\(\int \frac {1}{(d+e x^2)^3 \sqrt {a+c x^4}} \, dx\) [156]

Optimal result

Integrand size = 21, antiderivative size = 729 \[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a+c x^4}} \, dx=-\frac {3 \sqrt {c} e \left (3 c d^2+a e^2\right ) x \sqrt {a+c x^4}}{8 d^2 \left (c d^2+a e^2\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {e^2 x \sqrt {a+c x^4}}{4 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )^2}+\frac {3 e^2 \left (3 c d^2+a e^2\right ) x \sqrt {a+c x^4}}{8 d^2 \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac {3 \sqrt {e} \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right ) \arctan \left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{16 d^{5/2} \left (c d^2+a e^2\right )^{5/2}}+\frac {3 \sqrt [4]{a} \sqrt [4]{c} e \left (3 c d^2+a e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{8 d^2 \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {\sqrt [4]{c} \left (4 c d^2-\sqrt {a} \sqrt {c} d e+3 a e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{a} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \sqrt {a+c x^4}}-\frac {3 \left (\sqrt {c} d+\sqrt {a} e\right ) \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{32 \sqrt [4]{a} \sqrt [4]{c} d^3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}} \]

[Out]

3/16*(a^2*e^4+2*a*c*d^2*e^2+5*c^2*d^4)*arctan(x*(a*e^2+c*d^2)^(1/2)/d^(1/2)/e^(1/2)/(c*x^4+a)^(1/2))*e^(1/2)/d
^(5/2)/(a*e^2+c*d^2)^(5/2)+1/4*e^2*x*(c*x^4+a)^(1/2)/d/(a*e^2+c*d^2)/(e*x^2+d)^2+3/8*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)-3/8*e*(a*e^2+3*c*d^2)*x*c^(1/2)*(c*x^4+a)^(1/2)/d^2/(a*e^2+c*d^2)^2
/(a^(1/2)+x^2*c^(1/2))+3/8*a^(1/4)*c^(1/4)*e*(a*e^2+3*c*d^2)*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*
arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2*c^(1/2))*((c*x
^4+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/d^2/(a*e^2+c*d^2)^2/(c*x^4+a)^(1/2)-3/32*(a^2*e^4+2*a*c*d^2*e^2+5*c^2*d^4
)*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticPi(sin(2*arctan(c^(1/4)*
x/a^(1/4))),-1/4*(-e*a^(1/2)+d*c^(1/2))^2/d/e/a^(1/2)/c^(1/2),1/2*2^(1/2))*(e*a^(1/2)+d*c^(1/2))*(a^(1/2)+x^2*
c^(1/2))*((c*x^4+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/a^(1/4)/c^(1/4)/d^3/(a*e^2+c*d^2)^2/(-e*a^(1/2)+d*c^(1/2))/
(c*x^4+a)^(1/2)+1/8*c^(1/4)*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*Ellipt
icF(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2*c^(1/2))*(4*c*d^2+3*a*e^2-d*e*a^(1/2)*c^(1/2))*
((c*x^4+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/a^(1/4)/d^2/(a*e^2+c*d^2)/(-e*a^(1/2)+d*c^(1/2))/(c*x^4+a)^(1/2)

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 729, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1238, 1711, 1729, 1210, 1723, 226, 1721} \[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a+c x^4}} \, dx=\frac {3 \sqrt {e} \left (a^2 e^4+2 a c d^2 e^2+5 c^2 d^4\right ) \arctan \left (\frac {x \sqrt {a e^2+c d^2}}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{16 d^{5/2} \left (a e^2+c d^2\right )^{5/2}}-\frac {3 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (a^2 e^4+2 a c d^2 e^2+5 c^2 d^4\right ) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{32 \sqrt [4]{a} \sqrt [4]{c} d^3 \sqrt {a+c x^4} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (a e^2+c d^2\right )^2}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (-\sqrt {a} \sqrt {c} d e+3 a e^2+4 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{a} d^2 \sqrt {a+c x^4} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (a e^2+c d^2\right )}+\frac {3 \sqrt [4]{a} \sqrt [4]{c} e \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (a e^2+3 c d^2\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{8 d^2 \sqrt {a+c x^4} \left (a e^2+c d^2\right )^2}+\frac {3 e^2 x \sqrt {a+c x^4} \left (a e^2+3 c d^2\right )}{8 d^2 \left (d+e x^2\right ) \left (a e^2+c d^2\right )^2}+\frac {e^2 x \sqrt {a+c x^4}}{4 d \left (d+e x^2\right )^2 \left (a e^2+c d^2\right )}-\frac {3 \sqrt {c} e x \sqrt {a+c x^4} \left (a e^2+3 c d^2\right )}{8 d^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \left (a e^2+c d^2\right )^2} \]

