3.3.18 \(\int \frac {1}{(d+e x)^3 \sqrt {a+c x^4}} \, dx\) [218]

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

[Out]

3/2*c*d^2*e*(-a*e^4+c*d^4)*arctan(x*(-a*e^4-c*d^4)^(1/2)/d/e/(c*x^4+a)^(1/2))/(-a*e^4-c*d^4)^(5/2)-3/2*c*d^2*e
*(-a*e^4+c*d^4)*arctanh((c*d^2*x^2+a*e^2)/(a*e^4+c*d^4)^(1/2)/(c*x^4+a)^(1/2))/(a*e^4+c*d^4)^(5/2)-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)+3*c^(3/2)*d^3*e^2*x*(
c*x^4+a)^(1/2)/(a*e^4+c*d^4)^2/(a^(1/2)+x^2*c^(1/2))-3*a^(1/4)*c^(5/4)*d^3*e^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)/(a*e^4+c*d^4)^2/(c*x^4+a)^(1/2)+1/2*c^(3/4)*d*(cos(2*ar
ctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(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)/a^(1/4)/(a*e^4+c*d^4)/(c*x^4+a)^(
1/2)-3/4*c^(3/4)*d*(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^2*a^(1/2)+d^2*c^(1/2))^2/d^2/e^2/a^(1/2)/c^(1/2),1/2*2^(1/2))*(-e^2*a^(1/2
)+d^2*c^(1/2))^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)/(a*e^4+c*d^4)^2/(c*x^
4+a)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.78, antiderivative size = 659, normalized size of antiderivative = 1.00, 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 = {1741, 1753, 1756, 12, 1262, 739, 212, 1729, 1210, 1723, 226, 1721} \begin {gather*} \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 \text {ArcTan}\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 \text {ArcTan}\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 \text {ArcTan}\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^2 e \left (c d^4-a e^4\right ) \text {ArcTan}\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^{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 {e^3 \sqrt {a+c x^4}}{2 (d+e x)^2 \left (a e^4+c d^4\right )}-\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 {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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(e^3*Sqrt[a + c*x^4])/((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
*ArcTan[(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)*S
qrt[(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)/
(Sqrt[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 12

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

Rule 212

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

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 739

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 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 1262

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 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]

Rule 1741

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

Rule 1753

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 1756

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]

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 ) \text {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 ) \text {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] Result contains complex when optimal does not.
time = 11.41, size = 614, normalized size = 0.93 \begin {gather*} -\frac {\frac {c d^4 e^3 \sqrt {a+c x^4}}{(d+e x)^2}+\frac {a e^7 \sqrt {a+c x^4}}{(d+e x)^2}+\frac {6 c d^3 e^3 \sqrt {a+c x^4}}{d+e x}-\frac {6 c^2 d^6 e \tan ^{-1}\left (\frac {\sqrt {c} \left (d^2-e^2 x^2\right )+e^2 \sqrt {a+c x^4}}{\sqrt {-c d^4-a e^4}}\right )}{\sqrt {-c d^4-a e^4}}+\frac {6 a c d^2 e^5 \tan ^{-1}\left (\frac {\sqrt {c} \left (d^2-e^2 x^2\right )+e^2 \sqrt {a+c x^4}}{\sqrt {-c d^4-a e^4}}\right )}{\sqrt {-c d^4-a e^4}}+\frac {6 i a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c d^3 e^2 \sqrt {1+\frac {c x^4}{a}} E\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )}{\sqrt {a+c x^4}}+\frac {2 i c d \left (-2 c d^4-3 i \sqrt {a} \sqrt {c} d^2 e^2+a e^4\right ) \sqrt {1+\frac {c x^4}{a}} F\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} \sqrt {a+c x^4}}+\frac {6 \sqrt [4]{-1} \sqrt [4]{a} c^{7/4} d^5 \sqrt {1+\frac {c x^4}{a}} \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 )}{\sqrt {a+c x^4}}-\frac {6 \sqrt [4]{-1} a^{5/4} c^{3/4} d e^4 \sqrt {1+\frac {c x^4}{a}} \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 )}{\sqrt {a+c x^4}}}{2 \left (c d^4+a e^4\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((c*d^4*e^3*Sqrt[a + c*x^4])/(d + e*x)^2 + (a*e^7*Sqrt[a + c*x^4])/(d + e*x)^2 + (6*c*d^3*e^3*Sqrt[a + c*
x^4])/(d + e*x) - (6*c^2*d^6*e*ArcTan[(Sqrt[c]*(d^2 - e^2*x^2) + e^2*Sqrt[a + c*x^4])/Sqrt[-(c*d^4) - a*e^4]])
/Sqrt[-(c*d^4) - a*e^4] + (6*a*c*d^2*e^5*ArcTan[(Sqrt[c]*(d^2 - e^2*x^2) + e^2*Sqrt[a + c*x^4])/Sqrt[-(c*d^4)
- a*e^4]])/Sqrt[-(c*d^4) - a*e^4] + ((6*I)*a*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c*d^3*e^2*Sqrt[1 + (c*x^4)/a]*EllipticE
[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])/Sqrt[a + c*x^4] + ((2*I)*c*d*(-2*c*d^4 - (3*I)*Sqrt[a]*Sqrt[c]*d
^2*e^2 + a*e^4)*Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])/(Sqrt[(I*Sqrt[c])/S
qrt[a]]*Sqrt[a + c*x^4]) + (6*(-1)^(1/4)*a^(1/4)*c^(7/4)*d^5*Sqrt[1 + (c*x^4)/a]*EllipticPi[(I*Sqrt[a]*e^2)/(S
qrt[c]*d^2), ArcSin[((-1)^(3/4)*c^(1/4)*x)/a^(1/4)], -1])/Sqrt[a + c*x^4] - (6*(-1)^(1/4)*a^(5/4)*c^(3/4)*d*e^
4*Sqrt[1 + (c*x^4)/a]*EllipticPi[(I*Sqrt[a]*e^2)/(Sqrt[c]*d^2), ArcSin[((-1)^(3/4)*c^(1/4)*x)/a^(1/4)], -1])/S
qrt[a + c*x^4])/(c*d^4 + a*e^4)^2

