3.389 \(\int \frac{1}{(d+e x^2)^2 \sqrt{a+b x^2-c x^4}} \, dx\)

Optimal. Leaf size=718 \[ -\frac{\sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \left (e \left (b-\sqrt{4 a c+b^2}\right )+2 c d\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{4 a c+b^2}+b}}\right ),\frac{\sqrt{4 a c+b^2}+b}{b-\sqrt{4 a c+b^2}}\right )}{4 \sqrt{2} \sqrt{c} d \sqrt{a+b x^2-c x^4} \left (e (b d-a e)+c d^2\right )}+\frac{e \left (b-\sqrt{4 a c+b^2}\right ) \sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{4 \sqrt{2} \sqrt{c} d \sqrt{a+b x^2-c x^4} \left (e (b d-a e)+c d^2\right )}+\frac{\sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \left (e (2 b d-a e)+3 c d^2\right ) \Pi \left (-\frac{\left (b+\sqrt{b^2+4 a c}\right ) e}{2 c d};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{2 \sqrt{2} \sqrt{c} d^2 \sqrt{a+b x^2-c x^4} \left (e (b d-a e)+c d^2\right )}-\frac{e^2 x \sqrt{a+b x^2-c x^4}}{2 d \left (d+e x^2\right ) \left (-a e^2+b d e+c d^2\right )} \]

[Out]

-(e^2*x*Sqrt[a + b*x^2 - c*x^4])/(2*d*(c*d^2 + b*d*e - a*e^2)*(d + e*x^2)) + ((b - Sqrt[b^2 + 4*a*c])*Sqrt[b +
Sqrt[b^2 + 4*a*c]]*e*Sqrt[1 - (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*
EllipticE[ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 + 4*a*c]]], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a
*c])])/(4*Sqrt[2]*Sqrt[c]*d*(c*d^2 + e*(b*d - a*e))*Sqrt[a + b*x^2 - c*x^4]) - (Sqrt[b + Sqrt[b^2 + 4*a*c]]*(2
*c*d + (b - Sqrt[b^2 + 4*a*c])*e)*Sqrt[1 - (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2
+ 4*a*c])]*EllipticF[ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 + 4*a*c]]], (b + Sqrt[b^2 + 4*a*c])/(b - Sq
rt[b^2 + 4*a*c])])/(4*Sqrt[2]*Sqrt[c]*d*(c*d^2 + e*(b*d - a*e))*Sqrt[a + b*x^2 - c*x^4]) + (Sqrt[b + Sqrt[b^2
+ 4*a*c]]*(3*c*d^2 + e*(2*b*d - a*e))*Sqrt[1 - (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt
[b^2 + 4*a*c])]*EllipticPi[-((b + Sqrt[b^2 + 4*a*c])*e)/(2*c*d), ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2
+ 4*a*c]]], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])])/(2*Sqrt[2]*Sqrt[c]*d^2*(c*d^2 + e*(b*d - a*e))*S
qrt[a + b*x^2 - c*x^4])

________________________________________________________________________________________

Rubi [A]  time = 1.01523, antiderivative size = 718, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {1223, 1716, 1202, 524, 424, 419, 1220, 537} \[ -\frac{\sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \left (e \left (b-\sqrt{4 a c+b^2}\right )+2 c d\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{4 \sqrt{2} \sqrt{c} d \sqrt{a+b x^2-c x^4} \left (e (b d-a e)+c d^2\right )}+\frac{e \left (b-\sqrt{4 a c+b^2}\right ) \sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{4 \sqrt{2} \sqrt{c} d \sqrt{a+b x^2-c x^4} \left (e (b d-a e)+c d^2\right )}+\frac{\sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \left (e (2 b d-a e)+3 c d^2\right ) \Pi \left (-\frac{\left (b+\sqrt{b^2+4 a c}\right ) e}{2 c d};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{2 \sqrt{2} \sqrt{c} d^2 \sqrt{a+b x^2-c x^4} \left (e (b d-a e)+c d^2\right )}-\frac{e^2 x \sqrt{a+b x^2-c x^4}}{2 d \left (d+e x^2\right ) \left (e (b d-a e)+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x^2)^2*Sqrt[a + b*x^2 - c*x^4]),x]

