\(\int \frac {1}{(a+b x^2)^2 \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx\) [102]

Optimal result

Integrand size = 32, antiderivative size = 485 \[ \int \frac {1}{\left (a+b x^2\right )^2 \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=-\frac {b f x \sqrt {c+d x^2}}{2 a (b c-a d) (b e-a f) \sqrt {e+f x^2}}+\frac {b^2 x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 a (b c-a d) (b e-a f) \left (a+b x^2\right )}+\frac {b \sqrt {e} \sqrt {f} \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{2 a (b c-a d) (b e-a f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {d \sqrt {e} \sqrt {f} \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{2 c (b c-a d) (b e-a f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {\sqrt {-c} \left (b^2 c e+3 a^2 d f-2 a b (d e+c f)\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticPi}\left (\frac {b c}{a d},\arcsin \left (\frac {\sqrt {d} x}{\sqrt {-c}}\right ),\frac {c f}{d e}\right )}{2 a^2 \sqrt {d} (b c-a d) (b e-a f) \sqrt {c+d x^2} \sqrt {e+f x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {563, 552, 551, 545, 429, 506, 422} \[ \int \frac {1}{\left (a+b x^2\right )^2 \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\frac {\sqrt {-c} \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} \left (3 a^2 d f-2 a b (c f+d e)+b^2 c e\right ) \operatorname {EllipticPi}\left (\frac {b c}{a d},\arcsin \left (\frac {\sqrt {d} x}{\sqrt {-c}}\right ),\frac {c f}{d e}\right )}{2 a^2 \sqrt {d} \sqrt {c+d x^2} \sqrt {e+f x^2} (b c-a d) (b e-a f)}-\frac {d \sqrt {e} \sqrt {f} \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{2 c \sqrt {e+f x^2} (b c-a d) (b e-a f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {b \sqrt {e} \sqrt {f} \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{2 a \sqrt {e+f x^2} (b c-a d) (b e-a f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {b^2 x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right ) (b c-a d) (b e-a f)}-\frac {b f x \sqrt {c+d x^2}}{2 a \sqrt {e+f x^2} (b c-a d) (b e-a f)} \]

[In]

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

[Out]

-1/2*(b*f*x*Sqrt[c + d*x^2])/(a*(b*c - a*d)*(b*e - a*f)*Sqrt[e + f*x^2]) + (b^2*x*Sqrt[c + d*x^2]*Sqrt[e + f*x
^2])/(2*a*(b*c - a*d)*(b*e - a*f)*(a + b*x^2)) + (b*Sqrt[e]*Sqrt[f]*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*
x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(2*a*(b*c - a*d)*(b*e - a*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x
^2]) - (d*Sqrt[e]*Sqrt[f]*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(2*c*(b*c -
a*d)*(b*e - a*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (Sqrt[-c]*(b^2*c*e + 3*a^2*d*f - 2*
a*b*(d*e + c*f))*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^2*Sqrt[d]*(b*c - a*d)*(b*e - a*f)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

Rule 422

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sq
rt[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 429

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

Rule 506

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] - Dist[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 545

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

Rule 551

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

Rule 552

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[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 563

