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

Optimal. Leaf size=426 \[ \frac {\sqrt {c} \sqrt {1-\frac {d x^2}{c}} \sqrt {\frac {f x^2}{e}+1} \left (-3 a^2 d f+a b (2 d e-2 c f)+b^2 c e\right ) \Pi \left (-\frac {b c}{a d};\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {c f}{d e}\right )}{2 a^2 \sqrt {d} \sqrt {c-d x^2} \sqrt {e+f x^2} (a d+b c) (b e-a f)}+\frac {b^2 x \sqrt {c-d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right ) (a d+b c) (b e-a f)}-\frac {\sqrt {c} \sqrt {d} \sqrt {1-\frac {d x^2}{c}} \sqrt {\frac {f x^2}{e}+1} F\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {c f}{d e}\right )}{2 a \sqrt {c-d x^2} \sqrt {e+f x^2} (a d+b c)}+\frac {b \sqrt {c} \sqrt {d} \sqrt {1-\frac {d x^2}{c}} \sqrt {e+f x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {c f}{d e}\right )}{2 a \sqrt {c-d x^2} \sqrt {\frac {f x^2}{e}+1} (a d+b c) (b e-a f)} \]

[Out]

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

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Rubi [A]  time = 0.37, antiderivative size = 426, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {549, 524, 427, 426, 424, 421, 419, 538, 537} \[ \frac {\sqrt {c} \sqrt {1-\frac {d x^2}{c}} \sqrt {\frac {f x^2}{e}+1} \left (-3 a^2 d f+a b (2 d e-2 c f)+b^2 c e\right ) \Pi \left (-\frac {b c}{a d};\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {c f}{d e}\right )}{2 a^2 \sqrt {d} \sqrt {c-d x^2} \sqrt {e+f x^2} (a d+b c) (b e-a f)}+\frac {b^2 x \sqrt {c-d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right ) (a d+b c) (b e-a f)}-\frac {\sqrt {c} \sqrt {d} \sqrt {1-\frac {d x^2}{c}} \sqrt {\frac {f x^2}{e}+1} F\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {c f}{d e}\right )}{2 a \sqrt {c-d x^2} \sqrt {e+f x^2} (a d+b c)}+\frac {b \sqrt {c} \sqrt {d} \sqrt {1-\frac {d x^2}{c}} \sqrt {e+f x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {c f}{d e}\right )}{2 a \sqrt {c-d x^2} \sqrt {\frac {f x^2}{e}+1} (a d+b c) (b e-a f)} \]

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^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[c]*Sqrt[d]*Sqrt[1
- (d*x^2)/c]*Sqrt[e + f*x^2]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((c*f)/(d*e))])/(2*a*(b*c + a*d)*(b*e - a
*f)*Sqrt[c - d*x^2]*Sqrt[1 + (f*x^2)/e]) - (Sqrt[c]*Sqrt[d]*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[
ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((c*f)/(d*e))])/(2*a*(b*c + a*d)*Sqrt[c - d*x^2]*Sqrt[e + f*x^2]) + (Sqrt[c]*(b^
2*c*e - 3*a^2*d*f + a*b*(2*d*e - 2*c*f))*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[-((b*c)/(a*d)), Ar
cSin[(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 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rule 421

