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

Optimal. Leaf size=563 \[ -\frac{\sqrt [4]{a} \sqrt [4]{c} \sqrt{1-\frac{c x^4}{a}} \left (10 a^{3/2} \sqrt{c} d e^3+15 a^2 e^4-30 \sqrt{a} c^{3/2} d^3 e-32 a c d^2 e^2+57 c^2 d^4\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{48 d^3 \sqrt{a-c x^4} \left (\sqrt{c} d-\sqrt{a} e\right )^2 \left (\sqrt{a} e+\sqrt{c} d\right )^3}-\frac{e^2 x \sqrt{a-c x^4} \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right )}{16 d^3 \left (d+e x^2\right ) \left (c d^2-a e^2\right )^3}-\frac{a^{3/4} \sqrt [4]{c} e \sqrt{1-\frac{c x^4}{a}} \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right ) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{16 d^3 \sqrt{a-c x^4} \left (c d^2-a e^2\right )^3}+\frac{\sqrt [4]{a} \sqrt{1-\frac{c x^4}{a}} \left (17 a^2 c d^2 e^4-5 a^3 e^6-7 a c^2 d^4 e^2+35 c^3 d^6\right ) \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{16 \sqrt [4]{c} d^4 \sqrt{a-c x^4} \left (c d^2-a e^2\right )^3}-\frac{5 e^2 x \sqrt{a-c x^4} \left (3 c d^2-a e^2\right )}{24 d^2 \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )^2}-\frac{e^2 x \sqrt{a-c x^4}}{6 d \left (d+e x^2\right )^3 \left (c d^2-a e^2\right )} \]

[Out]

-(e^2*x*Sqrt[a - c*x^4])/(6*d*(c*d^2 - a*e^2)*(d + e*x^2)^3) - (5*e^2*(3*c*d^2 - a*e^2)*x*Sqrt[a - c*x^4])/(24
*d^2*(c*d^2 - a*e^2)^2*(d + e*x^2)^2) - (e^2*(29*c^2*d^4 - 14*a*c*d^2*e^2 + 5*a^2*e^4)*x*Sqrt[a - c*x^4])/(16*
d^3*(c*d^2 - a*e^2)^3*(d + e*x^2)) - (a^(3/4)*c^(1/4)*e*(29*c^2*d^4 - 14*a*c*d^2*e^2 + 5*a^2*e^4)*Sqrt[1 - (c*
x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(16*d^3*(c*d^2 - a*e^2)^3*Sqrt[a - c*x^4]) - (a^(1/4)*c^(1
/4)*(57*c^2*d^4 - 30*Sqrt[a]*c^(3/2)*d^3*e - 32*a*c*d^2*e^2 + 10*a^(3/2)*Sqrt[c]*d*e^3 + 15*a^2*e^4)*Sqrt[1 -
(c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(48*d^3*(Sqrt[c]*d - Sqrt[a]*e)^2*(Sqrt[c]*d + Sqrt[a]*
e)^3*Sqrt[a - c*x^4]) + (a^(1/4)*(35*c^3*d^6 - 7*a*c^2*d^4*e^2 + 17*a^2*c*d^2*e^4 - 5*a^3*e^6)*Sqrt[1 - (c*x^4
)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(16*c^(1/4)*d^4*(c*d^2 - a*e^2)^
3*Sqrt[a - c*x^4])