Int[1/(((a_) + (b_.)*(x_)^2)^2*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[b^2*x*S
qrt[c + d*x^2]*(Sqrt[e + f*x^2]/(2*a*(b*c - a*d)*(b*e - a*f)*(a + b*x^2))), x] + (Dist[(b^2*c*e + 3*a^2*d*f -
2*a*b*(d*e + c*f))/(2*a*(b*c - a*d)*(b*e - a*f)), Int[1/((a + b*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x]
- Dist[d*(f/(2*a*(b*c - a*d)*(b*e - a*f))), Int[(a + b*x^2)/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x]) /; Free
Q[{a, b, c, d, e, f}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {b^2 x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 a (b c-a d) (b e-a f) \left (a+b x^2\right )}-\frac {(d f) \int \frac {a+b x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{2 a (b c-a d) (b e-a f)}+\frac {\left (b^2 c e+3 a^2 d f-2 a b (d e+c f)\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{2 a (b c-a d) (b e-a f)} \\ & = \frac {b^2 x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 a (b c-a d) (b e-a f) \left (a+b x^2\right )}-\frac {(d f) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{2 (b c-a d) (b e-a f)}-\frac {(b d f) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{2 a (b c-a d) (b e-a f)}+\frac {\left (\left (b^2 c e+3 a^2 d f-2 a b (d e+c f)\right ) \sqrt {1+\frac {d x^2}{c}}\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {e+f x^2}} \, dx}{2 a (b c-a d) (b e-a f) \sqrt {c+d x^2}} \\ & = -\frac {b f x \sqrt {c+d x^2}}{2 a (b c-a d) (b e-a f) \sqrt {e+f x^2}}+\frac {b^2 x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 a (b c-a d) (b e-a f) \left (a+b x^2\right )}-\frac {d \sqrt {e} \sqrt {f} \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{2 c (b c-a d) (b e-a f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {(b e f) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{2 a (b c-a d) (b e-a f)}+\frac {\left (\left (b^2 c e+3 a^2 d f-2 a b (d e+c f)\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}}\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}}} \, dx}{2 a (b c-a d) (b e-a f) \sqrt {c+d x^2} \sqrt {e+f x^2}} \\ & = -\frac {b f x \sqrt {c+d x^2}}{2 a (b c-a d) (b e-a f) \sqrt {e+f x^2}}+\frac {b^2 x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 a (b c-a d) (b e-a f) \left (a+b x^2\right )}+\frac {b \sqrt {e} \sqrt {f} \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{2 a (b c-a d) (b e-a f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {d \sqrt {e} \sqrt {f} \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{2 c (b c-a d) (b e-a f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {\sqrt {-c} \left (b^2 c e+3 a^2 d f-2 a b (d e+c f)\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \Pi \left (\frac {b c}{a d};\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {-c}}\right )|\frac {c f}{d e}\right )}{2 a^2 \sqrt {d} (b c-a d) (b e-a f) \sqrt {c+d x^2} \sqrt {e+f x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.31 (sec) , antiderivative size = 587, normalized size of antiderivative = 1.21 \[ \int \frac {1}{\left (a+b x^2\right )^2 \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\frac {\frac {b^2 c e x}{a+b x^2}+\frac {b^2 d e x^3}{a+b x^2}+\frac {b^2 c f x^3}{a+b x^2}+\frac {b^2 d f x^5}{a+b x^2}+i b c \sqrt {\frac {d}{c}} e \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-i c \sqrt {\frac {d}{c}} (b e-a f) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )-\frac {i b^2 c e \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticPi}\left (\frac {b c}{a d},i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )}{a \sqrt {\frac {d}{c}}}+2 i b c \sqrt {\frac {d}{c}} e \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticPi}\left (\frac {b c}{a d},i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )+\frac {2 i b c f \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticPi}\left (\frac {b c}{a d},i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )}{\sqrt {\frac {d}{c}}}-3 i a c \sqrt {\frac {d}{c}} f \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticPi}\left (\frac {b c}{a d},i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )}{2 a (-b c+a d) (-b e+a f) \sqrt {c+d x^2} \sqrt {e+f x^2}} \]

[In]

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

[Out]

((b^2*c*e*x)/(a + b*x^2) + (b^2*d*e*x^3)/(a + b*x^2) + (b^2*c*f*x^3)/(a + b*x^2) + (b^2*d*f*x^5)/(a + b*x^2) +
I*b*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)] - 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)]
- (I*b^2*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]) + (2*I)*b*c*Sqrt[d/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)] + ((2*I)*b*c*f*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)])/Sqrt[d/c] - (3*I)*a*c*Sqrt[d/c]*f*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)])/(2*a*(-(b*c) + a*d)*(-(b*e) + a*f)*Sqrt[c
+ d*x^2]*Sqrt[e + f*x^2])

Maple [A] (verified)

Time = 4.89 (sec) , antiderivative size = 973, normalized size of antiderivative = 2.01

method	result	size
elliptic	\(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (\frac {b^{2} x \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{2 \left (a^{2} d f -a c f b -a b d e +b^{2} c e \right ) a \left (b \,x^{2}+a \right )}-\frac {d f \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{2 \left (a^{2} d f -a c f b -a b d e +b^{2} c e \right ) \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {b d e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{2 \left (a^{2} d f -a c f b -a b d e +b^{2} c e \right ) a \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {b d e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, E\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{2 \left (a^{2} d f -a c f b -a b d e +b^{2} c e \right ) a \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {3 \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \Pi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) d f}{2 \left (a^{2} d f -a c f b -a b d e +b^{2} c e \right ) \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {b \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \Pi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) c f}{\left (a^{2} d f -a c f b -a b d e +b^{2} c e \right ) a \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {b \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \Pi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) d e}{\left (a^{2} d f -a c f b -a b d e +b^{2} c e \right ) a \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {b^{2} \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \Pi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) c e}{2 \left (a^{2} d f -a c f b -a b d e +b^{2} c e \right ) a^{2} \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\)	\(973\)
default	\(\text {Expression too large to display}\)	\(1078\)

[In]

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

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^2 \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int \frac {1}{\left (a+b x^2\right )^2 \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{2} \sqrt {c + d x^{2}} \sqrt {e + f x^{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{\left (a+b x^2\right )^2 \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1}{\left (a+b x^2\right )^2 \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^2 \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^2\,\sqrt {d\,x^2+c}\,\sqrt {f\,x^2+e}} \,d x \]

[In]

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

[Out]

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