3.1.29 \(\int \frac {A+B x^2}{(d+e x^2)^2 (a+b x^2+c x^4)^{3/2}} \, dx\) [29]

Optimal. Leaf size=1301 \[ \frac {x \left (a b c \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right )+\left (b^2-2 a c\right ) \left (a B e (2 c d-b e)+A \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right )\right )-c \left (a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )+A \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right )\right ) x^2\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}}+\frac {\sqrt {c} \left (a B d \left (-4 c^2 d^2-3 b^2 e^2+4 c e (b d+2 a e)\right )+A \left (2 b^3 d e^2+2 b c d \left (c d^2-3 a e^2\right )-4 a c e \left (-2 c d^2+a e^2\right )+b^2 \left (-4 c d^2 e+a e^3\right )\right )\right ) x \sqrt {a+b x^2+c x^4}}{2 a \left (-b^2+4 a c\right ) d \left (c d^2+e (-b d+a e)\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {e^3 (B d-A e) x \sqrt {a+b x^2+c x^4}}{2 d \left (c d^2-b d e+a e^2\right )^2 \left (d+e x^2\right )}+\frac {e^{3/2} \left (A e \left (7 c d^2-e (4 b d-a e)\right )-B d \left (5 c d^2-e (2 b d+a e)\right )\right ) \tan ^{-1}\left (\frac {\sqrt {c d^2-b d e+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+c x^4}}\right )}{4 d^{3/2} \left (c d^2-b d e+a e^2\right )^{5/2}}-\frac {\sqrt [4]{c} \left (a B d \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right )-A \left (2 b^3 d e^2+2 b c d \left (c d^2-3 a e^2\right )-4 a c e \left (-2 c d^2+a e^2\right )+b^2 \left (-4 c d^2 e+a e^3\right )\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+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}-\frac {b}{4 \sqrt {a} \sqrt {c}}\right )}{2 a^{3/4} \left (b^2-4 a c\right ) d \left (c d^2+e (-b d+a e)\right )^2 \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{c} \left (a \sqrt {c} e (B d-2 A e)+\sqrt {a} (B d-A e) (c d-b e)+A \sqrt {c} d (-c d+b e)\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+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}-\frac {b}{4 \sqrt {a} \sqrt {c}}\right )}{2 a^{3/4} \left (b-2 \sqrt {a} \sqrt {c}\right ) d \left (-\sqrt {c} d+\sqrt {a} e\right ) \left (-c d^2+e (b d-a e)\right ) \sqrt {a+b x^2+c x^4}}-\frac {e \left (\sqrt {c} d+\sqrt {a} e\right ) \left (A e \left (7 c d^2-e (4 b d-a e)\right )-B d \left (5 c d^2-e (2 b d+a e)\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+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}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}} \]

[Out]

