\(\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [109]

Optimal result

Integrand size = 37, antiderivative size = 161 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 \sqrt {\frac {(f g-e h) (c+d x)}{(d g-c h) (e+f x)}} \sqrt {e+f x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b e-a f} \sqrt {g+h x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{\sqrt {b e-a f} \sqrt {f g-e h} \sqrt {c+d x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.23, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {176, 430} \[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 \sqrt {g+h x} \sqrt {\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt {c+d x} \sqrt {b g-a h} \sqrt {f g-e h} \sqrt {-\frac {(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}} \]

[In]

Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

(2*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)))])/(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)))])

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])

Rubi steps \begin{align*} \text {integral}& = \frac {\left (2 \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 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 \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 )}{\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)}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 23.05 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.41 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 \sqrt {a+b x} \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right ),\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )}{(b g-a h) \sqrt {c+d x} \sqrt {e+f x} \sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}} \]

[In]

Integrate[1/(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))]*Sqrt[((b*g - a*h)*(e + f*x))/((f*g - e
*h)*(a + b*x))]*Sqrt[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*g - a*h)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[((-(b*e)
+ a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))])

Maple [A] (verified)

Time = 1.50 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.68

method	result	size
default	\(-\frac {2 \sqrt {\frac {\left (c h -d g \right ) \left (b x +a \right )}{\left (a h -g b \right ) \left (d x +c \right )}}\, \sqrt {\frac {\left (a d -b c \right ) \left (f x +e \right )}{\left (a f -b e \right ) \left (d x +c \right )}}\, \sqrt {\frac {\left (a d -b c \right ) \left (h x +g \right )}{\left (a h -g b \right ) \left (d x +c \right )}}\, F\left (\sqrt {\frac {\left (c h -d g \right ) \left (b x +a \right )}{\left (a h -g b \right ) \left (d x +c \right )}}, \sqrt {\frac {\left (c f -d e \right ) \left (a h -g b \right )}{\left (a f -b e \right ) \left (c h -d g \right )}}\right ) \left (a \,d^{2} h \,x^{2}-b \,d^{2} g \,x^{2}+2 a c d h x -2 b c d g x +a \,c^{2} h -b \,c^{2} g \right )}{\sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}\, \left (c h -d g \right ) \left (a d -b c \right )}\)	\(270\)
elliptic	\(\frac {2 \sqrt {\left (b x +a \right ) \left (d x +c \right ) \left (f x +e \right ) \left (h x +g \right )}\, \left (\frac {g}{h}-\frac {a}{b}\right ) \sqrt {\frac {\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (x +\frac {a}{b}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}\, \left (x +\frac {c}{d}\right )^{2} \sqrt {\frac {\left (-\frac {c}{d}+\frac {a}{b}\right ) \left (x +\frac {e}{f}\right )}{\left (-\frac {e}{f}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}\, \sqrt {\frac {\left (-\frac {c}{d}+\frac {a}{b}\right ) \left (x +\frac {g}{h}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}\, F\left (\sqrt {\frac {\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (x +\frac {a}{b}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}, \sqrt {\frac {\left (\frac {e}{f}-\frac {c}{d}\right ) \left (\frac {g}{h}-\frac {a}{b}\right )}{\left (-\frac {a}{b}+\frac {e}{f}\right ) \left (-\frac {c}{d}+\frac {g}{h}\right )}}\right )}{\sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}\, \left (-\frac {g}{h}+\frac {c}{d}\right ) \left (-\frac {c}{d}+\frac {a}{b}\right ) \sqrt {b d f h \left (x +\frac {a}{b}\right ) \left (x +\frac {c}{d}\right ) \left (x +\frac {e}{f}\right ) \left (x +\frac {g}{h}\right )}}\)	\(374\)

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {1}{\sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]

[In]

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

[Out]

integral(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)/(b*d*f*h*x^4 + a*c*e*g + (b*d*f*g + (b*d*e +
(b*c + a*d)*f)*h)*x^3 + ((b*d*e + (b*c + a*d)*f)*g + (a*c*f + (b*c + a*d)*e)*h)*x^2 + (a*c*e*h + (a*c*f + (b*c
+ a*d)*e)*g)*x), x)

Sympy [F]

\[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {1}{\sqrt {a + b x} \sqrt {c + d x} \sqrt {e + f x} \sqrt {g + h x}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {1}{\sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]

[In]

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

Giac [F]

\[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {1}{\sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]

[In]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {1}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\sqrt {a+b\,x}\,\sqrt {c+d\,x}} \,d x \]

[In]

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

[Out]

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