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

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 426

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

Rule 427

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

Rule 524

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

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 \int \frac {1}{\sqrt {c-d x^2} \sqrt {e+f x^2}} \, dx}{2 a (b c+a d)}+\frac {(b d) \int \frac {\sqrt {e+f x^2}}{\sqrt {c-d 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^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 {\left (b d \sqrt {1-\frac {d x^2}{c}}\right ) \int \frac {\sqrt {e+f x^2}}{\sqrt {1-\frac {d x^2}{c}}} \, dx}{2 a (b c+a d) (b e-a f) \sqrt {c-d x^2}}-\frac {\left (d \sqrt {1+\frac {f x^2}{e}}\right ) \int \frac {1}{\sqrt {c-d x^2} \sqrt {1+\frac {f x^2}{e}}} \, dx}{2 a (b c+a d) \sqrt {e+f x^2}}+\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^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 {\sqrt {c} \left (b^2 c e-3 a^2 d f+a b (2 d e-2 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}}+\frac {\left (b d \sqrt {1-\frac {d x^2}{c}} \sqrt {e+f x^2}\right ) \int \frac {\sqrt {1+\frac {f x^2}{e}}}{\sqrt {1-\frac {d x^2}{c}}} \, dx}{2 a (b c+a d) (b e-a f) \sqrt {c-d x^2} \sqrt {1+\frac {f x^2}{e}}}-\frac {\left (d \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}}\right ) \int \frac {1}{\sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}}} \, dx}{2 a (b c+a d) \sqrt {c-d x^2} \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 {c} \sqrt {d} \sqrt {1-\frac {d x^2}{c}} \sqrt {e+f x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {c f}{d e}\right )}{2 a (b c+a d) (b e-a f) \sqrt {c-d x^2} \sqrt {1+\frac {f x^2}{e}}}-\frac {\sqrt {c} \sqrt {d} \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {c f}{d e}\right )}{2 a (b c+a d) \sqrt {c-d x^2} \sqrt {e+f x^2}}+\frac {\sqrt {c} \left (b^2 c e-3 a^2 d f+a b (2 d e-2 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 = 6.07, size = 617, normalized size = 1.45 \[ \frac {\frac {i b^2 c e \sqrt {1-\frac {d x^2}{c}} \sqrt {\frac {f x^2}{e}+1} \Pi \left (-\frac {b c}{a d};i \sinh ^{-1}\left (\sqrt {-\frac {d}{c}} x\right )|-\frac {c f}{d e}\right )}{a \sqrt {-\frac {d}{c}}}-\frac {b^2 c e x}{a+b x^2}-\frac {b^2 c f x^3}{a+b x^2}+\frac {b^2 d e x^3}{a+b x^2}+\frac {b^2 d f x^5}{a+b x^2}+i c \sqrt {-\frac {d}{c}} \sqrt {1-\frac {d x^2}{c}} \sqrt {\frac {f x^2}{e}+1} (b e-a f) F\left (i \sinh ^{-1}\left (\sqrt {-\frac {d}{c}} x\right )|-\frac {c f}{d e}\right )-2 i b c e \sqrt {-\frac {d}{c}} \sqrt {1-\frac {d x^2}{c}} \sqrt {\frac {f x^2}{e}+1} \Pi \left (-\frac {b c}{a d};i \sinh ^{-1}\left (\sqrt {-\frac {d}{c}} x\right )|-\frac {c f}{d e}\right )+3 i a c f \sqrt {-\frac {d}{c}} \sqrt {1-\frac {d x^2}{c}} \sqrt {\frac {f x^2}{e}+1} \Pi \left (-\frac {b c}{a d};i \sinh ^{-1}\left (\sqrt {-\frac {d}{c}} x\right )|-\frac {c f}{d e}\right )+\frac {2 i b d f \sqrt {1-\frac {d x^2}{c}} \sqrt {\frac {f x^2}{e}+1} \Pi \left (-\frac {b c}{a d};i \sinh ^{-1}\left (\sqrt {-\frac {d}{c}} x\right )|-\frac {c f}{d e}\right )}{\left (-\frac {d}{c}\right )^{3/2}}-i b c e \sqrt {-\frac {d}{c}} \sqrt {1-\frac {d x^2}{c}} \sqrt {\frac {f x^2}{e}+1} E\left (i \sinh ^{-1}\left (\sqrt {-\frac {d}{c}} x\right )|-\frac {c f}{d e}\right )}{2 a \sqrt {c-d x^2} \sqrt {e+f x^2} (a d+b c) (a f-b e)} \]

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*ArcSin
h[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*d*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))])/(-(d/c))^(3/2)
+ (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(-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="giac")

[Out]

Timed out

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maple [B]  time = 0.06, size = 1105, normalized size = 2.59 \[ \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)*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)*EllipticF((1/c*d)^(1/2)*x,(-c/d/e*f)^(1/2))*x^
2*a*b^2*d*e+(-(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)*EllipticPi((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+(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)*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+(-(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)*E
llipticPi((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+(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 \[ \int \frac {1}{{\left (b\,x^2+a\right )}^2\,\sqrt {c-d\,x^2}\,\sqrt {f\,x^2+e}} \,d x \]

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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