3.655 \(\int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+c x^2}} \, dx\)

Optimal. Leaf size=454 \[ -\frac {(d+e x) \sqrt [4]{a g^2+c f^2} \sqrt {\frac {\left (a+c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (a g^2+c f^2\right )}} \left (\frac {(f+g x) \sqrt {a e^2+c d^2}}{(d+e x) \sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {\frac {(f+g x)^2 \left (a e^2+c d^2\right )}{(d+e x)^2 \left (a g^2+c f^2\right )}-\frac {2 (f+g x) (a e g+c d f)}{(d+e x) \left (a g^2+c f^2\right )}+1}{\left (\frac {(f+g x) \sqrt {a e^2+c d^2}}{(d+e x) \sqrt {a g^2+c f^2}}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c d^2+a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2} \sqrt {d+e x}}\right )|\frac {1}{2} \left (\frac {c d f+a e g}{\sqrt {c d^2+a e^2} \sqrt {c f^2+a g^2}}+1\right )\right )}{\sqrt {a+c x^2} \sqrt [4]{a e^2+c d^2} (e f-d g) \sqrt {\frac {(f+g x)^2 \left (a e^2+c d^2\right )}{(d+e x)^2 \left (a g^2+c f^2\right )}-\frac {2 (f+g x) (a e g+c d f)}{(d+e x) \left (a g^2+c f^2\right )}+1}} \]

[Out]

-(a*g^2+c*f^2)^(1/4)*(e*x+d)*(cos(2*arctan((a*e^2+c*d^2)^(1/4)*(g*x+f)^(1/2)/(a*g^2+c*f^2)^(1/4)/(e*x+d)^(1/2)
))^2)^(1/2)/cos(2*arctan((a*e^2+c*d^2)^(1/4)*(g*x+f)^(1/2)/(a*g^2+c*f^2)^(1/4)/(e*x+d)^(1/2)))*EllipticF(sin(2
*arctan((a*e^2+c*d^2)^(1/4)*(g*x+f)^(1/2)/(a*g^2+c*f^2)^(1/4)/(e*x+d)^(1/2))),1/2*(2+2*(a*e*g+c*d*f)/(a*e^2+c*
d^2)^(1/2)/(a*g^2+c*f^2)^(1/2))^(1/2))*(1+(g*x+f)*(a*e^2+c*d^2)^(1/2)/(e*x+d)/(a*g^2+c*f^2)^(1/2))*((-d*g+e*f)
^2*(c*x^2+a)/(a*g^2+c*f^2)/(e*x+d)^2)^(1/2)*((1-2*(a*e*g+c*d*f)*(g*x+f)/(a*g^2+c*f^2)/(e*x+d)+(a*e^2+c*d^2)*(g
*x+f)^2/(a*g^2+c*f^2)/(e*x+d)^2)/(1+(g*x+f)*(a*e^2+c*d^2)^(1/2)/(e*x+d)/(a*g^2+c*f^2)^(1/2))^2)^(1/2)/(a*e^2+c
*d^2)^(1/4)/(-d*g+e*f)/(c*x^2+a)^(1/2)/(1-2*(a*e*g+c*d*f)*(g*x+f)/(a*g^2+c*f^2)/(e*x+d)+(a*e^2+c*d^2)*(g*x+f)^
2/(a*g^2+c*f^2)/(e*x+d)^2)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.63, antiderivative size = 454, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {936, 1103} \[ -\frac {(d+e x) \sqrt [4]{a g^2+c f^2} \sqrt {\frac {\left (a+c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (a g^2+c f^2\right )}} \left (\frac {(f+g x) \sqrt {a e^2+c d^2}}{(d+e x) \sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {\frac {(f+g x)^2 \left (a e^2+c d^2\right )}{(d+e x)^2 \left (a g^2+c f^2\right )}-\frac {2 (f+g x) (a e g+c d f)}{(d+e x) \left (a g^2+c f^2\right )}+1}{\left (\frac {(f+g x) \sqrt {a e^2+c d^2}}{(d+e x) \sqrt {a g^2+c f^2}}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c d^2+a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2} \sqrt {d+e x}}\right )|\frac {1}{2} \left (\frac {c d f+a e g}{\sqrt {c d^2+a e^2} \sqrt {c f^2+a g^2}}+1\right )\right )}{\sqrt {a+c x^2} \sqrt [4]{a e^2+c d^2} (e f-d g) \sqrt {\frac {(f+g x)^2 \left (a e^2+c d^2\right )}{(d+e x)^2 \left (a g^2+c f^2\right )}-\frac {2 (f+g x) (a e g+c d f)}{(d+e x) \left (a g^2+c f^2\right )}+1}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[d + e*x]*Sqrt[f + g*x]*Sqrt[a + c*x^2]),x]