[Out]

-(e^2*x*Sqrt[a + b*x^2 - c*x^4])/(2*d*(c*d^2 + e*(b*d - a*e))*(d + e*x^2)) + ((b - Sqrt[b^2 + 4*a*c])*Sqrt[b +
Sqrt[b^2 + 4*a*c]]*e*Sqrt[1 - (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*
EllipticE[ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 + 4*a*c]]], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a
*c])])/(4*Sqrt[2]*Sqrt[c]*d*(c*d^2 + e*(b*d - a*e))*Sqrt[a + b*x^2 - c*x^4]) - (Sqrt[b + Sqrt[b^2 + 4*a*c]]*(2
*c*d + (b - Sqrt[b^2 + 4*a*c])*e)*Sqrt[1 - (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2
+ 4*a*c])]*EllipticF[ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 + 4*a*c]]], (b + Sqrt[b^2 + 4*a*c])/(b - Sq
rt[b^2 + 4*a*c])])/(4*Sqrt[2]*Sqrt[c]*d*(c*d^2 + e*(b*d - a*e))*Sqrt[a + b*x^2 - c*x^4]) + (Sqrt[b + Sqrt[b^2
+ 4*a*c]]*(3*c*d^2 + e*(2*b*d - a*e))*Sqrt[1 - (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt
[b^2 + 4*a*c])]*EllipticPi[-((b + Sqrt[b^2 + 4*a*c])*e)/(2*c*d), ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2
+ 4*a*c]]], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])])/(2*Sqrt[2]*Sqrt[c]*d^2*(c*d^2 + e*(b*d - a*e))*S
qrt[a + b*x^2 - c*x^4])

Rule 1223

Int[((d_) + (e_.)*(x_)^2)^(q_)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> -Simp[(e^2*x*(d + e*x^2)
^(q + 1)*Sqrt[a + b*x^2 + c*x^4])/(2*d*(q + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(2*d*(q + 1)*(c*d^2 - b*d
*e + a*e^2)), Int[((d + e*x^2)^(q + 1)*Simp[a*e^2*(2*q + 3) + 2*d*(c*d - b*e)*(q + 1) - 2*e*(c*d*(q + 1) - b*e
*(q + 2))*x^2 + c*e^2*(2*q + 5)*x^4, x])/Sqrt[a + b*x^2 + c*x^4], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b
^2 - 4*a*c, 0] && ILtQ[q, -1]

Rule 1716

Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{A = Coeff[P4x,
x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, -Dist[(e^2)^(-1), Int[(C*d - B*e - C*e*x^2)/Sqrt[a + b*x^
2 + c*x^4], x], x] + Dist[(C*d^2 - B*d*e + A*e^2)/e^2, Int[1/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /;
FreeQ[{a, b, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && Ne
Q[c*d^2 - a*e^2, 0]

Rule 1202

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}
, Dist[(Sqrt[1 + (2*c*x^2)/(b - q)]*Sqrt[1 + (2*c*x^2)/(b + q)])/Sqrt[a + b*x^2 + c*x^4], Int[(d + e*x^2)/(Sqr
t[1 + (2*c*x^2)/(b - q)]*Sqrt[1 + (2*c*x^2)/(b + q)]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c
, 0] && NegQ[c/a]

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 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 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 1220

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c,
2]}, Dist[(Sqrt[1 + (2*c*x^2)/(b - q)]*Sqrt[1 + (2*c*x^2)/(b + q)])/Sqrt[a + b*x^2 + c*x^4], Int[1/((d + e*x^
2)*Sqrt[1 + (2*c*x^2)/(b - q)]*Sqrt[1 + (2*c*x^2)/(b + q)]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 -
4*a*c, 0] && NegQ[c/a]

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)
])

