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

Optimal. Leaf size=161 \[ -\frac {2 \sqrt {e+f x} \sqrt {\frac {(c+d x) (f g-e h)}{(e+f x) (d g-c h)}} \operatorname {EllipticF}\left (\tan ^{-1}\left (\frac {\sqrt {g+h x} \sqrt {b e-a f}}{\sqrt {a+b x} \sqrt {f g-e h}}\right ),\frac {(b g-a h) (d e-c f)}{(b e-a f) (d g-c h)}\right )}{\sqrt {c+d x} \sqrt {b e-a f} \sqrt {f g-e h}} \]

[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)

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Rubi [A]  time = 0.08, antiderivative size = 198, normalized size of antiderivative = 1.23, number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {170, 419} \[ \frac {2 \sqrt {g+h x} \sqrt {\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} F\left (\sin ^{-1}\left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right )|-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt {c+d x} \sqrt {b g-a h} \sqrt {f g-e h} \sqrt {-\frac {(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}} \]

Antiderivative was successfully verified.

[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 170

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 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx &=\frac {\left (2 \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}\right ) \operatorname {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*}

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Mathematica [A]  time = 1.34, size = 227, normalized size = 1.41 \[ -\frac {2 \sqrt {a+b x} \sqrt {g+h x} \sqrt {\frac {(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \sqrt {\frac {(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {(g+h x) (a f-b e)}{(a+b x) (f g-e h)}}\right ),\frac {(a d-b c) (e h-f g)}{(b e-a f) (d g-c h)}\right )}{\sqrt {c+d x} \sqrt {e+f x} (b g-a h) \sqrt {\frac {(g+h x) (a f-b e)}{(a+b x) (f g-e h)}}} \]

Antiderivative was successfully verified.

[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))])

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fricas [F]  time = 5.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}{b d f h x^{4} + a c e g + {\left (b d f g + {\left (b d e + {\left (b c + a d\right )} f\right )} h\right )} x^{3} + {\left ({\left (b d e + {\left (b c + a d\right )} f\right )} g + {\left (a c f + {\left (b c + a d\right )} e\right )} h\right )} x^{2} + {\left (a c e h + {\left (a c f + {\left (b c + a d\right )} e\right )} g\right )} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.07, size = 270, normalized size = 1.68 \[ \frac {2 \sqrt {\frac {\left (a f -b e \right ) \left (h x +g \right )}{\left (a h -b g \right ) \left (f x +e \right )}}\, \sqrt {\frac {\left (e h -f g \right ) \left (d x +c \right )}{\left (c h -d g \right ) \left (f x +e \right )}}\, \sqrt {\frac {\left (e h -f g \right ) \left (b x +a \right )}{\left (a h -b g \right ) \left (f x +e \right )}}\, \left (a \,f^{2} h \,x^{2}-b \,f^{2} g \,x^{2}+2 a e f h x -2 b e f g x +a \,e^{2} h -b \,e^{2} g \right ) \EllipticF \left (\sqrt {\frac {\left (a f -b e \right ) \left (h x +g \right )}{\left (a h -b g \right ) \left (f x +e \right )}}, \sqrt {\frac {\left (c f -d e \right ) \left (a h -b g \right )}{\left (c h -d g \right ) \left (a f -b e \right )}}\right )}{\sqrt {h x +g}\, \sqrt {f x +e}\, \sqrt {d x +c}\, \sqrt {b x +a}\, \left (e h -f g \right ) \left (a f -b e \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\sqrt {a+b\,x}\,\sqrt {c+d\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + b x} \sqrt {c + d x} \sqrt {e + f x} \sqrt {g + h x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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