\(\int \frac {\sqrt {c+d x^2} \sqrt {e+f x^2}}{(a+b x^2)^2} \, dx\) [162]

Optimal result

Integrand size = 32, antiderivative size = 413 \[ \int \frac {\sqrt {c+d x^2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^2} \, dx=-\frac {d x \sqrt {e+f x^2}}{2 a b \sqrt {c+d x^2}}+\frac {x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right )}+\frac {\sqrt {c} \sqrt {d} \sqrt {e+f x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{2 a b \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac {c^{3/2} \sqrt {d} (b e-a f) \sqrt {e+f x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {c f}{d e}\right )}{2 a b (b c-a d) e \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {c^{3/2} \left (b^2 c e-a^2 d f\right ) \sqrt {e+f x^2} \operatorname {EllipticPi}\left (1-\frac {b c}{a d},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {c f}{d e}\right )}{2 a^2 b \sqrt {d} (b c-a d) e \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}} \] Output:

-1/2*d*x*(f*x^2+e)^(1/2)/a/b/(d*x^2+c)^(1/2)+1/2*x*(d*x^2+c)^(1/2)*(f*x^2+ 
e)^(1/2)/a/(b*x^2+a)+1/2*c^(1/2)*d^(1/2)*(f*x^2+e)^(1/2)*EllipticE(d^(1/2) 
*x/c^(1/2)/(1+d*x^2/c)^(1/2),(1-c*f/d/e)^(1/2))/a/b/(d*x^2+c)^(1/2)/(c*(f* 
x^2+e)/e/(d*x^2+c))^(1/2)-1/2*c^(3/2)*d^(1/2)*(-a*f+b*e)*(f*x^2+e)^(1/2)*I 
nverseJacobiAM(arctan(d^(1/2)*x/c^(1/2)),(1-c*f/d/e)^(1/2))/a/b/(-a*d+b*c) 
/e/(d*x^2+c)^(1/2)/(c*(f*x^2+e)/e/(d*x^2+c))^(1/2)+1/2*c^(3/2)*(-a^2*d*f+b 
^2*c*e)*(f*x^2+e)^(1/2)*EllipticPi(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),1-b 
*c/a/d,(1-c*f/d/e)^(1/2))/a^2/b/d^(1/2)/(-a*d+b*c)/e/(d*x^2+c)^(1/2)/(c*(f 
*x^2+e)/e/(d*x^2+c))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.20 (sec) , antiderivative size = 401, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {c+d x^2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^2} \, dx=\frac {\frac {c e x}{a+b x^2}+\frac {d e x^3}{a+b x^2}+\frac {c f x^3}{a+b x^2}+\frac {d f x^5}{a+b x^2}+\frac {i c \sqrt {\frac {d}{c}} e \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )}{b}-\frac {i c \sqrt {\frac {d}{c}} (b e+a f) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )}{b^2}-\frac {i c e \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticPi}\left (\frac {b c}{a d},i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )}{a \sqrt {\frac {d}{c}}}+\frac {i a c \sqrt {\frac {d}{c}} f \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticPi}\left (\frac {b c}{a d},i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )}{b^2}}{2 a \sqrt {c+d x^2} \sqrt {e+f x^2}} \] Input:

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

Output:

((c*e*x)/(a + b*x^2) + (d*e*x^3)/(a + b*x^2) + (c*f*x^3)/(a + b*x^2) + (d* 
f*x^5)/(a + b*x^2) + (I*c*Sqrt[d/c]*e*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2) 
/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/b - (I*c*Sqrt[d/c]*(b* 
e + a*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[ 
d/c]*x], (c*f)/(d*e)])/b^2 - (I*c*e*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e 
]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(a*Sqrt[d/ 
c]) + (I*a*c*Sqrt[d/c]*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticP 
i[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/b^2)/(2*a*Sqrt[c + d* 
x^2]*Sqrt[e + f*x^2])
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 378, normalized size of antiderivative = 0.92, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {423, 406, 320, 388, 313, 413, 413, 412}

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.