Rubi steps

\begin{align*} \int \frac{1}{\left (d+e x^2\right )^2 \sqrt{a+b x^2-c x^4}} \, dx &=-\frac{e^2 x \sqrt{a+b x^2-c x^4}}{2 d \left (c d^2+e (b d-a e)\right ) \left (d+e x^2\right )}+\frac{\int \frac{2 c d^2+e (2 b d-a e)-2 c d e x^2-c e^2 x^4}{\left (d+e x^2\right ) \sqrt{a+b x^2-c x^4}} \, dx}{2 d \left (c d^2+e (b d-a e)\right )}\\ &=-\frac{e^2 x \sqrt{a+b x^2-c x^4}}{2 d \left (c d^2+e (b d-a e)\right ) \left (d+e x^2\right )}-\frac{\int \frac{c d e^2+c e^3 x^2}{\sqrt{a+b x^2-c x^4}} \, dx}{2 d e^2 \left (c d^2+e (b d-a e)\right )}+\frac{\left (3 c d^2+e (2 b d-a e)\right ) \int \frac{1}{\left (d+e x^2\right ) \sqrt{a+b x^2-c x^4}} \, dx}{2 d \left (c d^2+e (b d-a e)\right )}\\ &=-\frac{e^2 x \sqrt{a+b x^2-c x^4}}{2 d \left (c d^2+e (b d-a e)\right ) \left (d+e x^2\right )}-\frac{\left (\sqrt{1-\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}} \sqrt{1-\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}}\right ) \int \frac{c d e^2+c e^3 x^2}{\sqrt{1-\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}} \sqrt{1-\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}}} \, dx}{2 d e^2 \left (c d^2+e (b d-a e)\right ) \sqrt{a+b x^2-c x^4}}+\frac{\left (\left (3 c d^2+e (2 b d-a e)\right ) \sqrt{1-\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}} \sqrt{1-\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}}\right ) \int \frac{1}{\sqrt{1-\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}} \sqrt{1-\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}} \left (d+e x^2\right )} \, dx}{2 d \left (c d^2+e (b d-a e)\right ) \sqrt{a+b x^2-c x^4}}\\ &=-\frac{e^2 x \sqrt{a+b x^2-c x^4}}{2 d \left (c d^2+e (b d-a e)\right ) \left (d+e x^2\right )}+\frac{\sqrt{b+\sqrt{b^2+4 a c}} \left (3 c d^2+e (2 b d-a e)\right ) \sqrt{1-\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}} \sqrt{1-\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}} \Pi \left (-\frac{\left (b+\sqrt{b^2+4 a c}\right ) e}{2 c d};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{2 \sqrt{2} \sqrt{c} d^2 \left (c d^2+e (b d-a e)\right ) \sqrt{a+b x^2-c x^4}}+\frac{\left (\left (b-\sqrt{b^2+4 a c}\right ) e \sqrt{1-\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}} \sqrt{1-\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}}\right ) \int \frac{\sqrt{1-\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}}}{\sqrt{1-\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}}} \, dx}{4 d \left (c d^2+e (b d-a e)\right ) \sqrt{a+b x^2-c x^4}}-\frac{\left (\left (2 c d+\left (b-\sqrt{b^2+4 a c}\right ) e\right ) \sqrt{1-\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}} \sqrt{1-\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}}\right ) \int \frac{1}{\sqrt{1-\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}} \sqrt{1-\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}}} \, dx}{4 d \left (c d^2+e (b d-a e)\right ) \sqrt{a+b x^2-c x^4}}\\ &=-\frac{e^2 x \sqrt{a+b x^2-c x^4}}{2 d \left (c d^2+e (b d-a e)\right ) \left (d+e x^2\right )}+\frac{\left (b-\sqrt{b^2+4 a c}\right ) \sqrt{b+\sqrt{b^2+4 a c}} e \sqrt{1-\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}} \sqrt{1-\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}} E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{4 \sqrt{2} \sqrt{c} d \left (c d^2+e (b d-a e)\right ) \sqrt{a+b x^2-c x^4}}-\frac{\sqrt{b+\sqrt{b^2+4 a c}} \left (2 c d+\left (b-\sqrt{b^2+4 a c}\right ) e\right ) \sqrt{1-\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}} \sqrt{1-\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{4 \sqrt{2} \sqrt{c} d \left (c d^2+e (b d-a e)\right ) \sqrt{a+b x^2-c x^4}}+\frac{\sqrt{b+\sqrt{b^2+4 a c}} \left (3 c d^2+e (2 b d-a e)\right ) \sqrt{1-\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}} \sqrt{1-\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}} \Pi \left (-\frac{\left (b+\sqrt{b^2+4 a c}\right ) e}{2 c d};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{2 \sqrt{2} \sqrt{c} d^2 \left (c d^2+e (b d-a e)\right ) \sqrt{a+b x^2-c x^4}}\\ \end{align*}