________________________________________________________________________________________

Rubi [A]  time = 1.20551, antiderivative size = 563, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 11, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {1224, 1697, 1717, 1201, 224, 221, 1200, 1199, 424, 1219, 1218} \[ -\frac{e^2 x \sqrt{a-c x^4} \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right )}{16 d^3 \left (d+e x^2\right ) \left (c d^2-a e^2\right )^3}-\frac{\sqrt [4]{a} \sqrt [4]{c} \sqrt{1-\frac{c x^4}{a}} \left (10 a^{3/2} \sqrt{c} d e^3+15 a^2 e^4-30 \sqrt{a} c^{3/2} d^3 e-32 a c d^2 e^2+57 c^2 d^4\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{48 d^3 \sqrt{a-c x^4} \left (\sqrt{c} d-\sqrt{a} e\right )^2 \left (\sqrt{a} e+\sqrt{c} d\right )^3}-\frac{a^{3/4} \sqrt [4]{c} e \sqrt{1-\frac{c x^4}{a}} \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right ) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{16 d^3 \sqrt{a-c x^4} \left (c d^2-a e^2\right )^3}+\frac{\sqrt [4]{a} \sqrt{1-\frac{c x^4}{a}} \left (17 a^2 c d^2 e^4-5 a^3 e^6-7 a c^2 d^4 e^2+35 c^3 d^6\right ) \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{16 \sqrt [4]{c} d^4 \sqrt{a-c x^4} \left (c d^2-a e^2\right )^3}-\frac{5 e^2 x \sqrt{a-c x^4} \left (3 c d^2-a e^2\right )}{24 d^2 \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )^2}-\frac{e^2 x \sqrt{a-c x^4}}{6 d \left (d+e x^2\right )^3 \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e^2*x*Sqrt[a - c*x^4])/(6*d*(c*d^2 - a*e^2)*(d + e*x^2)^3) - (5*e^2*(3*c*d^2 - a*e^2)*x*Sqrt[a - c*x^4])/(24
*d^2*(c*d^2 - a*e^2)^2*(d + e*x^2)^2) - (e^2*(29*c^2*d^4 - 14*a*c*d^2*e^2 + 5*a^2*e^4)*x*Sqrt[a - c*x^4])/(16*
d^3*(c*d^2 - a*e^2)^3*(d + e*x^2)) - (a^(3/4)*c^(1/4)*e*(29*c^2*d^4 - 14*a*c*d^2*e^2 + 5*a^2*e^4)*Sqrt[1 - (c*
x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(16*d^3*(c*d^2 - a*e^2)^3*Sqrt[a - c*x^4]) - (a^(1/4)*c^(1
/4)*(57*c^2*d^4 - 30*Sqrt[a]*c^(3/2)*d^3*e - 32*a*c*d^2*e^2 + 10*a^(3/2)*Sqrt[c]*d*e^3 + 15*a^2*e^4)*Sqrt[1 -
(c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(48*d^3*(Sqrt[c]*d - Sqrt[a]*e)^2*(Sqrt[c]*d + Sqrt[a]*
e)^3*Sqrt[a - c*x^4]) + (a^(1/4)*(35*c^3*d^6 - 7*a*c^2*d^4*e^2 + 17*a^2*c*d^2*e^4 - 5*a^3*e^6)*Sqrt[1 - (c*x^4
)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(16*c^(1/4)*d^4*(c*d^2 - a*e^2)^
3*Sqrt[a - c*x^4])

Rule 1224

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

Rule 1697

Int[((P4x_)*((d_) + (e_.)*(x_)^2)^(q_))/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{A = Coeff[P4x, x, 0], B
= Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, -Simp[((C*d^2 - B*d*e + A*e^2)*x*(d + e*x^2)^(q + 1)*Sqrt[a + c*x^4
])/(2*d*(q + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(2*d*(q + 1)*(c*d^2 + a*e^2)), Int[((d + e*x^2)^(q + 1)*Simp[a*d
*(C*d - B*e) + A*(a*e^2*(2*q + 3) + 2*c*d^2*(q + 1)) + 2*d*(B*c*d - A*c*e + a*C*e)*(q + 1)*x^2 + c*(C*d^2 - B*
d*e + A*e^2)*(2*q + 5)*x^4, x])/Sqrt[a + c*x^4], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2] && LeQ[E
xpon[P4x, x], 4] && NeQ[c*d^2 + a*e^2, 0] && ILtQ[q, -1]

Rule 1717

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

Rule 1201

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

Rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + (b*x^4)/a]/Sqrt[a + b*x^4], Int[1/Sqrt[1 + (b*x^4)
/a], x], x] /; FreeQ[{a, b}, x] && NegQ[b/a] &&  !GtQ[a, 0]

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[(Rt[-b, 4]*x)/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 1200

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + (c*x^4)/a]/Sqrt[a + c*x^4], In
t[(d + e*x^2)/Sqrt[1 + (c*x^4)/a], x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] &&
!GtQ[a, 0]

Rule 1199

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Dist[d/Sqrt[a], Int[Sqrt[1 + (e*x^2)/d]/Sqrt
[1 - (e*x^2)/d], x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] && GtQ[a, 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 1219