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

\(\Big \downarrow \) 423

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

\(\displaystyle \frac {d f \left (\frac {a \sqrt {e} \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{c \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-b \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx\right )}{2 a b^2}+\frac {1}{2} \left (\frac {c e}{a}-\frac {a d f}{b^2}\right ) \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c} \sqrt {f x^2+e}}dx+\frac {x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right )}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {d f \left (\frac {a \sqrt {e} \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{c \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-b \left (\frac {x \sqrt {c+d x^2}}{d \sqrt {e+f x^2}}-\frac {e \int \frac {\sqrt {d x^2+c}}{\left (f x^2+e\right )^{3/2}}dx}{d}\right )\right )}{2 a b^2}+\frac {1}{2} \left (\frac {c e}{a}-\frac {a d f}{b^2}\right ) \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c} \sqrt {f x^2+e}}dx+\frac {x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right )}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {1}{2} \left (\frac {c e}{a}-\frac {a d f}{b^2}\right ) \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c} \sqrt {f x^2+e}}dx+\frac {d f \left (\frac {a \sqrt {e} \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{c \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-b \left (\frac {x \sqrt {c+d x^2}}{d \sqrt {e+f x^2}}-\frac {\sqrt {e} \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{d \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\right )\right )}{2 a b^2}+\frac {x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right )}\)

\(\Big \downarrow \) 413

\(\displaystyle \frac {\sqrt {\frac {d x^2}{c}+1} \left (\frac {c e}{a}-\frac {a d f}{b^2}\right ) \int \frac {1}{\left (b x^2+a\right ) \sqrt {\frac {d x^2}{c}+1} \sqrt {f x^2+e}}dx}{2 \sqrt {c+d x^2}}+\frac {d f \left (\frac {a \sqrt {e} \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{c \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-b \left (\frac {x \sqrt {c+d x^2}}{d \sqrt {e+f x^2}}-\frac {\sqrt {e} \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{d \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\right )\right )}{2 a b^2}+\frac {x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right )}\)

\(\Big \downarrow \) 413

\(\displaystyle \frac {\sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} \left (\frac {c e}{a}-\frac {a d f}{b^2}\right ) \int \frac {1}{\left (b x^2+a\right ) \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1}}dx}{2 \sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {d f \left (\frac {a \sqrt {e} \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{c \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-b \left (\frac {x \sqrt {c+d x^2}}{d \sqrt {e+f x^2}}-\frac {\sqrt {e} \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{d \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\right )\right )}{2 a b^2}+\frac {x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right )}\)

\(\Big \downarrow \) 412

\(\displaystyle \frac {\sqrt {-c} \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} \left (\frac {c e}{a}-\frac {a d f}{b^2}\right ) \operatorname {EllipticPi}\left (\frac {b c}{a d},\arcsin \left (\frac {\sqrt {d} x}{\sqrt {-c}}\right ),\frac {c f}{d e}\right )}{2 a \sqrt {d} \sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {d f \left (\frac {a \sqrt {e} \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{c \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-b \left (\frac {x \sqrt {c+d x^2}}{d \sqrt {e+f x^2}}-\frac {\sqrt {e} \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{d \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\right )\right )}{2 a b^2}+\frac {x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right )}\)

Input:

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

Output:

(x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(2*a*(a + b*x^2)) + (d*f*(-(b*((x*Sqrt 
[c + d*x^2])/(d*Sqrt[e + f*x^2]) - (Sqrt[e]*Sqrt[c + d*x^2]*EllipticE[ArcT 
an[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(d*Sqrt[f]*Sqrt[(e*(c + d*x^2)) 
/(c*(e + f*x^2))]*Sqrt[e + f*x^2]))) + (a*Sqrt[e]*Sqrt[c + d*x^2]*Elliptic 
F[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*Sqrt[f]*Sqrt[(e*(c + d 
*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])))/(2*a*b^2) + (Sqrt[-c]*((c*e)/a 
- (a*d*f)/b^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a 
*d), ArcSin[(Sqrt[d]*x)/Sqrt[-c]], (c*f)/(d*e)])/(2*a*Sqrt[d]*Sqrt[c + d*x 
^2]*Sqrt[e + f*x^2])
 

Defintions of rubi rules used

rule 313

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim 
p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c 
+ d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ 
[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
 

rule 320

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( 
c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre 
eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 388

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] 
 :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b   Int[Sqrt[ 
a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 406

