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

Optimal. Leaf size=485 \[ \frac {\sqrt {-c} \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} \left (3 a^2 d f-2 a b (c f+d e)+b^2 c e\right ) \Pi \left (\frac {b c}{a d};\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {-c}}\right )|\frac {c f}{d e}\right )}{2 a^2 \sqrt {d} \sqrt {c+d x^2} \sqrt {e+f x^2} (b c-a d) (b e-a f)}+\frac {b^2 x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right ) (b c-a d) (b e-a f)}-\frac {b f x \sqrt {c+d x^2}}{2 a \sqrt {e+f x^2} (b c-a d) (b e-a f)}-\frac {d \sqrt {e} \sqrt {f} \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{2 c \sqrt {e+f x^2} (b c-a d) (b e-a f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {b \sqrt {e} \sqrt {f} \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{2 a \sqrt {e+f x^2} (b c-a d) (b e-a f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \]

[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)

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Rubi [A]  time = 0.35, antiderivative size = 485, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {549, 531, 418, 492, 411, 538, 537} \[ \frac {\sqrt {-c} \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} \left (3 a^2 d f-2 a b (c f+d e)+b^2 c e\right ) \Pi \left (\frac {b c}{a d};\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {-c}}\right )|\frac {c f}{d e}\right )}{2 a^2 \sqrt {d} \sqrt {c+d x^2} \sqrt {e+f x^2} (b c-a d) (b e-a f)}+\frac {b^2 x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right ) (b c-a d) (b e-a f)}-\frac {b f x \sqrt {c+d x^2}}{2 a \sqrt {e+f x^2} (b c-a d) (b e-a f)}-\frac {d \sqrt {e} \sqrt {f} \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{2 c \sqrt {e+f x^2} (b c-a d) (b e-a f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {b \sqrt {e} \sqrt {f} \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{2 a \sqrt {e+f x^2} (b c-a d) (b e-a f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*f*x*Sqrt[c + d*x^2])/(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 411

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

Rule 418

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

Rule 492

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(x*Sqrt[a + b*x^2])/(b*Sqr
t[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 531

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 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), 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 538

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

Rule 549

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

Rubi steps

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

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Mathematica [C]  time = 2.88, size = 587, normalized size = 1.21 \[ \frac {-\frac {i b^2 c e \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} \Pi \left (\frac {b c}{a d};i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )}{a \sqrt {\frac {d}{c}}}+\frac {b^2 c e x}{a+b x^2}+\frac {b^2 c f x^3}{a+b x^2}+\frac {b^2 d e x^3}{a+b x^2}+\frac {b^2 d f x^5}{a+b x^2}-i c \sqrt {\frac {d}{c}} \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} (b e-a f) F\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )+2 i b c e \sqrt {\frac {d}{c}} \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} \Pi \left (\frac {b c}{a d};i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-3 i a c f \sqrt {\frac {d}{c}} \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} \Pi \left (\frac {b c}{a d};i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )+\frac {2 i b c f \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} \Pi \left (\frac {b c}{a d};i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )}{\sqrt {\frac {d}{c}}}+i b c e \sqrt {\frac {d}{c}} \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} E\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )}{2 a \sqrt {c+d x^2} \sqrt {e+f x^2} (a d-b c) (a f-b e)} \]

Antiderivative was successfully verified.

[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])

