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

Optimal. Leaf size=659 \[ \frac{3 c^{3/2} d^3 e^2 x \sqrt{a+c x^4}}{\left (\sqrt{a}+\sqrt{c} x^2\right ) \left (a e^4+c d^4\right )^2}+\frac{c^{3/4} d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt{a+c x^4} \left (a e^4+c d^4\right )}-\frac{3 \sqrt [4]{a} c^{5/4} d^3 e^2 \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{\sqrt{a+c x^4} \left (a e^4+c d^4\right )^2}-\frac{3 c^{3/4} d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{c} d^2-\sqrt{a} e^2\right )^2 \Pi \left (\frac{\left (\sqrt{c} d^2+\sqrt{a} e^2\right )^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} \sqrt{a+c x^4} \left (a e^4+c d^4\right )^2}-\frac{3 c d^3 e^3 \sqrt{a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )^2}-\frac{e^3 \sqrt{a+c x^4}}{2 (d+e x)^2 \left (a e^4+c d^4\right )}+\frac{3 c d^2 e \left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac{x \sqrt{-a e^4-c d^4}}{d e \sqrt{a+c x^4}}\right )}{2 \left (-a e^4-c d^4\right )^{5/2}}-\frac{3 c d^2 e \left (c d^4-a e^4\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right )}{2 \left (a e^4+c d^4\right )^{5/2}} \]

[Out]

-(e^3*Sqrt[a + c*x^4])/(2*(c*d^4 + a*e^4)*(d + e*x)^2) - (3*c*d^3*e^3*Sqrt[a + c*x^4])/((c*d^4 + a*e^4)^2*(d +
e*x)) + (3*c^(3/2)*d^3*e^2*x*Sqrt[a + c*x^4])/((c*d^4 + a*e^4)^2*(Sqrt[a] + Sqrt[c]*x^2)) + (3*c*d^2*e*(c*d^4
- a*e^4)*ArcTan[(Sqrt[-(c*d^4) - a*e^4]*x)/(d*e*Sqrt[a + c*x^4])])/(2*(-(c*d^4) - a*e^4)^(5/2)) - (3*c*d^2*e*
(c*d^4 - a*e^4)*ArcTanh[(a*e^2 + c*d^2*x^2)/(Sqrt[c*d^4 + a*e^4]*Sqrt[a + c*x^4])])/(2*(c*d^4 + a*e^4)^(5/2))
- (3*a^(1/4)*c^(5/4)*d^3*e^2*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*A
rcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/((c*d^4 + a*e^4)^2*Sqrt[a + c*x^4]) + (c^(3/4)*d*(Sqrt[a] + Sqrt[c]*x^2)*Sqr
t[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*(c*d^4 + a*
e^4)*Sqrt[a + c*x^4]) - (3*c^(3/4)*d*(Sqrt[c]*d^2 - Sqrt[a]*e^2)^2*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(S
qrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[(Sqrt[c]*d^2 + Sqrt[a]*e^2)^2/(4*Sqrt[a]*Sqrt[c]*d^2*e^2), 2*ArcTan[(c^(1/
4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^4])

