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

Optimal. Leaf size=299 \[ -\frac{\sqrt [4]{a} \sqrt [4]{c} \sqrt{1-\frac{c x^4}{a}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 d \sqrt{a-c x^4} \left (\sqrt{a} e+\sqrt{c} d\right )}-\frac{a^{3/4} \sqrt [4]{c} e \sqrt{1-\frac{c x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 d \sqrt{a-c x^4} \left (c d^2-a e^2\right )}-\frac{e^2 x \sqrt{a-c x^4}}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}+\frac{\sqrt [4]{a} \sqrt{1-\frac{c x^4}{a}} \left (3 c d^2-a e^2\right ) \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{c} d^2 \sqrt{a-c x^4} \left (c d^2-a e^2\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.356134, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {1224, 1717, 1201, 224, 221, 1200, 1199, 424, 1219, 1218} \[ -\frac{a^{3/4} \sqrt [4]{c} e \sqrt{1-\frac{c x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 d \sqrt{a-c x^4} \left (c d^2-a e^2\right )}-\frac{e^2 x \sqrt{a-c x^4}}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}+\frac{\sqrt [4]{a} \sqrt{1-\frac{c x^4}{a}} \left (3 c d^2-a e^2\right ) \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{c} d^2 \sqrt{a-c x^4} \left (c d^2-a e^2\right )}-\frac{\sqrt [4]{a} \sqrt [4]{c} \sqrt{1-\frac{c x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 d \sqrt{a-c x^4} \left (\sqrt{a} e+\sqrt{c} d\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [C]  time = 0.974078, size = 508, normalized size = 1.7 \[ \frac{-i \sqrt{c} d \sqrt{1-\frac{c x^4}{a}} \left (d+e x^2\right ) \left (\sqrt{a} e-\sqrt{c} d\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{-\frac{\sqrt{c}}{\sqrt{a}}}\right ),-1\right )-3 i c d^2 e x^2 \sqrt{1-\frac{c x^4}{a}} \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )-3 i c d^3 \sqrt{1-\frac{c x^4}{a}} \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )+c d e^2 x^5 \sqrt{-\frac{\sqrt{c}}{\sqrt{a}}}+i a e^3 x^2 \sqrt{1-\frac{c x^4}{a}} \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )+i a d e^2 \sqrt{1-\frac{c x^4}{a}} \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )-a d e^2 x \sqrt{-\frac{\sqrt{c}}{\sqrt{a}}}+i \sqrt{a} \sqrt{c} d e \sqrt{1-\frac{c x^4}{a}} \left (d+e x^2\right ) E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{2 d^2 \sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} \sqrt{a-c x^4} \left (d+e x^2\right ) \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(a*Sqrt[-(Sqrt[c]/Sqrt[a])]*d*e^2*x) + Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d*e^2*x^5 + I*Sqrt[a]*Sqrt[c]*d*e*(d + e*x
^2)*Sqrt[1 - (c*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - I*Sqrt[c]*d*(-(Sqrt[c]*d) + Sqr
t[a]*e)*(d + e*x^2)*Sqrt[1 - (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - (3*I)*c*d^3*Sqr
t[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + I*a*d*e^2
*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - (3*I)
*c*d^2*e*x^2*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x],
-1] + I*a*e^3*x^2*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a]
)]*x], -1])/(2*Sqrt[-(Sqrt[c]/Sqrt[a])]*d^2*(c*d^2 - a*e^2)*(d + e*x^2)*Sqrt[a - c*x^4])

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Maple [B]  time = 0.368, size = 523, normalized size = 1.8 \begin{align*}{\frac{{e}^{2}x}{ \left ( 2\,a{e}^{2}-2\,c{d}^{2} \right ) d \left ( e{x}^{2}+d \right ) }\sqrt{-c{x}^{4}+a}}+{\frac{c}{2\,a{e}^{2}-2\,c{d}^{2}}\sqrt{1-{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-c{x}^{4}+a}}}}-{\frac{e}{ \left ( 2\,a{e}^{2}-2\,c{d}^{2} \right ) d}\sqrt{a}\sqrt{c}\sqrt{1-{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-c{x}^{4}+a}}}}+{\frac{e}{ \left ( 2\,a{e}^{2}-2\,c{d}^{2} \right ) d}\sqrt{a}\sqrt{c}\sqrt{1-{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticE} \left ( x\sqrt{{\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-c{x}^{4}+a}}}}+{\frac{a{e}^{2}}{ \left ( 2\,a{e}^{2}-2\,c{d}^{2} \right ){d}^{2}}\sqrt{1-{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticPi} \left ( x\sqrt{{\sqrt{c}{\frac{1}{\sqrt{a}}}}},-{\frac{e}{d}\sqrt{a}{\frac{1}{\sqrt{c}}}},{\sqrt{-{\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\frac{1}{\sqrt{{\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}} \right ){\frac{1}{\sqrt{{\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-c{x}^{4}+a}}}}-{\frac{3\,c}{2\,a{e}^{2}-2\,c{d}^{2}}\sqrt{1-{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticPi} \left ( x\sqrt{{\sqrt{c}{\frac{1}{\sqrt{a}}}}},-{\frac{e}{d}\sqrt{a}{\frac{1}{\sqrt{c}}}},{\sqrt{-{\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\frac{1}{\sqrt{{\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}} \right ){\frac{1}{\sqrt{{\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-c{x}^{4}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )}^{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+a)^(1/2),x, algorithm="maxima")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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