[In]

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

[Out]

(-3*Sqrt[c]*e*(3*c*d^2 + a*e^2)*x*Sqrt[a + c*x^4])/(8*d^2*(c*d^2 + a*e^2)^2*(Sqrt[a] + Sqrt[c]*x^2)) + (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*Sqrt[e]*(5*c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*ArcTan[(Sqrt[c*d^2 + a*e^2]*x)/(
Sqrt[d]*Sqrt[e]*Sqrt[a + c*x^4])])/(16*d^(5/2)*(c*d^2 + a*e^2)^(5/2)) + (3*a^(1/4)*c^(1/4)*e*(3*c*d^2 + a*e^2)
*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)],
1/2])/(8*d^2*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4]) + (c^(1/4)*(4*c*d^2 - Sqrt[a]*Sqrt[c]*d*e + 3*a*e^2)*(Sqrt[a]
+ Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(8*a
^(1/4)*d^2*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)*Sqrt[a + c*x^4]) - (3*(Sqrt[c]*d + Sqrt[a]*e)*(5*c^2*d^4 +
2*a*c*d^2*e^2 + a^2*e^4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-1/4*(
Sqrt[c]*d - Sqrt[a]*e)^2/(Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(32*a^(1/4)*c^(1/4)*d^3*(
Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4])

Rule 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 1210

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

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 1721

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]
}, Simp[(-(B*d - A*e))*(ArcTan[Rt[c*(d/e) + a*(e/d), 2]*(x/Sqrt[a + c*x^4])]/(2*d*e*Rt[c*(d/e) + a*(e/d), 2]))
, x] + Simp[(B*d + A*e)*(A + B*x^2)*(Sqrt[A^2*((a + c*x^4)/(a*(A + B*x^2)^2))]/(4*d*e*A*q*Sqrt[a + c*x^4]))*El
lipticPi[Cancel[-(B*d - A*e)^2/(4*d*e*A*B)], 2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c
*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0]

Rule 1723

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

Rule 1729

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

Rubi steps \begin{align*} \text {integral}& = \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 \sqrt {a} c^{3/2} d e^2 \left (3 c d^2+a e^2\right )+c e \left (8 c^2 d^4+5 a c d^2 e^2+3 a^2 e^4\right )+\left (3 c e^2 \left (c d-\sqrt {a} \sqrt {c} e\right ) \left (3 c d^2+a e^2\right )-4 c^2 d e^2 \left (4 c d^2+a e^2\right )\right ) x^2}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx}{8 c d^2 e \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 {3 \sqrt {c} e \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 (\sqrt {a}+\sqrt {c} x^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 {3 \sqrt [4]{a} \sqrt [4]{c} e \left (3 c d^2+a e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{8 d^2 \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {\left (\sqrt {c} \left (4 c d^2-\sqrt {a} \sqrt {c} d e+3 a e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{4 d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )}-\frac {\left (3 \sqrt {a} e \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right )\right ) \int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx}{8 d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2} \\ & = -\frac {3 \sqrt {c} e \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 (\sqrt {a}+\sqrt {c} x^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 {3 \sqrt {e} \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{16 d^{5/2} \left (c d^2+a e^2\right )^{5/2}}+\frac {3 \sqrt [4]{a} \sqrt [4]{c} e \left (3 c d^2+a e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{8 d^2 \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {\sqrt [4]{c} \left (4 c d^2-\sqrt {a} \sqrt {c} d e+3 a e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{8 \sqrt [4]{a} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \sqrt {a+c x^4}}-\frac {3 \sqrt [4]{a} \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{32 \sqrt [4]{c} d^3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.85 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.46 \[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a+c x^4}} \, dx=\frac {\frac {d e^2 x \left (a+c x^4\right ) \left (a e^2 \left (5 d+3 e x^2\right )+c d^2 \left (11 d+9 e x^2\right )\right )}{\left (d+e x^2\right )^2}+\frac {\sqrt {1+\frac {c x^4}{a}} \left (-3 \sqrt {a} \sqrt {c} d e \left (3 c d^2+a e^2\right ) E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+i \left (\sqrt {c} d \left (7 c^{3/2} d^3-9 i \sqrt {a} c d^2 e+a \sqrt {c} d e^2-3 i a^{3/2} e^3\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-3 \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right ) \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}}{8 d^3 \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}} \]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.73 (sec) , antiderivative size = 1018, normalized size of antiderivative = 1.40