________________________________________________________________________________________

Rubi [A]  time = 1.15702, antiderivative size = 659, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 12, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.632, Rules used = {1727, 1739, 1742, 12, 1248, 725, 206, 1715, 1196, 1709, 220, 1707} \[ \frac{3 c^{3/2} d^3 e^2 x \sqrt{a+c x^4}}{\left (\sqrt{a}+\sqrt{c} x^2\right ) \left (a e^4+c d^4\right )^2}+\frac{c^{3/4} d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt{a+c x^4} \left (a e^4+c d^4\right )}-\frac{3 \sqrt [4]{a} c^{5/4} d^3 e^2 \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{\sqrt{a+c x^4} \left (a e^4+c d^4\right )^2}-\frac{3 c^{3/4} d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{c} d^2-\sqrt{a} e^2\right )^2 \Pi \left (\frac{\left (\sqrt{c} d^2+\sqrt{a} e^2\right )^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} \sqrt{a+c x^4} \left (a e^4+c d^4\right )^2}-\frac{3 c d^3 e^3 \sqrt{a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )^2}-\frac{e^3 \sqrt{a+c x^4}}{2 (d+e x)^2 \left (a e^4+c d^4\right )}+\frac{3 c d^2 e \left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac{x \sqrt{-a e^4-c d^4}}{d e \sqrt{a+c x^4}}\right )}{2 \left (-a e^4-c d^4\right )^{5/2}}-\frac{3 c d^2 e \left (c d^4-a e^4\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right )}{2 \left (a e^4+c d^4\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e^3*Sqrt[a + c*x^4])/(2*(c*d^4 + a*e^4)*(d + e*x)^2) - (3*c*d^3*e^3*Sqrt[a + c*x^4])/((c*d^4 + a*e^4)^2*(d +
e*x)) + (3*c^(3/2)*d^3*e^2*x*Sqrt[a + c*x^4])/((c*d^4 + a*e^4)^2*(Sqrt[a] + Sqrt[c]*x^2)) + (3*c*d^2*e*(c*d^4
- a*e^4)*ArcTan[(Sqrt[-(c*d^4) - a*e^4]*x)/(d*e*Sqrt[a + c*x^4])])/(2*(-(c*d^4) - a*e^4)^(5/2)) - (3*c*d^2*e*
(c*d^4 - a*e^4)*ArcTanh[(a*e^2 + c*d^2*x^2)/(Sqrt[c*d^4 + a*e^4]*Sqrt[a + c*x^4])])/(2*(c*d^4 + a*e^4)^(5/2))
- (3*a^(1/4)*c^(5/4)*d^3*e^2*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*A
rcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/((c*d^4 + a*e^4)^2*Sqrt[a + c*x^4]) + (c^(3/4)*d*(Sqrt[a] + Sqrt[c]*x^2)*Sqr
t[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*(c*d^4 + a*
e^4)*Sqrt[a + c*x^4]) - (3*c^(3/4)*d*(Sqrt[c]*d^2 - Sqrt[a]*e^2)^2*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(S
qrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[(Sqrt[c]*d^2 + Sqrt[a]*e^2)^2/(4*Sqrt[a]*Sqrt[c]*d^2*e^2), 2*ArcTan[(c^(1/
4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^4])

Rule 1727

Int[((d_) + (e_.)*(x_))^(q_)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[(e^3*(d + e*x)^(q + 1)*Sqrt[a + c*x^
4])/((q + 1)*(c*d^4 + a*e^4)), x] + Dist[c/((q + 1)*(c*d^4 + a*e^4)), Int[((d + e*x)^(q + 1)*Simp[d^3*(q + 1)
- d^2*e*(q + 1)*x + d*e^2*(q + 1)*x^2 - e^3*(q + 3)*x^3, x])/Sqrt[a + c*x^4], x], x] /; FreeQ[{a, c, d, e}, x]
&& NeQ[c*d^4 + a*e^4, 0] && ILtQ[q, -1]

Rule 1739

Int[((Px_)*((d_) + (e_.)*(x_))^(q_))/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Co
eff[Px, x, 1], C = Coeff[Px, x, 2], D = Coeff[Px, x, 3]}, -Simp[((d^3*D - C*d^2*e + B*d*e^2 - A*e^3)*(d + e*x)
^(q + 1)*Sqrt[a + c*x^4])/((q + 1)*(c*d^4 + a*e^4)), x] + Dist[1/((q + 1)*(c*d^4 + a*e^4)), Int[((d + e*x)^(q
+ 1)/Sqrt[a + c*x^4])*Simp[(q + 1)*(a*e*(d^2*D - C*d*e + B*e^2) + A*d*(c*d^2)) - (e*(q + 1)*(A*c*d^2 + a*e*(d*
D - C*e)) - B*d*(c*d^2*(q + 1)))*x + (q + 1)*(D*e*(a*e^2) + c*d*(C*d^2 - e*(B*d - A*e)))*x^2 + c*(q + 3)*(d^3*
D - C*d^2*e + B*d*e^2 - A*e^3)*x^3, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x] && LeQ[Expon[Px, x],
3] && NeQ[c*d^4 + a*e^4, 0] && LtQ[q, -1]

Rule 1742

Int[(Px_)/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[P
x, x, 1], C = Coeff[Px, x, 2], D = Coeff[Px, x, 3]}, Int[(x*(B*d - A*e + (d*D - C*e)*x^2))/((d^2 - e^2*x^2)*Sq
rt[a + c*x^4]), x] + Int[(A*d + (C*d - B*e)*x^2 - D*e*x^4)/((d^2 - e^2*x^2)*Sqrt[a + c*x^4]), x]] /; FreeQ[{a,
c, d, e}, x] && PolyQ[Px, x] && LeQ[Expon[Px, x], 3] && NeQ[c*d^4 + a*e^4, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1248

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 1715

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]

Rule 1196

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)]*EllipticE[2*ArcTan[
q*x], 1/2])/(q*Sqrt[a + c*x^4]), x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rule 1709