1/4*e^(3/2)*(A*e*(7*c*d^2-e*(-a*e+4*b*d))-B*d*(5*c*d^2-e*(a*e+2*b*d)))*arctan(x*(a*e^2-b*d*e+c*d^2)^(1/2)/d^(1
/2)/e^(1/2)/(c*x^4+b*x^2+a)^(1/2))/d^(3/2)/(a*e^2-b*d*e+c*d^2)^(5/2)+x*(a*b*c*(A*e*(-b*e+2*c*d)-B*(-a*e^2+c*d^
2))+(-2*a*c+b^2)*(a*B*e*(-b*e+2*c*d)+A*(c^2*d^2+b^2*e^2-c*e*(a*e+2*b*d)))-c*(a*B*(2*c^2*d^2+b^2*e^2-2*c*e*(a*e
+b*d))+A*(2*b^2*c*d*e-4*a*c^2*d*e-b^3*e^2-b*c*(-3*a*e^2+c*d^2)))*x^2)/a/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^2/(c*
x^4+b*x^2+a)^(1/2)-1/2*e^3*(-A*e+B*d)*x*(c*x^4+b*x^2+a)^(1/2)/d/(a*e^2-b*d*e+c*d^2)^2/(e*x^2+d)+1/2*(a*B*d*(-4
*c^2*d^2-3*b^2*e^2+4*c*e*(2*a*e+b*d))+A*(2*b^3*d*e^2+2*b*c*d*(-3*a*e^2+c*d^2)-4*a*c*e*(a*e^2-2*c*d^2)+b^2*(a*e
^3-4*c*d^2*e)))*x*c^(1/2)*(c*x^4+b*x^2+a)^(1/2)/a/(4*a*c-b^2)/d/(c*d^2+e*(a*e-b*d))^2/(a^(1/2)+x^2*c^(1/2))-1/
2*c^(1/4)*(a*B*d*(4*c^2*d^2+3*b^2*e^2-4*c*e*(2*a*e+b*d))-A*(2*b^3*d*e^2+2*b*c*d*(-3*a*e^2+c*d^2)-4*a*c*e*(a*e^
2-2*c*d^2)+b^2*(a*e^3-4*c*d^2*e)))*(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-b/a^(1/2)/c^(1/2))^(1/2))*(a^(1/2)+x^2*c^(1/2))*((c*x^4+b*x
^2+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/a^(3/4)/(-4*a*c+b^2)/d/(c*d^2+e*(a*e-b*d))^2/(c*x^4+b*x^2+a)^(1/2)-1/8*e*
(A*e*(7*c*d^2-e*(-a*e+4*b*d))-B*d*(5*c*d^2-e*(a*e+2*b*d)))*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*ar
ctan(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-b/a^(1/2)/c^(1/2))^(1/2))*(e*a^(1/2)+d*c^(1/2))*(a^(1/2)+x^2*c^(1/2))*((c*x^4+b*x^2+a)/(a^(1/2
)+x^2*c^(1/2))^2)^(1/2)/a^(1/4)/c^(1/4)/d^2/(a*e^2-b*d*e+c*d^2)^2/(-e*a^(1/2)+d*c^(1/2))/(c*x^4+b*x^2+a)^(1/2)
+1/2*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*arcta
n(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))*((-A*e+B*d)*(-b*e+c*d)*a^(1/2)+a*e*(-2*A*e+B*d)*c^(1/2)
+A*d*(b*e-c*d)*c^(1/2))*(a^(1/2)+x^2*c^(1/2))*((c*x^4+b*x^2+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/a^(3/4)/d/(-c*d^
2+e*(-a*e+b*d))/(e*a^(1/2)-d*c^(1/2))/(b-2*a^(1/2)*c^(1/2))/(c*x^4+b*x^2+a)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 2.21, antiderivative size = 2112, normalized size of antiderivative = 1.62, number of steps used = 15, number of rules used = 10, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {1734, 1192, 1211, 1117, 1209, 1237, 1728, 1722, 1720, 1230} \begin {gather*} -\frac {(B d-A e) x \sqrt {c x^4+b x^2+a} e^3}{2 d \left (c d^2-b e d+a e^2\right )^2 \left (e x^2+d\right )}-\frac {\sqrt [4]{a} \sqrt [4]{c} (B d-A e) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+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}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) e^2}{2 d \left (c d^2-b e d+a e^2\right )^2 \sqrt {c x^4+b x^2+a}}+\frac {\sqrt {c} (B d-A e) x \sqrt {c x^4+b x^2+a} e^2}{2 d \left (c d^2-b e d+a e^2\right )^2 \left (\sqrt {c} x^2+\sqrt {a}\right )}-\frac {(B d-A e) \left (3 c d^2-e (2 b d-a e)\right ) \text {ArcTan}\left (\frac {\sqrt {c d^2-b e d+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {c x^4+b x^2+a}}\right ) e^{3/2}}{4 d^{3/2} \left (c d^2-b e d+a e^2\right )^{5/2}}+\frac {\left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) \text {ArcTan}\left (\frac {\sqrt {c d^2-b e d+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {c x^4+b x^2+a}}\right ) e^{3/2}}{2 \sqrt {d} \left (c d^2-b e d+a e^2\right )^{5/2}}+\frac {\sqrt [4]{c} \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+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}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) e}{2 \sqrt [4]{a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b e d+a e^2\right )^2 \sqrt {c x^4+b x^2+a}}-\frac {\sqrt [4]{c} (B d-A e) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+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}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) e}{2 \sqrt [4]{a} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b e d+a e^2\right ) \sqrt {c x^4+b x^2+a}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) (B d-A e) \left (3 c d^2-e (2 b d-a e)\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+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}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) e}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b e d+a e^2\right )^2 \sqrt {c x^4+b x^2+a}}-\frac {a^{3/4} \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right )^2 \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+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}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) e}{4 \sqrt [4]{c} d \left (c d^2-a e^2\right ) \left (c d^2-b e d+a e^2\right )^2 \sqrt {c x^4+b x^2+a}}-\frac {\sqrt [4]{c} \left (a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )+A \left (-e^2 b^3+2 c d e b^2-c \left (c d^2-3 a e^2\right ) b-4 a