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[Sqrt[1 + (c*x^4)/a]/Sqrt[a + c*x^4]
, Int[1/((d + e*x^2)*Sqrt[1 + (c*x^4)/a]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] &&  !GtQ[a, 0]

Rule 1218

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

Rubi steps

\begin{align*} \int \frac{1}{\left (d+e x^2\right )^4 \sqrt{a-c x^4}} \, dx &=-\frac{e^2 x \sqrt{a-c x^4}}{6 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^3}+\frac{\int \frac{6 c d^2-5 a e^2-6 c d e x^2+3 c e^2 x^4}{\left (d+e x^2\right )^3 \sqrt{a-c x^4}} \, dx}{6 d \left (c d^2-a e^2\right )}\\ &=-\frac{e^2 x \sqrt{a-c x^4}}{6 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^3}-\frac{5 e^2 \left (3 c d^2-a e^2\right ) x \sqrt{a-c x^4}}{24 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )^2}+\frac{\int \frac{24 c^2 d^4-29 a c d^2 e^2+15 a^2 e^4-8 c d e \left (6 c d^2-a e^2\right ) x^2+5 c e^2 \left (3 c d^2-a e^2\right ) x^4}{\left (d+e x^2\right )^2 \sqrt{a-c x^4}} \, dx}{24 d^2 \left (c d^2-a e^2\right )^2}\\ &=-\frac{e^2 x \sqrt{a-c x^4}}{6 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^3}-\frac{5 e^2 \left (3 c d^2-a e^2\right ) x \sqrt{a-c x^4}}{24 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )^2}-\frac{e^2 \left (29 c^2 d^4-14 a c d^2 e^2+5 a^2 e^4\right ) x \sqrt{a-c x^4}}{16 d^3 \left (c d^2-a e^2\right )^3 \left (d+e x^2\right )}+\frac{\int \frac{48 c^3 d^6-19 a c^2 d^4 e^2+46 a^2 c d^2 e^4-15 a^3 e^6-4 c d e \left (36 c^2 d^4-11 a c d^2 e^2+5 a^2 e^4\right ) x^2-3 c e^2 \left (29 c^2 d^4-14 a c d^2 e^2+5 a^2 e^4\right ) x^4}{\left (d+e x^2\right ) \sqrt{a-c x^4}} \, dx}{48 d^3 \left (c d^2-a e^2\right )^3}\\ &=-\frac{e^2 x \sqrt{a-c x^4}}{6 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^3}-\frac{5 e^2 \left (3 c d^2-a e^2\right ) x \sqrt{a-c x^4}}{24 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )^2}-\frac{e^2 \left (29 c^2 d^4-14 a c d^2 e^2+5 a^2 e^4\right ) x \sqrt{a-c x^4}}{16 d^3 \left (c d^2-a e^2\right )^3 \left (d+e x^2\right )}-\frac{\int \frac{-3 c d e^2 \left (29 c^2 d^4-14 a c d^2 e^2+5 a^2 e^4\right )+4 c d e^2 \left (36 c^2 d^4-11 a c d^2 e^2+5 a^2 e^4\right )+3 c e^3 \left (29 c^2 d^4-14 a c d^2 e^2+5 a^2 e^4\right ) x^2}{\sqrt{a-c x^4}} \, dx}{48 d^3 e^2 \left (c d^2-a e^2\right )^3}+\frac{\left (35 c^3 d^6-7 a c^2 d^4 e^2+17 a^2 c d^2 e^4-5 a^3 e^6\right ) \int \frac{1}{\left (d+e x^2\right ) \sqrt{a-c x^4}} \, dx}{16 d^3 \left (c d^2-a e^2\right )^3}\\ &=-\frac{e^2 x \sqrt{a-c x^4}}{6 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^3}-\frac{5 e^2 \left (3 c d^2-a e^2\right ) x \sqrt{a-c x^4}}{24 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )^2}-\frac{e^2 \left (29 c^2 d^4-14 a c d^2 e^2+5 a^2 e^4\right ) x \sqrt{a-c x^4}}{16 d^3 \left (c d^2-a e^2\right )^3 \left (d+e x^2\right )}-\frac{\left (\sqrt{a} \sqrt{c} e \left (29 c^2 d^4-14 a c d^2 e^2+5 a^2 e^4\right )\right ) \int \frac{1+\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a-c x^4}} \, dx}{16 d^3 \left (c d^2-a e^2\right )^3}-\frac{\left (\sqrt{c} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (57 c^2 d^4-30 \sqrt{a} c^{3/2} d^3 e-32 a c d^2 e^2+10 a^{3/2} \sqrt{c} d e^3+15 a^2 e^4\right )\right ) \int \frac{1}{\sqrt{a-c x^4}} \, dx}{48 d^3 \left (c d^2-a e^2\right )^3}+\frac{\left (\left (35 c^3 d^6-7 a c^2 d^4 e^2+17 a^2 c d^2 e^4-5 a^3 e^6\right ) \sqrt{1-\frac{c x^4}{a}}\right ) \int \frac{1}{\left (d+e x^2\right ) \sqrt{1-\frac{c x^4}{a}}} \, dx}{16 