[Out]

-(((c*f^2 + a*g^2)^(1/4)*(d + e*x)*Sqrt[((e*f - d*g)^2*(a + c*x^2))/((c*f^2 + a*g^2)*(d + e*x)^2)]*(1 + (Sqrt[
c*d^2 + a*e^2]*(f + g*x))/(Sqrt[c*f^2 + a*g^2]*(d + e*x)))*Sqrt[(1 - (2*(c*d*f + a*e*g)*(f + g*x))/((c*f^2 + a
*g^2)*(d + e*x)) + ((c*d^2 + a*e^2)*(f + g*x)^2)/((c*f^2 + a*g^2)*(d + e*x)^2))/(1 + (Sqrt[c*d^2 + a*e^2]*(f +
g*x))/(Sqrt[c*f^2 + a*g^2]*(d + e*x)))^2]*EllipticF[2*ArcTan[((c*d^2 + a*e^2)^(1/4)*Sqrt[f + g*x])/((c*f^2 +
a*g^2)^(1/4)*Sqrt[d + e*x])], (1 + (c*d*f + a*e*g)/(Sqrt[c*d^2 + a*e^2]*Sqrt[c*f^2 + a*g^2]))/2])/((c*d^2 + a*
e^2)^(1/4)*(e*f - d*g)*Sqrt[a + c*x^2]*Sqrt[1 - (2*(c*d*f + a*e*g)*(f + g*x))/((c*f^2 + a*g^2)*(d + e*x)) + ((
c*d^2 + a*e^2)*(f + g*x)^2)/((c*f^2 + a*g^2)*(d + e*x)^2)]))

Rule 936

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[(-2*(d
+ e*x)*Sqrt[((e*f - d*g)^2*(a + c*x^2))/((c*f^2 + a*g^2)*(d + e*x)^2)])/((e*f - d*g)*Sqrt[a + c*x^2]), Subst[I
nt[1/Sqrt[1 - ((2*c*d*f + 2*a*e*g)*x^2)/(c*f^2 + a*g^2) + ((c*d^2 + a*e^2)*x^4)/(c*f^2 + a*g^2)], x], x, Sqrt[
f + g*x]/Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0]