________________________________________________________________________________________

Maple [C] Result contains complex when optimal does not.
time = 0.22, size = 483, normalized size = 0.73

method	result	size
default	\(-\frac {e^{3} \sqrt {c \,x^{4}+a}}{2 \left (e^{4} a +d^{4} c \right ) \left (e x +d \right )^{2}}-\frac {3 c \,d^{3} e^{3} \sqrt {c \,x^{4}+a}}{\left (e^{4} a +d^{4} c \right )^{2} \left (e x +d \right )}+\frac {c d \left (e^{4} a -2 d^{4} c \right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\left (e^{4} a +d^{4} c \right )^{2} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {3 i d^{3} e^{2} c^{\frac {3}{2}} \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\left (e^{4} a +d^{4} c \right )^{2} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {3 c \,d^{2} \left (e^{4} a -d^{4} c \right ) \left (-\frac {\arctanh \left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+a}\, \sqrt {c \,x^{4}+a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+a}}+\frac {e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticPi \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, e^{2}}{\sqrt {c}\, d^{2}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}+a}}\right )}{\left (e^{4} a +d^{4} c \right )^{2} e}\)	\(483\)
elliptic	\(-\frac {e^{3} \sqrt {c \,x^{4}+a}}{2 \left (e^{4} a +d^{4} c \right ) \left (e x +d \right )^{2}}-\frac {3 c \,d^{3} e^{3} \sqrt {c \,x^{4}+a}}{\left (e^{4} a +d^{4} c \right )^{2} \left (e x +d \right )}+\frac {c d \left (e^{4} a -2 d^{4} c \right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\left (e^{4} a +d^{4} c \right )^{2} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {3 i d^{3} e^{2} c^{\frac {3}{2}} \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\left (e^{4} a +d^{4} c \right )^{2} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {3 c \,d^{2} \left (e^{4} a -d^{4} c \right ) \left (-\frac {\arctanh \left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+a}\, \sqrt {c \,x^{4}+a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+a}}+\frac {e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticPi \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, e^{2}}{\sqrt {c}\, d^{2}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}+a}}\right )}{\left (e^{4} a +d^{4} c \right )^{2} e}\)	\(483\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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*d^3*e^2*c^(3/2)/(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)))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)*(x*e + d)^3), x)

________________________________________________________________________________________

Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a + c x^{4}} \left (d + e x\right )^{3}}\, dx \end {gather*}

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)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)*(x*e + d)^3), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{\sqrt {c\,x^4+a}\,{\left (d+e\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________