d^3 \left (c d^2-a e^2\right )^3 \sqrt{a-c x^4}}\\ &=-\frac{e^2 x \sqrt{a-c x^4}}{6 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^3}-\frac{5 e^2 \left (3 c d^2-a e^2\right ) x \sqrt{a-c x^4}}{24 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )^2}-\frac{e^2 \left (29 c^2 d^4-14 a c d^2 e^2+5 a^2 e^4\right ) x \sqrt{a-c x^4}}{16 d^3 \left (c d^2-a e^2\right )^3 \left (d+e x^2\right )}+\frac{\sqrt [4]{a} \left (35 c^3 d^6-7 a c^2 d^4 e^2+17 a^2 c d^2 e^4-5 a^3 e^6\right ) \sqrt{1-\frac{c x^4}{a}} \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{16 \sqrt [4]{c} d^4 \left (c d^2-a e^2\right )^3 \sqrt{a-c x^4}}-\frac{\left (\sqrt{a} \sqrt{c} e \left (29 c^2 d^4-14 a c d^2 e^2+5 a^2 e^4\right ) \sqrt{1-\frac{c x^4}{a}}\right ) \int \frac{1+\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{1-\frac{c x^4}{a}}} \, dx}{16 d^3 \left (c d^2-a e^2\right )^3 \sqrt{a-c x^4}}-\frac{\left (\sqrt{c} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (57 c^2 d^4-30 \sqrt{a} c^{3/2} d^3 e-32 a c d^2 e^2+10 a^{3/2} \sqrt{c} d e^3+15 a^2 e^4\right ) \sqrt{1-\frac{c x^4}{a}}\right ) \int \frac{1}{\sqrt{1-\frac{c x^4}{a}}} \, dx}{48 d^3 \left (c d^2-a e^2\right )^3 \sqrt{a-c x^4}}\\ &=-\frac{e^2 x \sqrt{a-c x^4}}{6 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^3}-\frac{5 e^2 \left (3 c d^2-a e^2\right ) x \sqrt{a-c x^4}}{24 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )^2}-\frac{e^2 \left (29 c^2 d^4-14 a c d^2 e^2+5 a^2 e^4\right ) x \sqrt{a-c x^4}}{16 d^3 \left (c d^2-a e^2\right )^3 \left (d+e x^2\right )}-\frac{\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (57 c^2 d^4-30 \sqrt{a} c^{3/2} d^3 e-32 a c d^2 e^2+10 a^{3/2} \sqrt{c} d e^3+15 a^2 e^4\right ) \sqrt{1-\frac{c x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{48 d^3 \left (c d^2-a e^2\right )^3 \sqrt{a-c x^4}}+\frac{\sqrt [4]{a} \left (35 c^3 d^6-7 a c^2 d^4 e^2+17 a^2 c d^2 e^4-5 a^3 e^6\right ) \sqrt{1-\frac{c x^4}{a}} \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{16 \sqrt [4]{c} d^4 \left (c d^2-a e^2\right )^3 \sqrt{a-c x^4}}-\frac{\left (\sqrt{a} \sqrt{c} e \left (29 c^2 d^4-14 a c d^2 e^2+5 a^2 e^4\right ) \sqrt{1-\frac{c x^4}{a}}\right ) \int \frac{\sqrt{1+\frac{\sqrt{c} x^2}{\sqrt{a}}}}{\sqrt{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}} \, dx}{16 d^3 \left (c d^2-a e^2\right )^3 \sqrt{a-c x^4}}\\ &=-\frac{e^2 x \sqrt{a-c x^4}}{6 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^3}-\frac{5 e^2 \left (3 c d^2-a e^2\right ) x \sqrt{a-c x^4}}{24 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )^2}-\frac{e^2 \left (29 c^2 d^4-14 a c d^2 e^2+5 a^2 e^4\right ) x \sqrt{a-c x^4}}{16 d^3 \left (c d^2-a e^2\right )^3 \left (d+e x^2\right )}-\frac{a^{3/4} \sqrt [4]{c} e \left (29 c^2 d^4-14 a c d^2 e^2+5 a^2 e^4\right ) \sqrt{1-\frac{c x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{16 d^3 \left (c d^2-a e^2\right )^3 \sqrt{a-c x^4}}-\frac{\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (57 c^2 d^4-30 \sqrt{a} c^{3/2} d^3 e-32 a c d^2 e^2+10 a^{3/2} \sqrt{c} d e^3+15 a^2 e^4\right ) \sqrt{1-\frac{c x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{48 d^3 \left (c d^2-a e^2\right )^3 \sqrt{a-c x^4}}+\frac{\sqrt [4]{a} \left (35 c^3 d^6-7 a c^2 d^4 e^2+17 a^2 c d^2 e^4-5 a^3 e^6\right ) \sqrt{1-\frac{c x^4}{a}} \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{16 \sqrt [4]{c} d^4 \left (c d^2-a e^2\right )^3 \sqrt{a-c x^4}}\\ \end{align*}

