\(\int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+c x^2}} \, dx\) [146]

Optimal result

Integrand size = 30, antiderivative size = 461 \[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=-\frac {2 \sqrt {c d f+a e g+\sqrt {-a} \sqrt {c} (e f-d g)} (d+e x) \sqrt {\frac {(e f-d g)^2 \left (a+c x^2\right )}{\left (c f^2+a g^2\right ) (d+e x)^2}} \sqrt {1-\frac {\left (c d^2+a e^2\right ) (f+g x)}{\left (c d f+a e g-\sqrt {-a} \sqrt {c} (e f-d g)\right ) (d+e x)}} \sqrt {1-\frac {\left (c d^2+a e^2\right ) (f+g x)}{\left (c d f+a e g+\sqrt {-a} \sqrt {c} (e f-d g)\right ) (d+e x)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c d^2+a e^2} \sqrt {f+g x}}{\sqrt {c d f+a e g+\sqrt {-a} \sqrt {c} (e f-d g)} \sqrt {d+e x}}\right ),\frac {c d f+a e g+\sqrt {-a} \sqrt {c} (e f-d g)}{c d f+a e g-\sqrt {-a} \sqrt {c} (e f-d g)}\right )}{\sqrt {c d^2+a e^2} (e f-d g) \sqrt {a+c x^2} \sqrt {1-\frac {2 (c d f+a e g) (f+g x)}{\left (c f^2+a g^2\right ) (d+e x)}+\frac {\left (c d^2+a e^2\right ) (f+g x)^2}{\left (c f^2+a g^2\right ) (d+e x)^2}}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 25.15 (sec) , antiderivative size = 344, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\frac {\sqrt {2} \left (i \sqrt {a}+\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {\frac {d-\frac {i \sqrt {a} e}{\sqrt {c}}+\frac {i \sqrt {c} d x}{\sqrt {a}}+e x}{d+e x}} \sqrt {\frac {\left (i \sqrt {c} d+\sqrt {a} e\right ) (f+g x)}{\left (i \sqrt {c} f+\sqrt {a} g\right ) (d+e x)}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {(e f-d g) \left (i \sqrt {a}+\sqrt {c} x\right )}{\left (\sqrt {c} f-i \sqrt {a} g\right ) (d+e x)}}\right ),-\frac {\frac {i \sqrt {c} d f}{\sqrt {a}}-e f+d g+\frac {i \sqrt {a} e g}{\sqrt {c}}}{2 e f-2 d g}\right )}{\left (\sqrt {c} d-i \sqrt {a} e\right ) \sqrt {\frac {(e f-d g) \left (i \sqrt {a}+\sqrt {c} x\right )}{\left (\sqrt {c} f-i \sqrt {a} g\right ) (d+e x)}} \sqrt {f+g x} \sqrt {a+c x^2}} \] Input:

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

Output:

(Sqrt[2]*(I*Sqrt[a] + Sqrt[c]*x)*Sqrt[d + e*x]*Sqrt[(d - (I*Sqrt[a]*e)/Sqr 
t[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[ArcSin[ 
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*Sqrt[ 
a] + Sqrt[c]*x))/((Sqrt[c]*f - I*Sqrt[a]*g)*(d + e*x))]*Sqrt[f + g*x]*Sqrt 
[a + c*x^2])
 

Rubi [A] (warning: unable to verify)

Time = 0.85 (sec) , antiderivative size = 454, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {732, 1416}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 732

\(\displaystyle -\frac {2 (d+e x) \sqrt {\frac {\left (a+c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (a g^2+c f^2\right )}} \int \frac {1}{\sqrt {\frac {\left (c d^2+a e^2\right ) (f+g x)^2}{\left (c f^2+a g^2\right ) (d+e x)^2}-\frac {2 (c d f+a e g) (f+g x)}{\left (c f^2+a g^2\right ) (d+e x)}+1}}d\frac {\sqrt {f+g x}}{\sqrt {d+e x}}}{\sqrt {a+c x^2} (e f-d g)}\)

\(\Big \downarrow \) 1416

\(\displaystyle -\frac {(d+e x) \sqrt [4]{a g^2+c f^2} \sqrt {\frac {\left (a+c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (a g^2+c f^2\right )}} \left (\frac {(f+g x) \sqrt {a e^2+c d^2}}{(d+e x) \sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {\frac {(f+g x)^2 \left (a e^2+c d^2\right )}{(d+e x)^2 \left (a g^2+c f^2\right )}-\frac {2 (f+g x) (a e g+c d f)}{(d+e x) \left (a g^2+c f^2\right )}+1}{\left (\frac {(f+g x) \sqrt {a e^2+c d^2}}{(d+e x) \sqrt {a g^2+c f^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2+a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2} \sqrt {d+e x}}\right ),\frac {1}{2} \left (\frac {c d f+a e g}{\sqrt {c d^2+a e^2} \sqrt {c f^2+a g^2}}+1\right )\right )}{\sqrt {a+c x^2} \sqrt [4]{a e^2+c d^2} (e f-d g) \sqrt {\frac {(f+g x)^2 \left (a e^2+c d^2\right )}{(d+e x)^2 \left (a g^2+c f^2\right )}-\frac {2 (f+g x) (a e g+c d f)}{(d+e x) \left (a g^2+c f^2\right )}+1}}\)

Input:

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

Output:

-(((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]*Sqr 
t[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)]))
 

Defintions of rubi rules used

rule 732

Int[1/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(a_) + (b_.)* 
(x_)^2]), x_Symbol] :> Simp[-2*(c + d*x)*(Sqrt[(d*e - c*f)^2*((a + b*x^2)/( 
(b*e^2 + a*f^2)*(c + d*x)^2))]/((d*e - c*f)*Sqrt[a + b*x^2]))   Subst[Int[1 
/Sqrt[Simp[1 - (2*b*c*e + 2*a*d*f)*(x^2/(b*e^2 + a*f^2)) + (b*c^2 + a*d^2)* 
(x^4/(b*e^2 + a*f^2)), x]], x], x, Sqrt[e + f*x]/Sqrt[c + d*x]], x] /; Free 
Q[{a, b, c, d, e, f}, x]
 

rule 1416

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)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

Maple [A] (verified)

Time = 11.06 (sec) , antiderivative size = 401, normalized size of antiderivative = 0.87

method	result	size
default	\(\frac {2 \left (c \,e^{2} f \,x^{2}-\sqrt {-a c}\, e^{2} g \,x^{2}+2 c d e f x -2 \sqrt {-a c}\, d e g x +c \,d^{2} f -\sqrt {-a c}\, d^{2} g \right ) \operatorname {EllipticF}\left (\sqrt {\frac {\left (e \sqrt {-a c}-c d \right ) \left (g x +f \right )}{\left (\sqrt {-a c}\, g -c f \right ) \left (e x +d \right )}}, \sqrt {\frac {\left (e \sqrt {-a c}+c d \right ) \left (\sqrt {-a c}\, g -c f \right )}{\left (\sqrt {-a c}\, g +c f \right ) \left (e \sqrt {-a c}-c d \right )}}\right ) \sqrt {\frac {\left (d g -e f \right ) \left (c x +\sqrt {-a c}\right )}{\left (\sqrt {-a c}\, g -c f \right ) \left (e x +d \right )}}\, \sqrt {\frac {\left (d g -e f \right ) \left (-c x +\sqrt {-a c}\right )}{\left (\sqrt {-a c}\, g +c f \right ) \left (e x +d \right )}}\, \sqrt {\frac {\left (e \sqrt {-a c}-c d \right ) \left (g x +f \right )}{\left (\sqrt {-a c}\, g -c f \right ) \left (e x +d \right )}}\, \sqrt {c \,x^{2}+a}\, \sqrt {g x +f}\, \sqrt {e x +d}}{\sqrt {-\frac {\left (g x +f \right ) \left (e x +d \right ) \left (-c x +\sqrt {-a c}\right ) \left (c x +\sqrt {-a c}\right )}{c}}\, \left (c d -e \sqrt {-a c}\right ) \left (d g -e f \right ) \sqrt {\left (g x +f \right ) \left (e x +d \right ) \left (c \,x^{2}+a \right )}}\)	\(401\)
elliptic	\(\frac {2 \sqrt {\left (g x +f \right ) \left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {f}{g}+\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {\left (-\frac {\sqrt {-a c}}{c}+\frac {d}{e}\right ) \left (x +\frac {f}{g}\right )}{\left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \left (x +\frac {d}{e}\right )}}\, \left (x +\frac {d}{e}\right )^{2} \sqrt {\frac {\left (-\frac {d}{e}+\frac {f}{g}\right ) \left (x -\frac {\sqrt {-a c}}{c}\right )}{\left (\frac {f}{g}+\frac {\sqrt {-a c}}{c}\right ) \left (x +\frac {d}{e}\right )}}\, \sqrt {\frac {\left (-\frac {d}{e}+\frac {f}{g}\right ) \left (x +\frac {\sqrt {-a c}}{c}\right )}{\left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \left (x +\frac {d}{e}\right )}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (-\frac {\sqrt {-a c}}{c}+\frac {d}{e}\right ) \left (x +\frac {f}{g}\right )}{\left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \left (x +\frac {d}{e}\right )}}, \sqrt {\frac {\left (-\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \left (-\frac {f}{g}+\frac {\sqrt {-a c}}{c}\right )}{\left (-\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \left (\frac {\sqrt {-a c}}{c}-\frac {d}{e}\right )}}\right )}{\sqrt {g x +f}\, \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}\, \left (-\frac {\sqrt {-a c}}{c}+\frac {d}{e}\right ) \left (-\frac {d}{e}+\frac {f}{g}\right ) \sqrt {c e g \left (x +\frac {f}{g}\right ) \left (x +\frac {d}{e}\right ) \left (x -\frac {\sqrt {-a c}}{c}\right ) \left (x +\frac {\sqrt {-a c}}{c}\right )}}\)	\(448\)

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} \sqrt {e x + d} \sqrt {g x + f}} \,d x } \] Input:

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

Output:

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)
 

Sympy [F]

\[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int \frac {1}{\sqrt {a + c x^{2}} \sqrt {d + e x} \sqrt {f + g x}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} \sqrt {e x + d} \sqrt {g x + f}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} \sqrt {e x + d} \sqrt {g x + f}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int \frac {1}{\sqrt {f+g\,x}\,\sqrt {c\,x^2+a}\,\sqrt {d+e\,x}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int \frac {1}{\sqrt {e x +d}\, \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}d x \] Input:

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

Output:

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