Mathematica [C]  time = 5.59278, size = 464, normalized size = 0.65 \[ -\frac{\sqrt{a+b x^2-c x^4} \left (4 d e^2 x+\frac{i \left (d+e x^2\right ) \sqrt{\frac{4 c x^2}{\sqrt{4 a c+b^2}-b}+2} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \left (d \left (e \left (b-\sqrt{4 a c+b^2}\right )+2 c d\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} x \sqrt{-\frac{c}{\sqrt{4 a c+b^2}+b}}\right ),\frac{\sqrt{4 a c+b^2}+b}{b-\sqrt{4 a c+b^2}}\right )+2 \left (e (a e-2 b d)-3 c d^2\right ) \Pi \left (-\frac{\left (b+\sqrt{b^2+4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2+4 a c}}} x\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )+d e \left (\sqrt{4 a c+b^2}-b\right ) E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2+4 a c}}} x\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )\right )}{\sqrt{-\frac{c}{\sqrt{4 a c+b^2}+b}} \left (-a-b x^2+c x^4\right )}\right )}{8 d^2 \left (d+e x^2\right ) \left (e (b d-a e)+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x^2)^2*Sqrt[a + b*x^2 - c*x^4]),x]

[Out]

-(Sqrt[a + b*x^2 - c*x^4]*(4*d*e^2*x + (I*Sqrt[2 + (4*c*x^2)/(-b + Sqrt[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b +
Sqrt[b^2 + 4*a*c])]*(d + e*x^2)*((-b + Sqrt[b^2 + 4*a*c])*d*e*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[-(c/(b + Sqrt[
b^2 + 4*a*c]))]*x], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])] + d*(2*c*d + (b - Sqrt[b^2 + 4*a*c])*e)*E
llipticF[I*ArcSinh[Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*x], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*
c])] + 2*(-3*c*d^2 + e*(-2*b*d + a*e))*EllipticPi[-((b + Sqrt[b^2 + 4*a*c])*e)/(2*c*d), I*ArcSinh[Sqrt[2]*Sqrt
[-(c/(b + Sqrt[b^2 + 4*a*c]))]*x], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])]))/(Sqrt[-(c/(b + Sqrt[b^2
+ 4*a*c]))]*(-a - b*x^2 + c*x^4))))/(8*d^2*(c*d^2 + e*(b*d - a*e))*(d + e*x^2))