Rule 1103

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(
a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(2*q*Sqrt[a + b*x^2 + c
*x^4]), x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+c x^2}} \, dx &=-\frac {\left (2 (d+e x) \sqrt {\frac {(e f-d g)^2 \left (a+c x^2\right )}{\left (c f^2+a g^2\right ) (d+e x)^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {(2 c d f+2 a e g) x^2}{c f^2+a g^2}+\frac {\left (c d^2+a e^2\right ) x^4}{c f^2+a g^2}}} \, dx,x,\frac {\sqrt {f+g x}}{\sqrt {d+e x}}\right )}{(e f-d g) \sqrt {a+c x^2}}\\ &=-\frac {\sqrt [4]{c f^2+a g^2} (d+e x) \sqrt {\frac {(e f-d g)^2 \left (a+c x^2\right )}{\left (c f^2+a g^2\right ) (d+e x)^2}} \left (1+\frac {\sqrt {c d^2+a e^2} (f+g x)}{\sqrt {c f^2+a g^2} (d+e x)}\right ) \sqrt {\frac {1-\frac {2 (c d f+a e g) (f+g x)}{\left (c f^2+a g^2\right ) (d+e x)}+\frac {\left (c d^2+a e^2\right ) (f+g x)^2}{\left (c f^2+a g^2\right ) (d+e x)^2}}{\left (1+\frac {\sqrt {c d^2+a e^2} (f+g x)}{\sqrt {c f^2+a g^2} (d+e x)}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c d^2+a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2} \sqrt {d+e x}}\right )|\frac {1}{2} \left (1+\frac {c d f+a e g}{\sqrt {c d^2+a e^2} \sqrt {c f^2+a g^2}}\right )\right )}{\sqrt [4]{c d^2+a e^2} (e f-d g) \sqrt {a+c x^2} \sqrt {1-\frac {2 (c d f+a e g) (f+g x)}{\left (c f^2+a g^2\right ) (d+e x)}+\frac {\left (c d^2+a e^2\right ) (f+g x)^2}{\left (c f^2+a g^2\right ) (d+e x)^2}}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 1.40, size = 344, normalized size = 0.76 \[ \frac {\sqrt {2} \left (\sqrt {c} x+i \sqrt {a}\right ) \sqrt {d+e x} \sqrt {\frac {\frac {i \sqrt {c} d x}{\sqrt {a}}-\frac {i \sqrt {a} e}{\sqrt {c}}+d+e x}{d+e x}} \sqrt {\frac {(f+g x) \left (\sqrt {a} e+i \sqrt {c} d\right )}{(d+e x) \left (\sqrt {a} g+i \sqrt {c} f\right )}} F\left (\sin ^{-1}\left (\sqrt {\frac {(e f-d g) \left (\sqrt {c} x+i \sqrt {a}\right )}{\left (\sqrt {c} f-i \sqrt {a} g\right ) (d+e x)}}\right )|-\frac {\frac {i \sqrt {c} d f}{\sqrt {a}}-e f+d g+\frac {i \sqrt {a} e g}{\sqrt {c}}}{2 e f-2 d g}\right )}{\sqrt {a+c x^2} \sqrt {f+g x} \left (\sqrt {c} d-i \sqrt {a} e\right ) \sqrt {\frac {\left (\sqrt {c} x+i \sqrt {a}\right ) (e f-d g)}{(d+e x) \left (\sqrt {c} f-i \sqrt {a} g\right )}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[d + e*x]*Sqrt[f + g*x]*Sqrt[a + c*x^2]),x]

[Out]

(Sqrt[2]*(I*Sqrt[a] + Sqrt[c]*x)*Sqrt[d + e*x]*Sqrt[(d - (I*Sqrt[a]*e)/Sqrt[c] + (I*Sqrt[c]*d*x)/Sqrt[a] + e*x
)/(d + e*x)]*Sqrt[((I*Sqrt[c]*d + Sqrt[a]*e)*(f + g*x))/((I*Sqrt[c]*f + Sqrt[a]*g)*(d + e*x))]*EllipticF[ArcSi
n[Sqrt[((e*f - d*g)*(I*Sqrt[a] + Sqrt[c]*x))/((Sqrt[c]*f - I*Sqrt[a]*g)*(d + e*x))]], -(((I*Sqrt[c]*d*f)/Sqrt[
a] - e*f + d*g + (I*Sqrt[a]*e*g)/Sqrt[c])/(2*e*f - 2*d*g))])/((Sqrt[c]*d - I*Sqrt[a]*e)*Sqrt[((e*f - d*g)*(I*S
qrt[a] + Sqrt[c]*x))/((Sqrt[c]*f - I*Sqrt[a]*g)*(d + e*x))]*Sqrt[f + g*x]*Sqrt[a + c*x^2])

