3.162 \(\int \frac{1}{(d+e x^2)^3 \sqrt{a-c x^4}} \, dx\)

Optimal. Leaf size=425 \[ -\frac{\sqrt [4]{a} \sqrt [4]{c} \sqrt{1-\frac{c x^4}{a}} \left (-2 \sqrt{a} \sqrt{c} d e-3 a e^2+7 c d^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{8 d^2 \sqrt{a-c x^4} \left (\sqrt{a} e+\sqrt{c} d\right ) \left (c d^2-a e^2\right )}+\frac{3 \sqrt [4]{a} \sqrt{1-\frac{c x^4}{a}} \left (a^2 e^4-2 a c d^2 e^2+5 c^2 d^4\right ) \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{c} d^3 \sqrt{a-c x^4} \left (c d^2-a e^2\right )^2}-\frac{3 a^{3/4} \sqrt [4]{c} e \sqrt{1-\frac{c x^4}{a}} \left (3 c d^2-a e^2\right ) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 d^2 \sqrt{a-c x^4} \left (c d^2-a e^2\right )^2}-\frac{3 e^2 x \sqrt{a-c x^4} \left (3 c d^2-a e^2\right )}{8 d^2 \left (d+e x^2\right ) \left (c d^2-a e^2\right )^2}-\frac{e^2 x \sqrt{a-c x^4}}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )} \]

[Out]

-(e^2*x*Sqrt[a - c*x^4])/(4*d*(c*d^2 - a*e^2)*(d + e*x^2)^2) - (3*e^2*(3*c*d^2 - a*e^2)*x*Sqrt[a - c*x^4])/(8*
d^2*(c*d^2 - a*e^2)^2*(d + e*x^2)) - (3*a^(3/4)*c^(1/4)*e*(3*c*d^2 - a*e^2)*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcS
in[(c^(1/4)*x)/a^(1/4)], -1])/(8*d^2*(c*d^2 - a*e^2)^2*Sqrt[a - c*x^4]) - (a^(1/4)*c^(1/4)*(7*c*d^2 - 2*Sqrt[a
]*Sqrt[c]*d*e - 3*a*e^2)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(8*d^2*(Sqrt[c]*d + S
qrt[a]*e)*(c*d^2 - a*e^2)*Sqrt[a - c*x^4]) + (3*a^(1/4)*(5*c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*Sqrt[1 - (c*x^4)
/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(8*c^(1/4)*d^3*(c*d^2 - a*e^2)^2*
Sqrt[a - c*x^4])

________________________________________________________________________________________

Rubi [A]  time = 0.750702, antiderivative size = 425, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ \frac{3 \sqrt [4]{a} \sqrt{1-\frac{c x^4}{a}} \left (a^2 e^4-2 a c d^2 e^2+5 c^2 d^4\right ) \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{c} d^3 \sqrt{a-c x^4} \left (c d^2-a e^2\right )^2}-\frac{3 a^{3/4} \sqrt [4]{c} e \sqrt{1-\frac{c x^4}{a}} \left (3 c d^2-a e^2\right ) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 d^2 \sqrt{a-c x^4} \left (c d^2-a e^2\right )^2}-\frac{3 e^2 x \sqrt{a-c x^4} \left (3 c d^2-a e^2\right )}{8 d^2 \left (d+e x^2\right ) \left (c d^2-a e^2\right )^2}-\frac{e^2 x \sqrt{a-c x^4}}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}-\frac{\sqrt [4]{a} \sqrt [4]{c} \sqrt{1-\frac{c x^4}{a}} \left (-2 \sqrt{a} \sqrt{c} d e-3 a e^2+7 c d^2\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 d^2 \sqrt{a-c x^4} \left (\sqrt{a} e+\sqrt{c} d\right ) \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x^2)^3*Sqrt[a - c*x^4]),x]

[Out]