Mathematica [C]  time = 1.99735, size = 458, normalized size = 0.81 \[ \frac{-\frac{d e^2 x \left (a-c x^4\right ) \left (3 \left (d+e x^2\right )^2 \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right )+10 d \left (d+e x^2\right ) \left (c d^2-a e^2\right ) \left (3 c d^2-a e^2\right )+8 \left (c d^3-a d e^2\right )^2\right )}{\left (d+e x^2\right )^3 \left (c d^2-a e^2\right )^3}-\frac{i \sqrt{1-\frac{c x^4}{a}} \left (\sqrt{c} d \left (42 a^{3/2} c d^2 e^3+5 a^2 \sqrt{c} d e^4-15 a^{5/2} e^5-2 a c^{3/2} d^3 e^2-87 \sqrt{a} c^2 d^4 e+57 c^{5/2} d^5\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{-\frac{\sqrt{c}}{\sqrt{a}}}\right ),-1\right )+3 \sqrt{a} \sqrt{c} d e \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right ) E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )+3 \left (-17 a^2 c d^2 e^4+5 a^3 e^6+7 a c^2 d^4 e^2-35 c^3 d^6\right ) \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )\right )}{\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} \left (a e^2-c d^2\right )^3}}{48 d^4 \sqrt{a-c x^4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((d*e^2*x*(a - c*x^4)*(8*(c*d^3 - a*d*e^2)^2 + 10*d*(c*d^2 - a*e^2)*(3*c*d^2 - a*e^2)*(d + e*x^2) + 3*(29*c^
2*d^4 - 14*a*c*d^2*e^2 + 5*a^2*e^4)*(d + e*x^2)^2))/((c*d^2 - a*e^2)^3*(d + e*x^2)^3)) - (I*Sqrt[1 - (c*x^4)/a
]*(3*Sqrt[a]*Sqrt[c]*d*e*(29*c^2*d^4 - 14*a*c*d^2*e^2 + 5*a^2*e^4)*EllipticE[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])
]*x], -1] + Sqrt[c]*d*(57*c^(5/2)*d^5 - 87*Sqrt[a]*c^2*d^4*e - 2*a*c^(3/2)*d^3*e^2 + 42*a^(3/2)*c*d^2*e^3 + 5*
a^2*Sqrt[c]*d*e^4 - 15*a^(5/2)*e^5)*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + 3*(-35*c^3*d^6 + 7*
a*c^2*d^4*e^2 - 17*a^2*c*d^2*e^4 + 5*a^3*e^6)*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/
Sqrt[a])]*x], -1]))/(Sqrt[-(Sqrt[c]/Sqrt[a])]*(-(c*d^2) + a*e^2)^3))/(48*d^4*Sqrt[a - c*x^4])

