3.1.7 \(\int \frac {A+B x^2}{(d+e x^2)^3 \sqrt {a+c x^4}} \, dx\) [7]

Optimal. Leaf size=875 \[ \frac {\sqrt {c} \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\right ) x \sqrt {a+c x^4}}{8 d^2 \left (c d^2+a e^2\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {e (B d-A e) x \sqrt {a+c x^4}}{4 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )^2}-\frac {e \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\right ) x \sqrt {a+c x^4}}{8 d^2 \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac {\left (3 A e \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right )-B \left (3 c^2 d^5-10 a c d^3 e^2-a^2 d e^4\right )\right ) \tan ^{-1}\left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{16 d^{5/2} \sqrt {e} \left (c d^2+a e^2\right )^{5/2}}-\frac {\sqrt [4]{a} \sqrt [4]{c} \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{8 d^2 \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {\sqrt [4]{c} \left (4 A c d^2+\sqrt {a} \sqrt {c} d (B d-A e)+a e (B d+3 A e)\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{8 \sqrt [4]{a} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \sqrt {a+c x^4}}-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \left (3 A e \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right )-B \left (3 c^2 d^5-10 a c d^3 e^2-a^2 d e^4\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{32 \sqrt [4]{a} \sqrt [4]{c} d^3 e \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}} \]

[Out]