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Maple [B]  time = 0.025, size = 1293, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*e^2/(a*e^2-b*d*e-c*d^2)/d*x*(-c*x^4+b*x^2+a)^(1/2)/(e*x^2+d)+1/8*c/(a*e^2-b*d*e-c*d^2)*2^(1/2)/(-b/a+1/a*(
4*a*c+b^2)^(1/2))^(1/2)*(4+2/a*x^2*b-2/a*x^2*(4*a*c+b^2)^(1/2))^(1/2)*(4+2/a*x^2*b+2/a*x^2*(4*a*c+b^2)^(1/2))^
(1/2)/(-c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1/2)*((-b+(4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4-2*b*(b+(4*a*c+b^
2)^(1/2))/a/c)^(1/2))-1/4*c*e/(a*e^2-b*d*e-c*d^2)/d*a*2^(1/2)/(-b/a+1/a*(4*a*c+b^2)^(1/2))^(1/2)*(4+2/a*x^2*b-
2/a*x^2*(4*a*c+b^2)^(1/2))^(1/2)*(4+2/a*x^2*b+2/a*x^2*(4*a*c+b^2)^(1/2))^(1/2)/(-c*x^4+b*x^2+a)^(1/2)/(b+(4*a*
c+b^2)^(1/2))*EllipticF(1/2*x*2^(1/2)*((-b+(4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4-2*b*(b+(4*a*c+b^2)^(1/2))/a/c)^
(1/2))+1/4*c*e/(a*e^2-b*d*e-c*d^2)/d*a*2^(1/2)/(-b/a+1/a*(4*a*c+b^2)^(1/2))^(1/2)*(4+2/a*x^2*b-2/a*x^2*(4*a*c+
b^2)^(1/2))^(1/2)*(4+2/a*x^2*b+2/a*x^2*(4*a*c+b^2)^(1/2))^(1/2)/(-c*x^4+b*x^2+a)^(1/2)/(b+(4*a*c+b^2)^(1/2))*E
llipticE(1/2*x*2^(1/2)*((-b+(4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4-2*b*(b+(4*a*c+b^2)^(1/2))/a/c)^(1/2))+1/2/(a*e
^2-b*d*e-c*d^2)/d^2*e^2*2^(1/2)/(-b/a+1/a*(4*a*c+b^2)^(1/2))^(1/2)*(1+1/2/a*x^2*b-1/2/a*x^2*(4*a*c+b^2)^(1/2))
^(1/2)*(1+1/2/a*x^2*b+1/2/a*x^2*(4*a*c+b^2)^(1/2))^(1/2)/(-c*x^4+b*x^2+a)^(1/2)*EllipticPi(1/2*x*2^(1/2)*((-b+
(4*a*c+b^2)^(1/2))/a)^(1/2),-2/(-b+(4*a*c+b^2)^(1/2))*a/d*e,(-1/2*(b+(4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/((-b+
(4*a*c+b^2)^(1/2))/a)^(1/2))*a-1/(a*e^2-b*d*e-c*d^2)/d*e*2^(1/2)/(-b/a+1/a*(4*a*c+b^2)^(1/2))^(1/2)*(1+1/2/a*x
^2*b-1/2/a*x^2*(4*a*c+b^2)^(1/2))^(1/2)*(1+1/2/a*x^2*b+1/2/a*x^2*(4*a*c+b^2)^(1/2))^(1/2)/(-c*x^4+b*x^2+a)^(1/
2)*EllipticPi(1/2*x*2^(1/2)*((-b+(4*a*c+b^2)^(1/2))/a)^(1/2),-2/(-b+(4*a*c+b^2)^(1/2))*a/d*e,(-1/2*(b+(4*a*c+b
^2)^(1/2))/a)^(1/2)*2^(1/2)/((-b+(4*a*c+b^2)^(1/2))/a)^(1/2))*b-3/2/(a*e^2-b*d*e-c*d^2)*2^(1/2)/(-b/a+1/a*(4*a
*c+b^2)^(1/2))^(1/2)*(1+1/2/a*x^2*b-1/2/a*x^2*(4*a*c+b^2)^(1/2))^(1/2)*(1+1/2/a*x^2*b+1/2/a*x^2*(4*a*c+b^2)^(1
/2))^(1/2)/(-c*x^4+b*x^2+a)^(1/2)*EllipticPi(1/2*x*2^(1/2)*((-b+(4*a*c+b^2)^(1/2))/a)^(1/2),-2/(-b+(4*a*c+b^2)
^(1/2))*a/d*e,(-1/2*(b+(4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/((-b+(4*a*c+b^2)^(1/2))/a)^(1/2))*c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-c x^{4} + b x^{2} + a}{\left (e x^{2} + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)^2/(-c*x^4+b*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-c*x^4 + b*x^2 + a)*(e*x^2 + d)^2), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)^2/(-c*x^4+b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d + e x^{2}\right )^{2} \sqrt{a + b x^{2} - c x^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x**2+d)**2/(-c*x**4+b*x**2+a)**(1/2),x)

[Out]

Integral(1/((d + e*x**2)**2*sqrt(a + b*x**2 - c*x**4)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-c x^{4} + b x^{2} + a}{\left (e x^{2} + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)^2/(-c*x^4+b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-c*x^4 + b*x^2 + a)*(e*x^2 + d)^2), x)