\(\int \frac {1}{(d+e x^2)^4 \sqrt {a-c x^4}} \, dx\) [163]

Optimal result

Integrand size = 22, antiderivative size = 563 \[ \int \frac {1}{\left (d+e x^2\right )^4 \sqrt {a-c x^4}} \, dx=-\frac {e^2 x \sqrt {a-c x^4}}{6 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^3}-\frac {5 e^2 \left (3 c d^2-a e^2\right ) x \sqrt {a-c x^4}}{24 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )^2}-\frac {e^2 \left (29 c^2 d^4-14 a c d^2 e^2+5 a^2 e^4\right ) x \sqrt {a-c x^4}}{16 d^3 \left (c d^2-a e^2\right )^3 \left (d+e x^2\right )}-\frac {a^{3/4} \sqrt [4]{c} e \left (29 c^2 d^4-14 a c d^2 e^2+5 a^2 e^4\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{16 d^3 \left (c d^2-a e^2\right )^3 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \sqrt [4]{c} \left (57 c^2 d^4-30 \sqrt {a} c^{3/2} d^3 e-32 a c d^2 e^2+10 a^{3/2} \sqrt {c} d e^3+15 a^2 e^4\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{48 d^3 \left (\sqrt {c} d-\sqrt {a} e\right )^2 \left (\sqrt {c} d+\sqrt {a} e\right )^3 \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (35 c^3 d^6-7 a c^2 d^4 e^2+17 a^2 c d^2 e^4-5 a^3 e^6\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{16 \sqrt [4]{c} d^4 \left (c d^2-a e^2\right )^3 \sqrt {a-c x^4}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1238, 1711, 1731, 1215, 230, 227, 1214, 1213, 435, 1233, 1232} \[ \int \frac {1}{\left (d+e x^2\right )^4 \sqrt {a-c x^4}} \, dx=-\frac {e^2 x \sqrt {a-c x^4} \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right )}{16 d^3 \left (d+e x^2\right ) \left (c d^2-a e^2\right )^3}-\frac {a^{3/4} \sqrt [4]{c} e \sqrt {1-\frac {c x^4}{a}} \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right ) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{16 d^3 \sqrt {a-c x^4} \left (c d^2-a e^2\right )^3}-\frac {\sqrt [4]{a} \sqrt [4]{c} \sqrt {1-\frac {c x^4}{a}} \left (10 a^{3/2} \sqrt {c} d e^3+15 a^2 e^4-30 \sqrt {a} c^{3/2} d^3 e-32 a c d^2 e^2+57 c^2 d^4\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{48 d^3 \sqrt {a-c x^4} \left (\sqrt {c} d-\sqrt {a} e\right )^2 \left (\sqrt {a} e+\sqrt {c} d\right )^3}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (-5 a^3 e^6+17 a^2 c d^2 e^4-7 a c^2 d^4 e^2+35 c^3 d^6\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{16 \sqrt [4]{c} d^4 \sqrt {a-c x^4} \left (c d^2-a e^2\right )^3}-\frac {5 e^2 x \sqrt {a-c x^4} \left (3 c d^2-a e^2\right )}{24 d^2 \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )^2}-\frac {e^2 x \sqrt {a-c x^4}}{6 d \left (d+e x^2\right )^3 \left (c d^2-a e^2\right )} \]

[In]

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

[Out]

-1/6*(e^2*x*Sqrt[a - c*x^4])/(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])/(1
6*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 227

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 230

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 435

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

Rule 1213

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 1214

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 1215

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-c/a, 2]}, Dist[(d*q - e)/q, In
t[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 1232

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

Rule 1233

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 1238

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
)/Sqrt[a + c*x^4])*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], x],
x] /; FreeQ[{a, c, d, e}, x] && ILtQ[q, -1]

Rule 1711

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)/Sqrt[a
+ c*x^4])*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], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2] && LeQ
[Expon[P4x, x], 4] && NeQ[c*d^2 + a*e^2, 0] && ILtQ[q, -1]

Rule 1731

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]

Rubi steps \begin{align*} \text {integral}& = -\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] (verified)

Result contains complex when optimal does not.

Time = 11.38 (sec) , antiderivative size = 458, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\left (d+e x^2\right )^4 \sqrt {a-c x^4}} \, dx=\frac {-\frac {d e^2 x \left (a-c x^4\right ) \left (8 \left (c d^3-a d e^2\right )^2+10 d \left (c d^2-a e^2\right ) \left (3 c d^2-a e^2\right ) \left (d+e x^2\right )+3 \left (29 c^2 d^4-14 a c d^2 e^2+5 a^2 e^4\right ) \left (d+e x^2\right )^2\right )}{\left (c d^2-a e^2\right )^3 \left (d+e x^2\right )^3}-\frac {i \sqrt {1-\frac {c x^4}{a}} \left (3 \sqrt {a} \sqrt {c} d e \left (29 c^2 d^4-14 a c d^2 e^2+5 a^2 e^4\right ) E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+\sqrt {c} d \left (57 c^{5/2} d^5-87 \sqrt {a} c^2 d^4 e-2 a c^{3/2} d^3 e^2+42 a^{3/2} c d^2 e^3+5 a^2 \sqrt {c} d e^4-15 a^{5/2} e^5\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+3 \left (-35 c^3 d^6+7 a c^2 d^4 e^2-17 a^2 c d^2 e^4+5 a^3 e^6\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} \left (-c d^2+a e^2\right )^3}}{48 d^4 \sqrt {a-c x^4}} \]

[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])

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1419 vs. \(2 (489 ) = 978\).

Time = 2.47 (sec) , antiderivative size = 1420, normalized size of antiderivative = 2.52

method	result	size
default	\(\text {Expression too large to display}\)	\(1420\)
elliptic	\(\text {Expression too large to display}\)	\(1420\)

[In]

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

[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)-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)*EllipticF(x*(1/a^(1/2)*c^(1/2))^(1/2),I)+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+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*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)*Elli
pticF(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)-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)-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

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^4 \sqrt {a-c x^4}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int \frac {1}{\left (d+e x^2\right )^4 \sqrt {a-c x^4}} \, dx=\int \frac {1}{\sqrt {a - c x^{4}} \left (d + e x^{2}\right )^{4}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {1}{\left (d+e x^2\right )^4 \sqrt {a-c x^4}} \, dx=\int { \frac {1}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}^{4}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {1}{\left (d+e x^2\right )^4 \sqrt {a-c x^4}} \, dx=\int { \frac {1}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}^{4}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^4 \sqrt {a-c x^4}} \, dx=\int \frac {1}{\sqrt {a-c\,x^4}\,{\left (e\,x^2+d\right )}^4} \,d x \]

[In]

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

[Out]

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