1/16*(3*A*e*(a^2*e^4+2*a*c*d^2*e^2+5*c^2*d^4)-B*(-a^2*d*e^4-10*a*c*d^3*e^2+3*c^2*d^5))*arctan(x*(a*e^2+c*d^2)^
(1/2)/d^(1/2)/e^(1/2)/(c*x^4+a)^(1/2))/d^(5/2)/(a*e^2+c*d^2)^(5/2)/e^(1/2)-1/4*e*(-A*e+B*d)*x*(c*x^4+a)^(1/2)/
d/(a*e^2+c*d^2)/(e*x^2+d)^2-1/8*e*(-3*A*a*e^3-9*A*c*d^2*e-B*a*d*e^2+5*B*c*d^3)*x*(c*x^4+a)^(1/2)/d^2/(a*e^2+c*
d^2)^2/(e*x^2+d)+1/8*(-3*A*a*e^3-9*A*c*d^2*e-B*a*d*e^2+5*B*c*d^3)*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))-1/8*a^(1/4)*c^(1/4)*(-3*A*a*e^3-9*A*c*d^2*e-B*a*d*e^2+5*B*c*d^3)*(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)-1/32*(3*A
*e*(a^2*e^4+2*a*c*d^2*e^2+5*c^2*d^4)-B*(-a^2*d*e^4-10*a*c*d^3*e^2+3*c^2*d^5))*(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/e/(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)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/
4))),1/2*2^(1/2))*(a^(1/2)+x^2*c^(1/2))*(4*A*c*d^2+a*e*(3*A*e+B*d)+d*(-A*e+B*d)*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]
time = 1.17, antiderivative size = 875, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1711, 1729, 1210, 1723, 226, 1721} \begin {gather*} \frac {\sqrt {c} \left (5 B c d^3-9 A c e d^2-a B e^2 d-3 a A e^3\right ) \sqrt {c x^4+a} x}{8 d^2 \left (c d^2+a e^2\right )^2 \left (\sqrt {c} x^2+\sqrt {a}\right )}-\frac {e \left (5 B c d^3-9 A c e d^2-a B e^2 d-3 a A e^3\right ) \sqrt {c x^4+a} x}{8 d^2 \left (c d^2+a e^2\right )^2 \left (e x^2+d\right )}-\frac {e (B d-A e) \sqrt {c x^4+a} x}{4 d \left (c d^2+a e^2\right ) \left (e x^2+d\right )^2}+\frac {\left (3 A e \left (5 c^2 d^4+2 a c e^2 d^2+a^2 e^4\right )-B \left (3 c^2 d^5-10 a c e^2 d^3-a^2 e^4 d\right )\right ) \text {ArcTan}\left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {c x^4+a}}\right )}{16 d^{5/2} \sqrt {e} \left (c d^2+a e^2\right )^{5/2}}-\frac {\sqrt [4]{a} \sqrt [4]{c} \left (5 B c d^3-9 A c e d^2-a B e^2 d-3 a A e^3\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} E\left (2 \text {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 {c x^4+a}}+\frac {\sqrt [4]{c} \left (4 A c d^2+\sqrt {a} \sqrt {c} (B d-A e) d+a e (B d+3 A e)\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} F\left (2 \text {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 {c x^4+a}}-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \left (3 A e \left (5 c^2 d^4+2 a c e^2 d^2+a^2 e^4\right )-B \left (3 c^2 d^5-10 a c e^2 d^3-a^2 e^4 d\right )\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{32 \sqrt [4]{a} \sqrt [4]{c} d^3 e \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c]*(5*B*c*d^3 - 9*A*c*d^2*e - a*B*d*e^2 - 3*a*A*e^3)*x*Sqrt[a + c*x^4])/(8*d^2*(c*d^2 + a*e^2)^2*(Sqrt[a
] + Sqrt[c]*x^2)) - (e*(B*d - A*e)*x*Sqrt[a + c*x^4])/(4*d*(c*d^2 + a*e^2)*(d + e*x^2)^2) - (e*(5*B*c*d^3 - 9*
A*c*d^2*e - a*B*d*e^2 - 3*a*A*e^3)*x*Sqrt[a + c*x^4])/(8*d^2*(c*d^2 + a*e^2)^2*(d + e*x^2)) + ((3*A*e*(5*c^2*d
^4 + 2*a*c*d^2*e^2 + a^2*e^4) - B*(3*c^2*d^5 - 10*a*c*d^3*e^2 - a^2*d*e^4))*ArcTan[(Sqrt[c*d^2 + a*e^2]*x)/(Sq
rt[d]*Sqrt[e]*Sqrt[a + c*x^4])])/(16*d^(5/2)*Sqrt[e]*(c*d^2 + a*e^2)^(5/2)) - (a^(1/4)*c^(1/4)*(5*B*c*d^3 - 9*
A*c*d^2*e - a*B*d*e^2 - 3*a*A*e^3)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Ellipti
cE[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*A*c*d^2 + Sqrt
[a]*Sqrt[c]*d*(B*d - A*e) + a*e*(B*d + 3*A*e))*(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)*Sqr
t[a + c*x^4]) - ((Sqrt[c]*d + Sqrt[a]*e)*(3*A*e*(5*c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4) - B*(3*c^2*d^5 - 10*a*c*
d^3*e^2 - a^2*d*e^4))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-1/4*(Sqr
t[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*e*(S
qrt[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 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*} \int \frac {A+B x^2}{\left (d+e x^2\right )^3 \sqrt {a+c x^4}} \, dx &=-\frac {e (B d-A e) 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 A c d^2-a B d e-3 a A e^2-4 c d (B d-A e) x^2+c e (B d-A e) 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 (B d-A e) x \sqrt {a+c x^4}}{4 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )^2}-\frac {e \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\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 {a B d e \left (7 c d^2+a e^2\right )+A \left (8 c^2 d^4+5 a c d^2 e^2+3 a^2 e^4\right )+4 c d \left (2 B c d^3-4 A c d^2 e-a B d e^2-a A e^3\right ) x^2+c e \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\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 (B d-A e) x \sqrt {a+c x^4}}{4 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )^2}-\frac {e \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\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 {\sqrt {a} c^{3/2} d e \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\right )+c e \left (a B d e \left (7 c d^2+a e^2\right )+A \left (8 c^2 d^4+5 a c d^2 e^2+3 a^2 e^4\right )\right )+\left (-c e \left (c d-\sqrt {a} \sqrt {c} e\right ) \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\right )+4 c^2 d e \left (2 B c d^3-4 A c d^2 e-a B d e^2-a A e^3\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 (\sqrt {a} \sqrt {c} \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\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 {\sqrt {c} \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\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 (B d-A e) x \sqrt {a+c x^4}}{4 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )^2}-\frac {e \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\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 (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\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 A c d^2+\sqrt {a} \sqrt {c} d (B d-A e)+a e (B d+3 A e)\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 (\sqrt {a} \left (3 A e \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right )-B \left (3 c^2 d^5-10 a c d^3 e^2-a^2 d e^4\right )\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 {\sqrt {c} \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\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 (B d-A e) x \sqrt {a+c x^4}}{4 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )^2}-\frac {e \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\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 (3 A e \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right )-B \left (3 c^2 d^5-10 a c d^3 e^2-a^2 d e^4\right )\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} \sqrt {e} \left (c d^2+a e^2\right )^{5/2}}-\frac {\sqrt [4]{a} \sqrt [4]{c} \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\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 A c d^2+\sqrt {a} \sqrt {c} d (B d-A e)+a e (B d+3 A e)\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 {\left (\sqrt {c} d+\sqrt {a} e\right ) \left (3 A e \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right )-B \left (3 c^2 d^5-10 a c d^3 e^2-a^2 d e^4\right )\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]{a} \sqrt [4]{c} d^3 e \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 11.13, size = 453, normalized size = 0.52 \begin {gather*} \frac {-\frac {d e^2 x \left (a+c x^4\right ) \left (2 d (B d-A e) \left (c d^2+a e^2\right )+\left (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\right ) \left (d+e x^2\right )\right )}{\left (d+e x^2\right )^2}-\frac {i \sqrt {1+\frac {c x^4}{a}} \left (-i \sqrt {a} \sqrt {c} d e \left (-5 B c d^3+9 A c d^2 e+a B d e^2+3 a A e^3\right ) E\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+\sqrt {c} d \left (\sqrt {c} d-i \sqrt {a} e\right ) \left (A e \left (-7 c d^2+2 i \sqrt {a} \sqrt {c} d e-3 a e^2\right )+B d \left (3 c d^2-2 i \sqrt {a} \sqrt {c} d e-a e^2\right )\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+\left (3 A e \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right )+B \left (-3 c^2 d^5+10 a c d^3 e^2+a^2 d e^4\right )\right ) \Pi \left (-\frac {i \sqrt {a} e}{\sqrt {c} d};\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}}{8 d^3 e \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-((d*e^2*x*(a + c*x^4)*(2*d*(B*d - A*e)*(c*d^2 + a*e^2) + (5*B*c*d^3 - 9*A*c*d^2*e - a*B*d*e^2 - 3*a*A*e^3)*(
d + e*x^2)))/(d + e*x^2)^2) - (I*Sqrt[1 + (c*x^4)/a]*((-I)*Sqrt[a]*Sqrt[c]*d*e*(-5*B*c*d^3 + 9*A*c*d^2*e + a*B
*d*e^2 + 3*a*A*e^3)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + Sqrt[c]*d*(Sqrt[c]*d - I*Sqrt[a]*e
)*(A*e*(-7*c*d^2 + (2*I)*Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2) + B*d*(3*c*d^2 - (2*I)*Sqrt[a]*Sqrt[c]*d*e - a*e^2))*E
llipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + (3*A*e*(5*c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4) + B*(-3*c^
2*d^5 + 10*a*c*d^3*e^2 + a^2*d*e^4))*EllipticPi[((-I)*Sqrt[a]*e)/(Sqrt[c]*d), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[
a]]*x], -1]))/Sqrt[(I*Sqrt[c])/Sqrt[a]])/(8*d^3*e*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4])