-(e^2*x*Sqrt[a - c*x^4])/(4*d*(c*d^2 - a*e^2)*(d + e*x^2)^2) - (3*e^2*(3*c*d^2 - a*e^2)*x*Sqrt[a - c*x^4])/(8*
d^2*(c*d^2 - a*e^2)^2*(d + e*x^2)) - (3*a^(3/4)*c^(1/4)*e*(3*c*d^2 - a*e^2)*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcS
in[(c^(1/4)*x)/a^(1/4)], -1])/(8*d^2*(c*d^2 - a*e^2)^2*Sqrt[a - c*x^4]) - (a^(1/4)*c^(1/4)*(7*c*d^2 - 2*Sqrt[a
]*Sqrt[c]*d*e - 3*a*e^2)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(8*d^2*(Sqrt[c]*d + S
qrt[a]*e)*(c*d^2 - a*e^2)*Sqrt[a - c*x^4]) + (3*a^(1/4)*(5*c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*Sqrt[1 - (c*x^4)
/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(8*c^(1/4)*d^3*(c*d^2 - a*e^2)^2*
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 )^3 \sqrt{a-c x^4}} \, dx &=-\frac{e^2 x \sqrt{a-c x^4}}{4 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^2}+\frac{\int \frac{4 c d^2-3 a e^2-4 c d e x^2+c e^2 x^4}{\left (d+e x^2\right )^2 \sqrt{a-c x^4}} \, dx}{4 d \left (c d^2-a e^2\right )}\\ &=-\frac{e^2 x \sqrt{a-c x^4}}{4 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^2}-\frac{3 e^2 \left (3 c d^2-a e^2\right ) x \sqrt{a-c x^4}}{8 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )}+\frac{\int \frac{8 c^2 d^4-5 a c d^2 e^2+3 a^2 e^4-4 c d e \left (4 c d^2-a e^2\right ) x^2-3 c e^2 \left (3 c d^2-a e^2\right ) x^4}{\left (d+e x^2\right ) \sqrt{a-c x^4}} \, dx}{8 d^2 \left (c d^2-a e^2\right )^2}\\ &=-\frac{e^2 x \sqrt{a-c x^4}}{4 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^2}-\frac{3 e^2 \left (3 c d^2-a e^2\right ) x \sqrt{a-c x^4}}{8 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )}-\frac{\int \frac{-3 c d e^2 \left (3 c d^2-a e^2\right )+4 c d e^2 \left (4 c d^2-a e^2\right )+3 c e^3 \left (3 c d^2-a e^2\right ) x^2}{\sqrt{a-c x^4}} \, dx}{8 d^2 e^2 \left (c d^2-a e^2\right )^2}+\frac{\left (3 \left (5 c^2 d^4-2 a c d^2 e^2+a^2 e^4\right )\right ) \int \frac{1}{\left (d+e x^2\right ) \sqrt{a-c x^4}} \, dx}{8 d^2 \left (c d^2-a e^2\right )^2}\\ &=-\frac{e^2 x \sqrt{a-c x^4}}{4 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^2}-\frac{3 e^2 \left (3 c d^2-a e^2\right ) x \sqrt{a-c x^4}}{8 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )}-\frac{\left (\sqrt{c} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (7 c d^2-2 \sqrt{a} \sqrt{c} d e-3 a e^2\right )\right ) \int \frac{1}{\sqrt{a-c x^4}} \, dx}{8 d^2 \left (c d^2-a e^2\right )^2}-\frac{\left (3 \sqrt{a} \sqrt{c} e \left (3 c d^2-a e^2\right )\right ) \int \frac{1+\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a-c x^4}} \, dx}{8 d^2 \left (c d^2-a e^2\right )^2}+\frac{\left (3 \left (5 c^2 d^4-2 a c d^2 e^2+a^2 e^4\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}{8 d^2 \left (c d^2-a e^2\right )^2 \sqrt{a-c x^4}}\\ &=-\frac{e^2 x \sqrt{a-c x^4}}{4 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^2}-\frac{3 e^2 \left (3 c d^2-a e^2\right ) x \sqrt{a-c x^4}}{8 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )}+\frac{3 \sqrt [4]{a} \left (5 c^2 d^4-2 a c d^2 e^2+a^2 e^4\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 )}{8 \sqrt [4]{c} d^3 \left (c d^2-a e^2\right )^2 \sqrt{a-c x^4}}-\frac{\left (\sqrt{c} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (7 c d^2-2 \sqrt{a} \sqrt{c} d e-3 a e^2\right ) \sqrt{1-\frac{c x^4}{a}}\right ) \int \frac{1}{\sqrt{1-\frac{c x^4}{a}}} \, dx}{8 d^2 \left (c d^2-a e^2\right )^2 \sqrt{a-c x^4}}-\frac{\left (3 \sqrt{a} \sqrt{c} e \left (3 c d^2-a e^2\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}{8 d^2 \left (c d^2-a e^2\right )^2 \sqrt{a-c x^4}}\\ &=-\frac{e^2 x \sqrt{a-c x^4}}{4 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^2}-\frac{3 e^2 \left (3 c d^2-a e^2\right ) x \sqrt{a-c x^4}}{8 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )}-\frac{\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (7 c d^2-2 \sqrt{a} \sqrt{c} d e-3 a e^2\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 )}{8 d^2 \left (c d^2-a e^2\right )^2 \sqrt{a-c x^4}}+\frac{3 \sqrt [4]{a} \left (5 c^2 d^4-2 a c d^2 e^2+a^2 e^4\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 )}{8 \sqrt [4]{c} d^3 \left (c d^2-a e^2\right )^2 \sqrt{a-c x^4}}-\frac{\left (3 \sqrt{a} \sqrt{c} e \left (3 c d^2-a e^2\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}{8 d^2 \left (c d^2-a e^2\right )^2 \sqrt{a-c x^4}}\\ &=-\frac{e^2 x \sqrt{a-c x^4}}{4 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^2}-\frac{3 e^2 \left (3 c d^2-a e^2\right ) x \sqrt{a-c x^4}}{8 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )}-\frac{3 a^{3/4} \sqrt [4]{c} e \left (3 c d^2-a e^2\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 )}{8 d^2 \left (c d^2-a e^2\right )^2 \sqrt{a-c x^4}}-\frac{\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (7 c d^2-2 \sqrt{a} \sqrt{c} d e-3 a e^2\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 )}{8 d^2 \left (c d^2-a e^2\right )^2 \sqrt{a-c x^4}}+\frac{3 \sqrt [4]{a} \left (5 c^2 d^4-2 a c d^2 e^2+a^2 e^4\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 )}{8 \sqrt [4]{c} d^3 \left (c d^2-a e^2\right )^2 \sqrt{a-c x^4}}\\ \end{align*}

