\(\int \frac {\sqrt {a+b x} \sqrt {c+d x}}{\sqrt {e+f x} \sqrt {g+h x}} \, dx\) [99]

Optimal result

Integrand size = 37, antiderivative size = 721 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{\sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \sqrt {g+h x}}{h \sqrt {e+f x}}-\frac {\sqrt {d g-c h} \sqrt {f g-e h} \sqrt {a+b x} \sqrt {\frac {(d e-c f) (g+h x)}{(d g-c h) (e+f x)}} E\left (\arcsin \left (\frac {\sqrt {f g-e h} \sqrt {c+d x}}{\sqrt {d g-c h} \sqrt {e+f x}}\right )|\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{f h \sqrt {-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}} \sqrt {g+h x}}+\frac {(d e-c f) (b f g+b e h-2 a f h) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{f^2 h \sqrt {b g-a h} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac {\sqrt {b g-a h} (a d f h-b (d f g+d e h-c f h)) \sqrt {\frac {(f g-e h) (a+b x)}{(b g-a h) (e+f x)}} \sqrt {\frac {(f g-e h) (c+d x)}{(d g-c h) (e+f x)}} (e+f x) \operatorname {EllipticPi}\left (\frac {f (b g-a h)}{(b e-a f) h},\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {g+h x}}{\sqrt {b g-a h} \sqrt {e+f x}}\right ),\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{f^2 \sqrt {b e-a f} h^2 \sqrt {a+b x} \sqrt {c+d x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 721, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {179, 182, 435, 171, 551, 176, 430} \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{\sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {(e+f x) \sqrt {b g-a h} \sqrt {\frac {(a+b x) (f g-e h)}{(e+f x) (b g-a h)}} \sqrt {\frac {(c+d x) (f g-e h)}{(e+f x) (d g-c h)}} (a d f h-b (-c f h+d e h+d f g)) \operatorname {EllipticPi}\left (\frac {f (b g-a h)}{(b e-a f) h},\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {g+h x}}{\sqrt {b g-a h} \sqrt {e+f x}}\right ),\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{f^2 h^2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {b e-a f}}+\frac {\sqrt {g+h x} (d e-c f) (-2 a f h+b e h+b f g) \sqrt {\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{f^2 h \sqrt {c+d x} \sqrt {b g-a h} \sqrt {f g-e h} \sqrt {-\frac {(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}-\frac {\sqrt {a+b x} \sqrt {d g-c h} \sqrt {f g-e h} \sqrt {\frac {(g+h x) (d e-c f)}{(e+f x) (d g-c h)}} E\left (\arcsin \left (\frac {\sqrt {f g-e h} \sqrt {c+d x}}{\sqrt {d g-c h} \sqrt {e+f x}}\right )|\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{f h \sqrt {g+h x} \sqrt {-\frac {(a+b x) (d e-c f)}{(e+f x) (b c-a d)}}}+\frac {\sqrt {a+b x} \sqrt {c+d x} \sqrt {g+h x}}{h \sqrt {e+f x}} \]

[In]

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[g + h*x])/(h*Sqrt[e + f*x]) - (Sqrt[d*g - c*h]*Sqrt[f*g - e*h]*Sqrt[a + b*x]
*Sqrt[((d*e - c*f)*(g + h*x))/((d*g - c*h)*(e + f*x))]*EllipticE[ArcSin[(Sqrt[f*g - e*h]*Sqrt[c + d*x])/(Sqrt[
d*g - c*h]*Sqrt[e + f*x])], ((b*e - a*f)*(d*g - c*h))/((b*c - a*d)*(f*g - e*h))])/(f*h*Sqrt[-(((d*e - c*f)*(a
+ b*x))/((b*c - a*d)*(e + f*x)))]*Sqrt[g + h*x]) + ((d*e - c*f)*(b*f*g + b*e*h - 2*a*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)))])/(f^2*h*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)))]) + (Sqrt[b*g - a*h]*(a*d*f*h - b
*(d*f*g + d*e*h - c*f*h))*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))])/(f^2*Sqrt[b*e -
a*f]*h^2*Sqrt[a + b*x]*Sqrt[c + d*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 179

Int[(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)])/(Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x
_Symbol] :> Simp[Sqrt[a + b*x]*Sqrt[c + d*x]*(Sqrt[g + h*x]/(h*Sqrt[e + f*x])), x] + (-Dist[(d*e - c*f)*((f*g
- e*h)/(2*f*h)), Int[Sqrt[a + b*x]/(Sqrt[c + d*x]*(e + f*x)^(3/2)*Sqrt[g + h*x]), x], x] + Dist[(a*d*f*h - b*(
d*f*g + d*e*h - c*f*h))/(2*f^2*h), Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[g + h*x]), x], x] + Dis
t[(d*e - c*f)*((b*f*g + b*e*h - 2*a*f*h)/(2*f^2*h)), Int[1/(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}, x]

Rule 182

Int[Sqrt[(c_.) + (d_.)*(x_)]/(((a_.) + (b_.)*(x_))^(3/2)*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x
_Symbol] :> Dist[-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]*Sqrt[(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 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 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[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 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])