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[e   Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim 
p[f   Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, 
f, p, q}, x]
 

rule 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[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] && S 
implerSqrtQ[-f/e, -d/c])
 

rule 413

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/((a + 
 b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, 
e, f}, x] &&  !GtQ[c, 0]
 

rule 423

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

Maple [A] (verified)

Time = 1.55 (sec) , antiderivative size = 559, normalized size of antiderivative = 1.35

method	result	size
elliptic	\(\frac {\sqrt {\left (x^{2} d +c \right ) \left (f \,x^{2}+e \right )}\, \left (\frac {x \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{2 a \left (b \,x^{2}+a \right )}+\frac {d f \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{2 b^{2} \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {d e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{2 a b \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {d e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{2 a b \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {\sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) d f}{2 b^{2} \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {\sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) c e}{2 a^{2} \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}\right )}{\sqrt {x^{2} d +c}\, \sqrt {f \,x^{2}+e}}\)	\(559\)
default	\(\frac {\sqrt {x^{2} d +c}\, \sqrt {f \,x^{2}+e}\, \left (\sqrt {-\frac {d}{c}}\, a \,b^{2} d f \,x^{5}+\sqrt {\frac {x^{2} d +c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a^{2} b d f \,x^{2}+\sqrt {\frac {x^{2} d +c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a \,b^{2} d e \,x^{2}-\sqrt {\frac {x^{2} d +c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a \,b^{2} d e \,x^{2}-\sqrt {\frac {x^{2} d +c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) a^{2} b d f \,x^{2}+\sqrt {\frac {x^{2} d +c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) b^{3} c e \,x^{2}+\sqrt {-\frac {d}{c}}\, a \,b^{2} c f \,x^{3}+\sqrt {-\frac {d}{c}}\, a \,b^{2} d e \,x^{3}+\sqrt {\frac {x^{2} d +c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a^{3} d f +\sqrt {\frac {x^{2} d +c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a^{2} b d e -\sqrt {\frac {x^{2} d +c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a^{2} b d e -\sqrt {\frac {x^{2} d +c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) a^{3} d f +\sqrt {\frac {x^{2} d +c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) a \,b^{2} c e +\sqrt {-\frac {d}{c}}\, a \,b^{2} c e x \right )}{2 \left (d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e \right ) a^{2} \left (b \,x^{2}+a \right ) b^{2} \sqrt {-\frac {d}{c}}}\)	\(765\)

Input:

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

Output:

((d*x^2+c)*(f*x^2+e))^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2)*(1/2*x/a*(d*f* 
x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)/(b*x^2+a)+1/2*d*f/b^2/(-1/c*d)^(1/2)*(1+d*x 
^2/c)^(1/2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)*Elliptic 
F(x*(-1/c*d)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))+1/2*d/a/b*e/(-1/c*d)^(1/2)*(1 
+d*x^2/c)^(1/2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)*Elli 
pticF(x*(-1/c*d)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))-1/2*d/a/b*e/(-1/c*d)^(1/2 
)*(1+d*x^2/c)^(1/2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)* 
EllipticE(x*(-1/c*d)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))-1/2/b^2/(-1/c*d)^(1/2 
)*(1+d*x^2/c)^(1/2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)* 
EllipticPi(x*(-1/c*d)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-1/c*d)^(1/2))*d*f+1/2/a 
^2/(-1/c*d)^(1/2)*(1+d*x^2/c)^(1/2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e 
*x^2+c*e)^(1/2)*EllipticPi(x*(-1/c*d)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-1/c*d)^ 
(1/2))*c*e)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x^2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\sqrt {c+d x^2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^2} \, dx=\int \frac {\sqrt {c + d x^{2}} \sqrt {e + f x^{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\sqrt {c+d x^2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^2} \, dx=\int { \frac {\sqrt {d x^{2} + c} \sqrt {f x^{2} + e}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\sqrt {c+d x^2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^2} \, dx=\int { \frac {\sqrt {d x^{2} + c} \sqrt {f x^{2} + e}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x^2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^2} \, dx=\int \frac {\sqrt {d\,x^2+c}\,\sqrt {f\,x^2+e}}{{\left (b\,x^2+a\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {c+d x^2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^2} \, dx=\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}}{b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \] Input:

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

Output:

int((sqrt(e + f*x**2)*sqrt(c + d*x**2))/(a**2 + 2*a*b*x**2 + b**2*x**4),x)