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 220

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)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 1707

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)]*EllipticPi[Cancel[-((B*d - A*e)^2
/(4*d*e*A*B))], 2*ArcTan[q*x], 1/2])/(4*d*e*A*q*Sqrt[a + c*x^4]), 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]

Rubi steps

\begin{align*} \int \frac{1}{(d+e x)^3 \sqrt{a+c x^4}} \, dx &=-\frac{e^3 \sqrt{a+c x^4}}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac{c \int \frac{-2 d^3+2 d^2 e x-2 d e^2 x^2}{(d+e x)^2 \sqrt{a+c x^4}} \, dx}{2 \left (c d^4+a e^4\right )}\\ &=-\frac{e^3 \sqrt{a+c x^4}}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac{3 c d^3 e^3 \sqrt{a+c x^4}}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac{c \int \frac{2 d^2 \left (c d^4-2 a e^4\right )-2 d e \left (2 c d^4-a e^4\right ) x+6 c d^4 e^2 x^2+6 c d^3 e^3 x^3}{(d+e x) \sqrt{a+c x^4}} \, dx}{2 \left (c d^4+a e^4\right )^2}\\ &=-\frac{e^3 \sqrt{a+c x^4}}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac{3 c d^3 e^3 \sqrt{a+c x^4}}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac{c \int \frac{\left (-2 d^2 e \left (c d^4-2 a e^4\right )-2 d^2 e \left (2 c d^4-a e^4\right )\right ) x}{\left (d^2-e^2 x^2\right ) \sqrt{a+c x^4}} \, dx}{2 \left (c d^4+a e^4\right )^2}+\frac{c \int \frac{2 d^3 \left (c d^4-2 a e^4\right )+\left (6 c d^5 e^2+2 d e^2 \left (2 c d^4-a e^4\right )\right ) x^2-6 c d^3 e^4 x^4}{\left (d^2-e^2 x^2\right ) \sqrt{a+c x^4}} \, dx}{2 \left (c d^4+a e^4\right )^2}\\ &=-\frac{e^3 \sqrt{a+c x^4}}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac{3 c d^3 e^3 \sqrt{a+c x^4}}{\left (c d^4+a e^4\right )^2 (d+e x)}-\frac{\int \frac{-6 \sqrt{a} c^{3/2} d^5 e^4-2 c d^3 e^2 \left (c d^4-2 a e^4\right )+\left (6 c d^3 e^4 \left (c d^2+\sqrt{a} \sqrt{c} e^2\right )-c e^2 \left (6 c d^5 e^2+2 d e^2 \left (2 c d^4-a e^4\right )\right )\right ) x^2}{\left (d^2-e^2 x^2\right ) \sqrt{a+c x^4}} \, dx}{2 e^2 \left (c d^4+a e^4\right )^2}-\frac{\left (3 \sqrt{a} c^{3/2} d^3 e^2\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+c x^4}} \, dx}{\left (c d^4+a e^4\right )^2}-\frac{\left (3 c d^2 e \left (c d^4-a e^4\right )\right ) \int \frac{x}{\left (d^2-e^2 x^2\right ) \sqrt{a+c x^4}} \, dx}{\left (c d^4+a e^4\right )^2}\\ &=-\frac{e^3 \sqrt{a+c x^4}}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac{3 c d^3 e^3 \sqrt{a+c x^4}}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac{3 c^{3/2} d^3 e^2 x \sqrt{a+c x^4}}{\left (c d^4+a e^4\right )^2 \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{3 \sqrt [4]{a} c^{5/4} d^3 e^2 \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 )}{\left (c d^4+a e^4\right )^2 \sqrt{a+c x^4}}+\frac{\left (3 \sqrt{a} c d^3 e^2 \left (\sqrt{c} d^2-\sqrt{a} e^2\right )\right ) \int \frac{1+\frac{\sqrt{c} x^2}{\sqrt{a}}}{\left (d^2-e^2 x^2\right ) \sqrt{a+c x^4}} \, dx}{\left (c d^4+a e^4\right )^2}-\frac{\left (3 c d^2 e \left (c d^4-a e^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d^2-e^2 x\right ) \sqrt{a+c x^2}} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )^2}+\frac{(c d) \int \frac{1}{\sqrt{a+c x^4}} \, dx}{c d^4+a e^4}\\ &=-\frac{e^3 \sqrt{a+c x^4}}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac{3 c d^3 e^3 \sqrt{a+c x^4}}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac{3 c^{3/2} d^3 e^2 x \sqrt{a+c x^4}}{\left (c d^4+a e^4\right )^2 \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{3 c d^2 e \left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{-c d^4-a e^4} x}{d e \sqrt{a+c x^4}}\right )}{2 \left (-c d^4-a e^4\right )^{5/2}}-\frac{3 \sqrt [4]{a} c^{5/4} d^3 e^2 \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 )}{\left (c d^4+a e^4\right )^2 \sqrt{a+c x^4}}+\frac{c^{3/4} d \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 )}{2 \sqrt [4]{a} \left (c d^4+a e^4\right ) \sqrt{a+c x^4}}-\frac{3 c^{3/4} d \left (\sqrt{c} d^2-\sqrt{a} e^2\right )^2 \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^2+\sqrt{a} e^2\right )^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} \left (c d^4+a e^4\right )^2 \sqrt{a+c x^4}}+\frac{\left (3 c d^2 e \left (c d^4-a e^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c d^4+a e^4-x^2} \, dx,x,\frac{-a e^2-c d^2 x^2}{\sqrt{a+c x^4}}\right )}{2 \left (c d^4+a e^4\right )^2}\\ &=-\frac{e^3 \sqrt{a+c x^4}}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac{3 c d^3 e^3 \sqrt{a+c x^4}}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac{3 c^{3/2} d^3 e^2 x \sqrt{a+c x^4}}{\left (c d^4+a e^4\right )^2 \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{3 c d^2 e \left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{-c d^4-a e^4} x}{d e \sqrt{a+c x^4}}\right )}{2 \left (-c d^4-a e^4\right )^{5/2}}-\frac{3 c d^2 e \left (c d^4-a e^4\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2 x^2}{\sqrt{c d^4+a e^4} \sqrt{a+c x^4}}\right )}{2 \left (c d^4+a e^4\right )^{5/2}}-\frac{3 \sqrt [4]{a} c^{5/4} d^3 e^2 \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 )}{\left (c d^4+a e^4\right )^2 \sqrt{a+c x^4}}+\frac{c^{3/4} d \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 )}{2 \sqrt [4]{a} \left (c d^4+a e^4\right ) \sqrt{a+c x^4}}-\frac{3 c^{3/4} d \left (\sqrt{c} d^2-\sqrt{a} e^2\right )^2 \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^2+\sqrt{a} e^2\right )^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} \left (c d^4+a e^4\right )^2 \sqrt{a+c x^4}}\\ \end{align*}