c^2 d e\right )\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+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}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{a^{3/4} \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )^2 \sqrt {c x^4+b x^2+a}}-\frac {\sqrt [4]{c} \left (a^{3/2} B \sqrt {c} e^2+a (2 B c d-b B e-A c e) e+A (c d-b e)^2-\sqrt {a} \sqrt {c} \left (B c d^2-A e (2 c d-b e)\right )\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+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}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 a^{3/4} \left (b-2 \sqrt {a} \sqrt {c}\right ) \left (c d^2-b e d+a e^2\right )^2 \sqrt {c x^4+b x^2+a}}+\frac {\sqrt {c} \left (a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )+A \left (-e^2 b^3+2 c d e b^2-c \left (c d^2-3 a e^2\right ) b-4 a c^2 d e\right )\right ) x \sqrt {c x^4+b x^2+a}}{a \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )^2 \left (\sqrt {c} x^2+\sqrt {a}\right )}+\frac {x \left (-c \left (a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )+A \left (-e^2 b^3+2 c d e b^2-c \left (c d^2-3 a e^2\right ) b-4 a c^2 d e\right )\right ) x^2+a b c \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right )+\left (b^2-2 a c\right ) \left (a B e (2 c d-b e)+A \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right )\right )\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )^2 \sqrt {c x^4+b x^2+a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a*b*c*(A*e*(2*c*d - b*e) - B*(c*d^2 - a*e^2)) + (b^2 - 2*a*c)*(a*B*e*(2*c*d - b*e) + A*(c^2*d^2 + b^2*e^2
- c*e*(2*b*d + a*e))) - c*(a*B*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e)) + A*(2*b^2*c*d*e - 4*a*c^2*d*e - b^3*
e^2 - b*c*(c*d^2 - 3*a*e^2)))*x^2))/(a*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*x^2 + c*x^4]) + (Sqr
t[c]*e^2*(B*d - A*e)*x*Sqrt[a + b*x^2 + c*x^4])/(2*d*(c*d^2 - b*d*e + a*e^2)^2*(Sqrt[a] + Sqrt[c]*x^2)) + (Sqr
t[c]*(a*B*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e)) + A*(2*b^2*c*d*e - 4*a*c^2*d*e - b^3*e^2 - b*c*(c*d^2 - 3*
a*e^2)))*x*Sqrt[a + b*x^2 + c*x^4])/(a*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*(Sqrt[a] + Sqrt[c]*x^2)) - (e^3
*(B*d - A*e)*x*Sqrt[a + b*x^2 + c*x^4])/(2*d*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x^2)) - (e^(3/2)*(B*d - A*e)*(3*
c*d^2 - e*(2*b*d - a*e))*ArcTan[(Sqrt[c*d^2 - b*d*e + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + b*x^2 + c*x^4])])/(4
*d^(3/2)*(c*d^2 - b*d*e + a*e^2)^(5/2)) + (e^(3/2)*(A*e*(2*c*d - b*e) - B*(c*d^2 - a*e^2))*ArcTan[(Sqrt[c*d^2
- b*d*e + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + b*x^2 + c*x^4])])/(2*Sqrt[d]*(c*d^2 - b*d*e + a*e^2)^(5/2)) - (a
^(1/4)*c^(1/4)*e^2*(B*d - A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Ell
ipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*d*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*
x^2 + c*x^4]) - (c^(1/4)*(a*B*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e)) + A*(2*b^2*c*d*e - 4*a*c^2*d*e - b^3*e
^2 - b*c*(c*d^2 - 3*a*e^2)))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Ellip
ticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(a^(3/4)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^
2)^2*Sqrt[a + b*x^2 + c*x^4]) - (c^(1/4)*e*(B*d - A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[
a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*a^(1/4)*d*(Sqrt
[c]*d - Sqrt[a]*e)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x^2 + c*x^4]) + (c^(1/4)*e*(A*e*(2*c*d - b*e) - B*(c*d^2
- a*e^2))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(
1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*a^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 - b*d*e + a*e^2)^2*S
qrt[a + b*x^2 + c*x^4]) - (c^(1/4)*(a^(3/2)*B*Sqrt[c]*e^2 + A*(c*d - b*e)^2 + a*e*(2*B*c*d - b*B*e - A*c*e) -
Sqrt[a]*Sqrt[c]*(B*c*d^2 - A*e*(2*c*d - b*e)))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqr
t[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*a^(3/4)*(b - 2*Sqrt[a]*
Sqrt[c])*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*x^2 + c*x^4]) + (e*(Sqrt[c]*d + Sqrt[a]*e)*(B*d - A*e)*(3*c*d^2
- e*(2*b*d - a*e))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + 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)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])
/(8*a^(1/4)*c^(1/4)*d^2*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*x^2 + c*x^4]) - (a^(3/4)*
e*((Sqrt[c]*d)/Sqrt[a] + e)^2*(A*e*(2*c*d - b*e) - B*(c*d^2 - a*e^2))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2
+ 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)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(4*c^(1/4)*d*(c*d^2 - a*e^2)*(c*d^2 - b*d*e + a*e^2)^2*Sqr
t[a + b*x^2 + c*x^4])