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Maple [C] Result contains complex when optimal does not.
time = 0.16, size = 1591, normalized size = 1.82

method	result	size
default	\(\text {Expression too large to display}\)	\(1591\)
elliptic	\(\text {Expression too large to display}\)	\(1982\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(A*e-B*d)/e*(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)*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)-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)-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)*El
lipticF(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/d*e^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)*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)*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))*c^2)+B/e*(1/2*e^2*x*(c*x^4+a)^(1/2)/d/(a*e^2+c*d^2)/(e*x^2+d)-1/2*c/(a*e^2+c*d^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)-1/2*I*c^(1/2)*e/d/(a*e^2+c*d^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)+1/2*I*c
^(1/2)*e/d/(a*e^2+c*d^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)+1/2/d^2/(a*e^2+c*d^2)*e^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)*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+3/2/(a*e^2+
c*d^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)*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))*c)

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

[Out]

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

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

[Out]

integral(sqrt(c*x^4 + a)*(B*x^2 + A)/(c*d^3*x^4 + a*d^3 + (c*x^10 + a*x^6)*e^3 + 3*(c*d*x^8 + a*d*x^4)*e^2 + 3
*(c*d^2*x^6 + a*d^2*x^2)*e), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x^{2}}{\sqrt {a + c x^{4}} \left (d + e x^{2}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((A + B*x**2)/(sqrt(a + c*x**4)*(d + e*x**2)**3), x)

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {B\,x^2+A}{\sqrt {c\,x^4+a}\,{\left (e\,x^2+d\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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