Mathematica [C]  time = 2.33085, size = 513, normalized size = 0.78 \[ \frac{-\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \left (e^3 \sqrt{a e^4+c d^4} \left (a^2 e^4+a c \left (6 d^3 e x+7 d^4+e^4 x^4\right )+c^2 d^3 x^4 (7 d+6 e x)\right )+6 \sqrt [4]{-1} \sqrt [4]{a} c^{3/4} d \sqrt{\frac{c x^4}{a}+1} (d+e x)^2 \left (c d^4-a e^4\right ) \sqrt{a e^4+c d^4} \Pi \left (\frac{i \sqrt{a} e^2}{\sqrt{c} d^2};\left .\sin ^{-1}\left (\frac{(-1)^{3/4} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )+3 c d^2 e \sqrt{a+c x^4} (d+e x)^2 \left (c d^4-a e^4\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right )\right )+6 \sqrt{a} c^{3/2} d^3 e^2 \sqrt{\frac{c x^4}{a}+1} (d+e x)^2 \sqrt{a e^4+c d^4} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )+2 i c d \sqrt{\frac{c x^4}{a}+1} (d+e x)^2 \left (3 i \sqrt{a} \sqrt{c} d^2 e^2-a e^4+2 c d^4\right ) \sqrt{a e^4+c d^4} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{2 \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{a+c x^4} (d+e x)^2 \left (a e^4+c d^4\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*Sqrt[a]*c^(3/2)*d^3*e^2*Sqrt[c*d^4 + a*e^4]*(d + e*x)^2*Sqrt[1 + (c*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqr
t[c])/Sqrt[a]]*x], -1] + (2*I)*c*d*(2*c*d^4 + (3*I)*Sqrt[a]*Sqrt[c]*d^2*e^2 - a*e^4)*Sqrt[c*d^4 + a*e^4]*(d +
e*x)^2*Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] - Sqrt[(I*Sqrt[c])/Sqrt[a]]*(
e^3*Sqrt[c*d^4 + a*e^4]*(a^2*e^4 + c^2*d^3*x^4*(7*d + 6*e*x) + a*c*(7*d^4 + 6*d^3*e*x + e^4*x^4)) + 3*c*d^2*e*
(c*d^4 - a*e^4)*(d + e*x)^2*Sqrt[a + c*x^4]*ArcTanh[(a*e^2 + c*d^2*x^2)/(Sqrt[c*d^4 + a*e^4]*Sqrt[a + c*x^4])]
+ 6*(-1)^(1/4)*a^(1/4)*c^(3/4)*d*(c*d^4 - a*e^4)*Sqrt[c*d^4 + a*e^4]*(d + e*x)^2*Sqrt[1 + (c*x^4)/a]*Elliptic
Pi[(I*Sqrt[a]*e^2)/(Sqrt[c]*d^2), ArcSin[((-1)^(3/4)*c^(1/4)*x)/a^(1/4)], -1]))/(2*Sqrt[(I*Sqrt[c])/Sqrt[a]]*(
c*d^4 + a*e^4)^(5/2)*(d + e*x)^2*Sqrt[a + c*x^4])

