3.12 \(\int \frac{A+B x^2}{(d+e x^2) (a+c x^4)^{3/2}} \, dx\)

Optimal. Leaf size=732 \[ \frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right ) \left (-\sqrt{a} \sqrt{c} (B d-A e)+a B e+A c d\right )}{4 a^{5/4} \sqrt [4]{c} \sqrt{a+c x^4} \left (a e^2+c d^2\right )}-\frac{\sqrt [4]{c} e \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} (B d-A e) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt{a+c x^4} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (a e^2+c d^2\right )}+\frac{a^{3/4} e \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right )^2 (B d-A e) \Pi \left (-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{c} d \sqrt{a+c x^4} \left (c^2 d^4-a^2 e^4\right )}+\frac{\sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} (B d-A e) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{3/4} \sqrt{a+c x^4} \left (a e^2+c d^2\right )}-\frac{\sqrt{c} x \sqrt{a+c x^4} (B d-A e)}{2 a \left (\sqrt{a}+\sqrt{c} x^2\right ) \left (a e^2+c d^2\right )}+\frac{x \left (a B e+c x^2 (B d-A e)+A c d\right )}{2 a \sqrt{a+c x^4} \left (a e^2+c d^2\right )}-\frac{e^{3/2} (B d-A e) \tan ^{-1}\left (\frac{x \sqrt{a e^2+c d^2}}{\sqrt{d} \sqrt{e} \sqrt{a+c x^4}}\right )}{2 \sqrt{d} \left (a e^2+c d^2\right )^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.781061, antiderivative size = 732, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1721, 1179, 1198, 220, 1196, 1217, 1707} \[ \frac{a^{3/4} e \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right )^2 (B d-A e) \Pi \left (-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{c} d \sqrt{a+c x^4} \left (c^2 d^4-a^2 e^4\right )}+\frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right ) \left (-\sqrt{a} \sqrt{c} (B d-A e)+a B e+A c d\right )}{4 a^{5/4} \sqrt [4]{c} \sqrt{a+c x^4} \left (a e^2+c d^2\right )}+\frac{\sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} (B d-A e) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{3/4} \sqrt{a+c x^4} \left (a e^2+c d^2\right )}-\frac{\sqrt{c} x \sqrt{a+c x^4} (B d-A e)}{2 a \left (\sqrt{a}+\sqrt{c} x^2\right ) \left (a e^2+c d^2\right )}+\frac{x \left (a B e+c x^2 (B d-A e)+A c d\right )}{2 a \sqrt{a+c x^4} \left (a e^2+c d^2\right )}-\frac{e^{3/2} (B d-A e) \tan ^{-1}\left (\frac{x \sqrt{a e^2+c d^2}}{\sqrt{d} \sqrt{e} \sqrt{a+c x^4}}\right )}{2 \sqrt{d} \left (a e^2+c d^2\right )^{3/2}}-\frac{\sqrt [4]{c} e \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} (B d-A e) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt{a+c x^4} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1721

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a +
c*x^4], Px*(d + e*x^2)^q*(a + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x^2] && NeQ[c*d^
2 + a*e^2, 0] && IntegerQ[p + 1/2] && IntegerQ[q]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> -Simp[(x*(d + e*x^2)*(a + c*x^4)^(p + 1))/(
4*a*(p + 1)), x] + Dist[1/(4*a*(p + 1)), Int[Simp[d*(4*p + 5) + e*(4*p + 7)*x^2, x]*(a + c*x^4)^(p + 1), x], x
] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1198

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

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 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 1217