Mathematica [C]  time = 1.27414, size = 321, normalized size = 0.76 \[ \frac{\frac{d e^2 x \left (a-c x^4\right ) \left (a e^2 \left (5 d+3 e x^2\right )-c d^2 \left (11 d+9 e x^2\right )\right )}{\left (d+e x^2\right )^2}-\frac{i \sqrt{1-\frac{c x^4}{a}} \left (\left (-3 a^{3/2} \sqrt{c} d e^3+9 \sqrt{a} c^{3/2} d^3 e+a c d^2 e^2-7 c^2 d^4\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{-\frac{\sqrt{c}}{\sqrt{a}}}\right ),-1\right )+3 \left (a^2 e^4-2 a c d^2 e^2+5 c^2 d^4\right ) \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )+3 \sqrt{a} \sqrt{c} d e \left (a e^2-3 c d^2\right ) E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )\right )}{\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}}}}{8 d^3 \sqrt{a-c x^4} \left (c d^2-a e^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x^2)^3*Sqrt[a - c*x^4]),x]

[Out]

((d*e^2*x*(a - c*x^4)*(a*e^2*(5*d + 3*e*x^2) - c*d^2*(11*d + 9*e*x^2)))/(d + e*x^2)^2 - (I*Sqrt[1 - (c*x^4)/a]
*(3*Sqrt[a]*Sqrt[c]*d*e*(-3*c*d^2 + a*e^2)*EllipticE[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + (-7*c^2*d^4
+ 9*Sqrt[a]*c^(3/2)*d^3*e + a*c*d^2*e^2 - 3*a^(3/2)*Sqrt[c]*d*e^3)*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])
]*x], -1] + 3*(5*c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sq
rt[c]/Sqrt[a])]*x], -1]))/Sqrt[-(Sqrt[c]/Sqrt[a])])/(8*d^3*(c*d^2 - a*e^2)^2*Sqrt[a - c*x^4])

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Maple [B]  time = 0.286, size = 961, normalized size = 2.3 \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)^3/(-c*x^4+a)^(1/2),x)

[Out]

1/4*e^2/(a*e^2-c*d^2)/d*x*(-c*x^4+a)^(1/2)/(e*x^2+d)^2+3/8*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)+1/8*c/d/(a*e^2-c*d^2)^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)*a*e^2-7/8*c^2*d/(a*e^2-c*d^2)^
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)*E
llipticF(x*(1/a^(1/2)*c^(1/2))^(1/2),I)-3/8*c^(1/2)*e^3/(a*e^2-c*d^2)^2/d^2*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)+9/8*c^(3/2)*e/(a*e^2-c*d^2)^2*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)+3/8*c^(1/2)*e^3/(a*e^2
-c*d^2)^2/d^2*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)-9/8*c^(3/2)*e/(a*e^2-c*d^2)^2*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)+3/8/(a*e^2-c*d^2)^2/d^3*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^2-3/4/(a*e^2-c*d^2)^2/d*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+15/8/(a*e^2-c*d^2)^2
*d/(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^2

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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 )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)^3/(-c*x^4+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-c*x^4 + a)*(e*x^2 + d)^3), 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)^3/(-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 )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x**2+d)**3/(-c*x**4+a)**(1/2),x)

[Out]

Integral(1/(sqrt(a - c*x**4)*(d + e*x**2)**3), 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 )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)^3/(-c*x^4+a)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-c*x^4 + a)*(e*x^2 + d)^3), x)