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a+b x} \sqrt {c+d x} \sqrt {g+h x}}{h \sqrt {e+f x}}-\frac {((d e-c f) (f g-e h)) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x} (e+f x)^{3/2} \sqrt {g+h x}} \, dx}{2 f h}+\frac {((d e-c f) (b f g+b e h-2 a f h)) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{2 f^2 h}+\frac {(a d f h-b (d f g+d e h-c f h)) \int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {g+h x}} \, dx}{2 f^2 h} \\ & = \frac {\sqrt {a+b x} \sqrt {c+d x} \sqrt {g+h x}}{h \sqrt {e+f x}}+\frac {\left ((a d f h-b (d f g+d e h-c f h)) \sqrt {\frac {(f g-e h) (a+b x)}{(b g-a h) (e+f x)}} \sqrt {\frac {(f g-e h) (c+d x)}{(d g-c h) (e+f x)}} (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\left (h-f x^2\right ) \sqrt {1+\frac {(-b e+a f) x^2}{b g-a h}} \sqrt {1+\frac {(-d e+c f) x^2}{d g-c h}}} \, dx,x,\frac {\sqrt {g+h x}}{\sqrt {e+f x}}\right )}{f^2 h \sqrt {a+b x} \sqrt {c+d x}}+\frac {\left ((d e-c f) (b f g+b e h-2 a f h) \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 )}{f^2 h (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 {\left ((d e-c f) (f g-e h) \sqrt {a+b x} \sqrt {\frac {(d e-c f) (g+h x)}{(d g-c h) (e+f x)}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {(-b e+a f) x^2}{b c-a d}}}{\sqrt {1-\frac {(f g-e h) x^2}{d g-c h}}} \, dx,x,\frac {\sqrt {c+d x}}{\sqrt {e+f x}}\right )}{f (-d e+c f) h \sqrt {\frac {(-d e+c f) (a+b x)}{(b c-a d) (e+f x)}} \sqrt {g+h x}} \\ & = \frac {\sqrt {a+b x} \sqrt {c+d x} \sqrt {g+h x}}{h \sqrt {e+f x}}-\frac {\sqrt {d g-c h} \sqrt {f g-e h} \sqrt {a+b x} \sqrt {\frac {(d e-c f) (g+h x)}{(d g-c h) (e+f x)}} E\left (\sin ^{-1}\left (\frac {\sqrt {f g-e h} \sqrt {c+d x}}{\sqrt {d g-c h} \sqrt {e+f x}}\right )|\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{f h \sqrt {-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}} \sqrt {g+h x}}+\frac {(d e-c f) (b f g+b e h-2 a f h) \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 )}{f^2 h \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 {\sqrt {b g-a h} (a d f h-b (d f g+d e h-c f h)) \sqrt {\frac {(f g-e h) (a+b x)}{(b g-a h) (e+f x)}} \sqrt {\frac {(f g-e h) (c+d x)}{(d g-c h) (e+f x)}} (e+f x) \Pi \left (\frac {f (b g-a h)}{(b e-a f) h};\sin ^{-1}\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 )}{f^2 \sqrt {b e-a f} h^2 \sqrt {a+b x} \sqrt {c+d x}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 48.04 (sec) , antiderivative size = 484, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{\sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-f^2 h (g+h x)+\frac {\sqrt {\frac {(f g-e h) (a+b x)}{(b g-a h) (e+f x)}} (g+h x) \left (-f (-d e+c f) h (-b g+a h) E\left (\arcsin \left (\sqrt {\frac {(d e-c f) (g+h x)}{(d g-c h) (e+f x)}}\right )|\frac {(b e-a f) (d g-c h)}{(d e-c f) (b g-a h)}\right )+(d e-c f) h (b f g+b e h-2 a f h) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {(d e-c f) (g+h x)}{(d g-c h) (e+f x)}}\right ),\frac {(b e-a f) (d g-c h)}{(d e-c f) (b g-a h)}\right )+(f g-e h) (-a d f h+b (d f g+d e h-c f h)) \operatorname {EllipticPi}\left (\frac {d f g-c f h}{d e h-c f h},\arcsin \left (\sqrt {\frac {(d e-c f) (g+h x)}{(d g-c h) (e+f x)}}\right ),\frac {(b e-a f) (d g-c h)}{(d e-c f) (b g-a h)}\right )\right )}{(d g-c h) (a+b x) \sqrt {\frac {(-d e+c f) (-f g+e h) (c+d x) (g+h x)}{(d g-c h)^2 (e+f x)^2}}}\right )}{f^2 h^2 \sqrt {e+f x} \sqrt {g+h x}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1543\) vs. \(2(656)=1312\).

Time = 4.03 (sec) , antiderivative size = 1544, normalized size of antiderivative = 2.14

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1544\)
default	\(\text {Expression too large to display}\)	\(15274\)

[In]

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

Fricas [F(-1)]

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

[In]

integrate((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]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

Giac [F]

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

[In]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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