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[c/a, 2]}, Dist[(c*d + a*e*q
)/(c*d^2 - a*e^2), Int[1/Sqrt[a + c*x^4], x], x] - Dist[(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}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0]
&& PosQ[c/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{A+B x^2}{\left (d+e x^2\right ) \left (a+c x^4\right )^{3/2}} \, dx &=\int \left (\frac{A c d+a B e+c (B d-A e) x^2}{\left (c d^2+a e^2\right ) \left (a+c x^4\right )^{3/2}}+\frac{e (-B d+A e)}{\left (c d^2+a e^2\right ) \left (d+e x^2\right ) \sqrt{a+c x^4}}\right ) \, dx\\ &=\frac{\int \frac{A c d+a B e+c (B d-A e) x^2}{\left (a+c x^4\right )^{3/2}} \, dx}{c d^2+a e^2}-\frac{(e (B d-A e)) \int \frac{1}{\left (d+e x^2\right ) \sqrt{a+c x^4}} \, dx}{c d^2+a e^2}\\ &=\frac{x \left (A c d+a B e+c (B d-A e) x^2\right )}{2 a \left (c d^2+a e^2\right ) \sqrt{a+c x^4}}-\frac{\int \frac{-A c d-a B e+c (B d-A e) x^2}{\sqrt{a+c x^4}} \, dx}{2 a \left (c d^2+a e^2\right )}-\frac{\left (\sqrt{c} e (B d-A e)\right ) \int \frac{1}{\sqrt{a+c x^4}} \, dx}{\left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right )}+\frac{\left (\sqrt{a} e^2 (B d-A e)\right ) \int \frac{1+\frac{\sqrt{c} x^2}{\sqrt{a}}}{\left (d+e x^2\right ) \sqrt{a+c x^4}} \, dx}{\left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right )}\\ &=\frac{x \left (A c d+a B e+c (B d-A e) x^2\right )}{2 a \left (c d^2+a e^2\right ) \sqrt{a+c x^4}}-\frac{e^{3/2} (B d-A e) \tan ^{-1}\left (\frac{\sqrt{c d^2+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{a+c x^4}}\right )}{2 \sqrt{d} \left (c d^2+a e^2\right )^{3/2}}-\frac{\sqrt [4]{c} e (B d-A e) \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 (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right ) \sqrt{a+c x^4}}+\frac{\sqrt [4]{a} e \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right ) (B d-A e) \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 )}{4 \sqrt [4]{c} d \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} (B d-A e)\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+c x^4}} \, dx}{2 \sqrt{a} \left (c d^2+a e^2\right )}+\frac{\left (A c d+a B e-\sqrt{a} \sqrt{c} (B d-A e)\right ) \int \frac{1}{\sqrt{a+c x^4}} \, dx}{2 a \left (c d^2+a e^2\right )}\\ &=\frac{x \left (A c d+a B e+c (B d-A e) x^2\right )}{2 a \left (c d^2+a e^2\right ) \sqrt{a+c x^4}}-\frac{\sqrt{c} (B d-A e) x \sqrt{a+c x^4}}{2 a \left (c d^2+a e^2\right ) \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{e^{3/2} (B d-A e) \tan ^{-1}\left (\frac{\sqrt{c d^2+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{a+c x^4}}\right )}{2 \sqrt{d} \left (c d^2+a e^2\right )^{3/2}}+\frac{\sqrt [4]{c} (B d-A e) \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 )}{2 a^{3/4} \left (c d^2+a e^2\right ) \sqrt{a+c x^4}}-\frac{\sqrt [4]{c} e (B d-A e) \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 (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right ) \sqrt{a+c x^4}}+\frac{\left (A c d+a B e-\sqrt{a} \sqrt{c} (B d-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 )}{4 a^{5/4} \sqrt [4]{c} \left (c d^2+a e^2\right ) \sqrt{a+c x^4}}+\frac{\sqrt [4]{a} e \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right ) (B d-A e) \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 )}{4 \sqrt [4]{c} d \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right ) \sqrt{a+c x^4}}\\ \end{align*}

Mathematica [C]  time = 0.80458, size = 432, normalized size = 0.59 \[ \frac{d \sqrt{\frac{c x^4}{a}+1} \left (\sqrt{a} B-i A \sqrt{c}\right ) \left (\sqrt{c} d-i \sqrt{a} e\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}\right ),-1\right )-\sqrt{a} \sqrt{c} d \sqrt{\frac{c x^4}{a}+1} (B d-A e) E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )+A c d^2 x \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}-2 i a A e^2 \sqrt{\frac{c x^4}{a}+1} \Pi \left (-\frac{i \sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )-A c d e x^3 \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}+B c d^2 x^3 \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}+2 i a B d e \sqrt{\frac{c x^4}{a}+1} \Pi \left (-\frac{i \sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )+a B d e x \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}}{2 a d \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{a+c x^4} \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c*d^2*x + a*B*Sqrt[(I*Sqrt[c])/Sqrt[a]]*d*e*x + B*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c*d^2
*x^3 - A*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c*d*e*x^3 - Sqrt[a]*Sqrt[c]*d*(B*d - A*e)*Sqrt[1 + (c*x^4)/a]*EllipticE[I*A
rcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + (Sqrt[a]*B - I*A*Sqrt[c])*d*(Sqrt[c]*d - I*Sqrt[a]*e)*Sqrt[1 + (c*x
^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + (2*I)*a*B*d*e*Sqrt[1 + (c*x^4)/a]*EllipticPi[((
-I)*Sqrt[a]*e)/(Sqrt[c]*d), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] - (2*I)*a*A*e^2*Sqrt[1 + (c*x^4)/a]*El
lipticPi[((-I)*Sqrt[a]*e)/(Sqrt[c]*d), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])/(2*a*Sqrt[(I*Sqrt[c])/Sqrt
[a]]*d*(c*d^2 + a*e^2)*Sqrt[a + c*x^4])

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Maple [C]  time = 0.023, size = 564, normalized size = 0.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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