________________________________________________________________________________________

fricas [F]  time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} \sqrt {e x + d} \sqrt {g x + f}}{c e g x^{4} + {\left (c e f + c d g\right )} x^{3} + a d f + {\left (c d f + a e g\right )} x^{2} + {\left (a e f + a d g\right )} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c x^{2} + a} \sqrt {e x + d} \sqrt {g x + f}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.20, size = 433, normalized size = 0.95 \[ \frac {2 \left (c \,e^{2} f \,x^{2}+2 c d e f x -\sqrt {-a c}\, e^{2} g \,x^{2}+c \,d^{2} f -2 \sqrt {-a c}\, d e g x -\sqrt {-a c}\, d^{2} g \right ) \sqrt {\frac {\left (d g -e f \right ) \left (c x +\sqrt {-a c}\right )}{\left (-c f +\sqrt {-a c}\, g \right ) \left (e x +d \right )}}\, \sqrt {\frac {\left (d g -e f \right ) \left (-c x +\sqrt {-a c}\right )}{\left (c f +\sqrt {-a c}\, g \right ) \left (e x +d \right )}}\, \sqrt {\frac {\left (-c d +\sqrt {-a c}\, e \right ) \left (g x +f \right )}{\left (-c f +\sqrt {-a c}\, g \right ) \left (e x +d \right )}}\, \sqrt {e x +d}\, \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, \EllipticF \left (\sqrt {\frac {\left (-c d +\sqrt {-a c}\, e \right ) \left (g x +f \right )}{\left (-c f +\sqrt {-a c}\, g \right ) \left (e x +d \right )}}, \sqrt {\frac {\left (c d +\sqrt {-a c}\, e \right ) \left (-c f +\sqrt {-a c}\, g \right )}{\left (c f +\sqrt {-a c}\, g \right ) \left (-c d +\sqrt {-a c}\, e \right )}}\right )}{\sqrt {-\frac {\left (g x +f \right ) \left (e x +d \right ) \left (-c x +\sqrt {-a c}\right ) \left (c x +\sqrt {-a c}\right )}{c}}\, \left (d g -e f \right ) \left (c d -\sqrt {-a c}\, e \right ) \sqrt {c e g \,x^{4}+c d g \,x^{3}+c e f \,x^{3}+a e g \,x^{2}+c d f \,x^{2}+a d g x +a e f x +a d f}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(c*e^2*f*x^2-(-a*c)^(1/2)*x^2*e^2*g+2*c*d*e*f*x-2*(-a*c)^(1/2)*x*d*e*g+c*d^2*f-(-a*c)^(1/2)*d^2*g)*EllipticF
((((-a*c)^(1/2)*e-c*d)*(g*x+f)/(-c*f+(-a*c)^(1/2)*g)/(e*x+d))^(1/2),((c*d+(-a*c)^(1/2)*e)*(-c*f+(-a*c)^(1/2)*g
)/(c*f+(-a*c)^(1/2)*g)/((-a*c)^(1/2)*e-c*d))^(1/2))*((d*g-e*f)*(c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)/(e*x+d
))^(1/2)*((d*g-e*f)*(-c*x+(-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)/(e*x+d))^(1/2)*(((-a*c)^(1/2)*e-c*d)*(g*x+f)/(-c*
f+(-a*c)^(1/2)*g)/(e*x+d))^(1/2)*(e*x+d)^(1/2)*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/(-1/c*(g*x+f)*(e*x+d)*(-c*x+(-a*c
)^(1/2))*(c*x+(-a*c)^(1/2)))^(1/2)/(d*g-e*f)/(c*d-(-a*c)^(1/2)*e)/(c*e*g*x^4+c*d*g*x^3+c*e*f*x^3+a*e*g*x^2+c*d
*f*x^2+a*d*g*x+a*e*f*x+a*d*f)^(1/2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c x^{2} + a} \sqrt {e x + d} \sqrt {g x + f}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\sqrt {f+g\,x}\,\sqrt {c\,x^2+a}\,\sqrt {d+e\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + c x^{2}} \sqrt {d + e x} \sqrt {f + g x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(a + c*x**2)*sqrt(d + e*x)*sqrt(f + g*x)), x)

________________________________________________________________________________________