) \sqrt {b d f h \left (x +\frac {a}{b}\right ) \left (x +\frac {c}{d}\right ) \left (x +\frac {e}{f}\right ) \left (x +\frac {g}{h}\right )}}+\frac {4 d f \left (\frac {g}{h}-\frac {a}{b}\right ) \sqrt {\frac {\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (x +\frac {a}{b}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}\, \left (x +\frac {c}{d}\right )^{2} \sqrt {\frac {\left (-\frac {c}{d}+\frac {a}{b}\right ) \left (x +\frac {e}{f}\right )}{\left (-\frac {e}{f}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}\, \sqrt {\frac {\left (-\frac {c}{d}+\frac {a}{b}\right ) \left (x +\frac {g}{h}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}\, \left (-\frac {c F\left (\sqrt {\frac {\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (x +\frac {a}{b}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}, \sqrt {\frac {\left (\frac {e}{f}-\frac {c}{d}\right ) \left (\frac {g}{h}-\frac {a}{b}\right )}{\left (-\frac {a}{b}+\frac {e}{f}\right ) \left (-\frac {c}{d}+\frac {g}{h}\right )}}\right )}{d}+\left (\frac {c}{d}-\frac {a}{b}\right ) \Pi \left (\sqrt {\frac {\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (x +\frac {a}{b}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}, \frac {-\frac {g}{h}+\frac {a}{b}}{-\frac {g}{h}+\frac {c}{d}}, \sqrt {\frac {\left (\frac {e}{f}-\frac {c}{d}\right ) \left (\frac {g}{h}-\frac {a}{b}\right )}{\left (-\frac {a}{b}+\frac {e}{f}\right ) \left (-\frac {c}{d}+\frac {g}{h}\right )}}\right )\right )}{\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (-\frac {c}{d}+\frac {a}{b}\right ) \sqrt {b d f h \left (x +\frac {a}{b}\right ) \left (x +\frac {c}{d}\right ) \left (x +\frac {e}{f}\right ) \left (x +\frac {g}{h}\right )}}\right )}{\sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}\)	\(855\)
default	\(\text {Expression too large to display}\)	\(2454\)

[In]

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)

[Out]

((b*x+a)*(d*x+c)*(f*x+e)*(h*x+g))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)*(2*(c*f+d*e)*(
g/h-a/b)*((-g/h+c/d)*(x+a/b)/(-g/h+a/b)/(x+c/d))^(1/2)*(x+c/d)^2*((-c/d+a/b)*(x+e/f)/(-e/f+a/b)/(x+c/d))^(1/2)
*((-c/d+a/b)*(x+g/h)/(-g/h+a/b)/(x+c/d))^(1/2)/(-g/h+c/d)/(-c/d+a/b)/(b*d*f*h*(x+a/b)*(x+c/d)*(x+e/f)*(x+g/h))
^(1/2)*EllipticF(((-g/h+c/d)*(x+a/b)/(-g/h+a/b)/(x+c/d))^(1/2),((e/f-c/d)*(g/h-a/b)/(-a/b+e/f)/(-c/d+g/h))^(1/
2))+4*d*f*(g/h-a/b)*((-g/h+c/d)*(x+a/b)/(-g/h+a/b)/(x+c/d))^(1/2)*(x+c/d)^2*((-c/d+a/b)*(x+e/f)/(-e/f+a/b)/(x+
c/d))^(1/2)*((-c/d+a/b)*(x+g/h)/(-g/h+a/b)/(x+c/d))^(1/2)/(-g/h+c/d)/(-c/d+a/b)/(b*d*f*h*(x+a/b)*(x+c/d)*(x+e/
f)*(x+g/h))^(1/2)*(-c/d*EllipticF(((-g/h+c/d)*(x+a/b)/(-g/h+a/b)/(x+c/d))^(1/2),((e/f-c/d)*(g/h-a/b)/(-a/b+e/f
)/(-c/d+g/h))^(1/2))+(c/d-a/b)*EllipticPi(((-g/h+c/d)*(x+a/b)/(-g/h+a/b)/(x+c/d))^(1/2),(-g/h+a/b)/(-g/h+c/d),
((e/f-c/d)*(g/h-a/b)/(-a/b+e/f)/(-c/d+g/h))^(1/2))))

Fricas [F(-1)]

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

[In]

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")

[Out]

Timed out

Sympy [F]

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

[In]

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)

[Out]

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]

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

[In]

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")

[Out]

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]

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

[In]

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")

[Out]

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

[In]

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)

[Out]

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)