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Maple [B]  time = 0.195, size = 1420, normalized size = 2.5 \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)^4/(-c*x^4+a)^(1/2),x)

[Out]

1/6*e^2/(a*e^2-c*d^2)/d*x*(-c*x^4+a)^(1/2)/(e*x^2+d)^3+5/24*e^2*(a*e^2-3*c*d^2)/(a*e^2-c*d^2)^2/d^2*x*(-c*x^4+
a)^(1/2)/(e*x^2+d)^2+1/16*e^2*(5*a^2*e^4-14*a*c*d^2*e^2+29*c^2*d^4)/(a*e^2-c*d^2)^3/d^3*x*(-c*x^4+a)^(1/2)/(e*
x^2+d)-7/8*c^(3/2)*e^3/(a*e^2-c*d^2)^3/d*a^(3/2)/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+
1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*EllipticE(x*(1/a^(1/2)*c^(1/2))^(1/2),I)+29/16*c^(5/2)*e/(a*e^2-
c*d^2)^3*d*a^(1/2)/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(
-c*x^4+a)^(1/2)*EllipticE(x*(1/a^(1/2)*c^(1/2))^(1/2),I)+5/48*c/d^2/(a*e^2-c*d^2)^3/(1/a^(1/2)*c^(1/2))^(1/2)*
(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x*(1/a^(1/2)*c^(1/2
))^(1/2),I)*a^2*e^4-1/24*c^2/(a*e^2-c*d^2)^3/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^
(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x*(1/a^(1/2)*c^(1/2))^(1/2),I)*a*e^2+5/16/(a*e^2-c*d^2)^3/
d^4*e^6/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(
1/2)*EllipticPi(x*(1/a^(1/2)*c^(1/2))^(1/2),-e*a^(1/2)/d/c^(1/2),(-1/a^(1/2)*c^(1/2))^(1/2)/(1/a^(1/2)*c^(1/2)
)^(1/2))*a^3+7/16/(a*e^2-c*d^2)^3*e^2/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c
^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*EllipticPi(x*(1/a^(1/2)*c^(1/2))^(1/2),-e*a^(1/2)/d/c^(1/2),(-1/a^(1/2)*c^(
1/2))^(1/2)/(1/a^(1/2)*c^(1/2))^(1/2))*a*c^2-35/16/(a*e^2-c*d^2)^3*d^2/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*
c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*EllipticPi(x*(1/a^(1/2)*c^(1/2))^(1/2),-e*
a^(1/2)/d/c^(1/2),(-1/a^(1/2)*c^(1/2))^(1/2)/(1/a^(1/2)*c^(1/2))^(1/2))*c^3+19/16*c^3*d^2/(a*e^2-c*d^2)^3/(1/a
^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*Ellipti
cF(x*(1/a^(1/2)*c^(1/2))^(1/2),I)-5/16*c^(1/2)*e^5/(a*e^2-c*d^2)^3/d^3*a^(5/2)/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/
a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x*(1/a^(1/2)*c^(1/2))^(1
/2),I)+7/8*c^(3/2)*e^3/(a*e^2-c*d^2)^3/d*a^(3/2)/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+
1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x*(1/a^(1/2)*c^(1/2))^(1/2),I)-29/16*c^(5/2)*e/(a*e^2-
c*d^2)^3*d*a^(1/2)/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(
-c*x^4+a)^(1/2)*EllipticF(x*(1/a^(1/2)*c^(1/2))^(1/2),I)+5/16*c^(1/2)*e^5/(a*e^2-c*d^2)^3/d^3*a^(5/2)/(1/a^(1/
2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*EllipticE(x
*(1/a^(1/2)*c^(1/2))^(1/2),I)-17/16/(a*e^2-c*d^2)^3/d^2*e^4/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2
)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*EllipticPi(x*(1/a^(1/2)*c^(1/2))^(1/2),-e*a^(1/2)/d/c
^(1/2),(-1/a^(1/2)*c^(1/2))^(1/2)/(1/a^(1/2)*c^(1/2))^(1/2))*a^2*c

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(-c*x^4 + a)*(e*x^2 + d)^4), 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)^4/(-c*x^4+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}{\sqrt{a - c x^{4}} \left (d + e x^{2}\right )^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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