method	result	size
default	\(\text {Expression too large to display}\)	\(1018\)
elliptic	\(\text {Expression too large to display}\)	\(1018\)

[In]

int(1/(e*x^2+d)^3/(c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/4*e^2*x*(c*x^4+a)^(1/2)/d/(a*e^2+c*d^2)/(e*x^2+d)^2+3/8*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)-1/8*c/d/(a*e^2+c*d^2)^2/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)
*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)*a*e^2-7/8*c^2*d/(a*e^2+c*d^2)^2/(
I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*Ellip
ticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)-9/8*I*c^(3/2)*e/(a*e^2+c*d^2)^2*a^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(
1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),
I)+9/8*I*c^(3/2)*e/(a*e^2+c*d^2)^2*a^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1
/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticE(x*(I/a^(1/2)*c^(1/2))^(1/2),I)+3/8*I*c^(1/2)*e^3/d^2/(a*e^2+c
*d^2)^2*a^(3/2)/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x
^4+a)^(1/2)*EllipticE(x*(I/a^(1/2)*c^(1/2))^(1/2),I)-3/8*I*c^(1/2)*e^3/d^2/(a*e^2+c*d^2)^2*a^(3/2)/(I/a^(1/2)*
c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I/
a^(1/2)*c^(1/2))^(1/2),I)+3/8/d^3/(a*e^2+c*d^2)^2*e^4/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2
)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticPi(x*(I/a^(1/2)*c^(1/2))^(1/2),I*a^(1/2)/c^(1/2)*e/d
,(-I/a^(1/2)*c^(1/2))^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2))*a^2+3/4/(a*e^2+c*d^2)^2*e^2/d/(I/a^(1/2)*c^(1/2))^(1/2)
*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticPi(x*(I/a^(1/2)*c^(1/
2))^(1/2),I*a^(1/2)/c^(1/2)*e/d,(-I/a^(1/2)*c^(1/2))^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2))*a*c+15/8*d/(a*e^2+c*d^2)
^2/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*E
llipticPi(x*(I/a^(1/2)*c^(1/2))^(1/2),I*a^(1/2)/c^(1/2)*e/d,(-I/a^(1/2)*c^(1/2))^(1/2)/(I/a^(1/2)*c^(1/2))^(1/
2))*c^2

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a+c x^4}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a+c x^4}} \, dx=\int \frac {1}{\sqrt {a + c x^{4}} \left (d + e x^{2}\right )^{3}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a+c x^4}} \, dx=\int { \frac {1}{\sqrt {c x^{4} + a} {\left (e x^{2} + d\right )}^{3}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a+c x^4}} \, dx=\int { \frac {1}{\sqrt {c x^{4} + a} {\left (e x^{2} + d\right )}^{3}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a+c x^4}} \, dx=\int \frac {1}{\sqrt {c\,x^4+a}\,{\left (e\,x^2+d\right )}^3} \,d x \]

[In]

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

[Out]

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