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Maple [C]  time = 0.016, size = 483, normalized size = 0.7 \begin{align*} -{\frac{{e}^{3}}{ \left ( 2\,a{e}^{4}+2\,c{d}^{4} \right ) \left ( ex+d \right ) ^{2}}\sqrt{c{x}^{4}+a}}-3\,{\frac{c{d}^{3}{e}^{3}\sqrt{c{x}^{4}+a}}{ \left ( a{e}^{4}+c{d}^{4} \right ) ^{2} \left ( ex+d \right ) }}+{\frac{cd \left ( a{e}^{4}-2\,c{d}^{4} \right ) }{ \left ( a{e}^{4}+c{d}^{4} \right ) ^{2}}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}}+{\frac{3\,i{e}^{2}{d}^{3}}{ \left ( a{e}^{4}+c{d}^{4} \right ) ^{2}}{c}^{{\frac{3}{2}}}\sqrt{a}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}}-3\,{\frac{c{d}^{2} \left ( a{e}^{4}-c{d}^{4} \right ) }{ \left ( a{e}^{4}+c{d}^{4} \right ) ^{2}e} \left ( -1/2\,{{\it Artanh} \left ( 1/2\,{\frac{1}{\sqrt{c{x}^{4}+a}} \left ( 2\,{\frac{c{d}^{2}{x}^{2}}{{e}^{2}}}+2\,a \right ){\frac{1}{\sqrt{{\frac{c{d}^{4}}{{e}^{4}}}+a}}}} \right ){\frac{1}{\sqrt{{\frac{c{d}^{4}}{{e}^{4}}}+a}}}}+{\frac{e}{d\sqrt{c{x}^{4}+a}}\sqrt{1-{\frac{i\sqrt{c}{x}^{2}}{\sqrt{a}}}}\sqrt{1+{\frac{i\sqrt{c}{x}^{2}}{\sqrt{a}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{i\sqrt{c}}{\sqrt{a}}}},{\frac{-i\sqrt{a}{e}^{2}}{{d}^{2}\sqrt{c}}},{\sqrt{{\frac{-i\sqrt{c}}{\sqrt{a}}}}{\frac{1}{\sqrt{{\frac{i\sqrt{c}}{\sqrt{a}}}}}}} \right ){\frac{1}{\sqrt{{\frac{i\sqrt{c}}{\sqrt{a}}}}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*e^3*(c*x^4+a)^(1/2)/(a*e^4+c*d^4)/(e*x+d)^2-3*c*d^3*e^3*(c*x^4+a)^(1/2)/(a*e^4+c*d^4)^2/(e*x+d)+c*d*(a*e^
4-2*c*d^4)/(a*e^4+c*d^4)^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*I*c^(3/2)*e^2*d^3/(a*e^4+c*d^4)^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)*(Ellip
ticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)-EllipticE(x*(I/a^(1/2)*c^(1/2))^(1/2),I))-3*c*d^2*(a*e^4-c*d^4)/(a*e^4+c*d
^4)^2/e*(-1/2/(c*d^4/e^4+a)^(1/2)*arctanh(1/2*(2*c*x^2*d^2/e^2+2*a)/(c*d^4/e^4+a)^(1/2)/(c*x^4+a)^(1/2))+1/(I/
a^(1/2)*c^(1/2))^(1/2)/d*e*(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)*Ell
ipticPi(x*(I/a^(1/2)*c^(1/2))^(1/2),-I*a^(1/2)/c^(1/2)/d^2*e^2,(-I/a^(1/2)*c^(1/2))^(1/2)/(I/a^(1/2)*c^(1/2))^
(1/2)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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