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [B]  time = 0.03, size = 1078, normalized size = 2.22 \[ -\frac {\left (-\sqrt {-\frac {d}{c}}\, a \,b^{2} d f \,x^{5}+\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, a^{2} b d f \,x^{2} \EllipticF \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {\frac {c f}{d e}}\right )-3 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, a^{2} b d f \,x^{2} \EllipticPi \left (\sqrt {-\frac {d}{c}}\, x , \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right )-\sqrt {-\frac {d}{c}}\, a \,b^{2} c f \,x^{3}+2 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, a \,b^{2} c f \,x^{2} \EllipticPi \left (\sqrt {-\frac {d}{c}}\, x , \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right )-\sqrt {-\frac {d}{c}}\, a \,b^{2} d e \,x^{3}+\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, a \,b^{2} d e \,x^{2} \EllipticE \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {\frac {c f}{d e}}\right )-\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, a \,b^{2} d e \,x^{2} \EllipticF \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {\frac {c f}{d e}}\right )+2 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, a \,b^{2} d e \,x^{2} \EllipticPi \left (\sqrt {-\frac {d}{c}}\, x , \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right )-\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, b^{3} c e \,x^{2} \EllipticPi \left (\sqrt {-\frac {d}{c}}\, x , \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right )+\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, a^{3} d f \EllipticF \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {\frac {c f}{d e}}\right )-3 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, a^{3} d f \EllipticPi \left (\sqrt {-\frac {d}{c}}\, x , \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right )+2 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, a^{2} b c f \EllipticPi \left (\sqrt {-\frac {d}{c}}\, x , \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right )+\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, a^{2} b d e \EllipticE \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {\frac {c f}{d e}}\right )-\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, a^{2} b d e \EllipticF \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {\frac {c f}{d e}}\right )+2 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, a^{2} b d e \EllipticPi \left (\sqrt {-\frac {d}{c}}\, x , \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right )-\sqrt {-\frac {d}{c}}\, a \,b^{2} c e x -\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, a \,b^{2} c e \EllipticPi \left (\sqrt {-\frac {d}{c}}\, x , \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right )\right ) \sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}}{2 \left (b \,x^{2}+a \right ) \sqrt {-\frac {d}{c}}\, \left (a f -b e \right ) \left (a d -b c \right ) \left (d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e \right ) a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(-(-1/c*d)^(1/2)*a*b^2*d*f*x^5+((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticE((-1/c*d)^(1/2)*x,(c/d/e*
f)^(1/2))*x^2*a*b^2*d*e-3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi((-1/c*d)^(1/2)*x,1/a*b*c/d,(-1/e*
f)^(1/2)/(-1/c*d)^(1/2))*x^2*a^2*b*d*f+2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi((-1/c*d)^(1/2)*x,1
/a*b*c/d,(-1/e*f)^(1/2)/(-1/c*d)^(1/2))*x^2*a*b^2*c*f+2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi((-1
/c*d)^(1/2)*x,1/a*b*c/d,(-1/e*f)^(1/2)/(-1/c*d)^(1/2))*x^2*a*b^2*d*e-((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*E
llipticPi((-1/c*d)^(1/2)*x,1/a*b*c/d,(-1/e*f)^(1/2)/(-1/c*d)^(1/2))*x^2*b^3*c*e+((d*x^2+c)/c)^(1/2)*((f*x^2+e)
/e)^(1/2)*EllipticF((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*x^2*a^2*b*d*f-((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*El
lipticF((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*x^2*a*b^2*d*e-(-1/c*d)^(1/2)*a*b^2*c*f*x^3-(-1/c*d)^(1/2)*a*b^2*d*e*
x^3+((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticE((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*a^2*b*d*e-3*((d*x^2+c)
/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi((-1/c*d)^(1/2)*x,1/a*b*c/d,(-1/e*f)^(1/2)/(-1/c*d)^(1/2))*a^3*d*f+2*(
(d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi((-1/c*d)^(1/2)*x,1/a*b*c/d,(-1/e*f)^(1/2)/(-1/c*d)^(1/2))*a^
2*b*c*f+2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi((-1/c*d)^(1/2)*x,1/a*b*c/d,(-1/e*f)^(1/2)/(-1/c*d
)^(1/2))*a^2*b*d*e-((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi((-1/c*d)^(1/2)*x,1/a*b*c/d,(-1/e*f)^(1/2
)/(-1/c*d)^(1/2))*a*b^2*c*e+((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticF((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2)
)*a^3*d*f-((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticF((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*a^2*b*d*e-(-1/c*
d)^(1/2)*a*b^2*c*e*x)*(f*x^2+e)^(1/2)*(d*x^2+c)^(1/2)/(b*x^2+a)/a^2/(-1/c*d)^(1/2)/(a*f-b*e)/(a*d-b*c)/(d*f*x^
4+c*f*x^2+d*e*x^2+c*e)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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