Rule 1117

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(
a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(
4*c))], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1192

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*(a*b*e - d*(b^2 - 2*a
*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1209

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

Rule 1211

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

Rule 1230

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

Rule 1237

Int[((d_) + (e_.)*(x_)^2)^(q_)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> Simp[(-e^2)*x*(d + e*x^2
)^(q + 1)*(Sqrt[a + b*x^2 + c*x^4]/(2*d*(q + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/(2*d*(q + 1)*(c*d^2 - b
*d*e + a*e^2)), Int[((d + e*x^2)^(q + 1)/Sqrt[a + b*x^2 + c*x^4])*Simp[a*e^2*(2*q + 3) + 2*d*(c*d - b*e)*(q +
1) - 2*e*(c*d*(q + 1) - b*e*(q + 2))*x^2 + c*e^2*(2*q + 5)*x^4, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ
[b^2 - 4*a*c, 0] && ILtQ[q, -1]

Rule 1720

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

Rule 1722

Int[((A_.) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (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 + b*x^2 + 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 + b*x^2 + c*x^4]), x]
, x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2
- a*e^2, 0] && PosQ[c/a] && NeQ[c*A^2 - a*B^2, 0]

Rule 1728

Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (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 + b
*x^2 + 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 + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[b^2 - 4*a*c, 0] && NeQ[
c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] &&  !GtQ[b^2 - 4*a*c, 0]

Rule 1734

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

Rubi steps

\begin {align*} \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\int \left (\frac {a B e (2 c d-b e)+A \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right )-c \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) x^2}{\left (c d^2-b d e+a e^2\right )^2 \left (a+b x^2+c x^4\right )^{3/2}}+\frac {e (-B d+A e)}{\left (c d^2-b d e+a e^2\right ) \left (d+e x^2\right )^2 \sqrt {a+b x^2+c x^4}}+\frac {e \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right )}{\left (c d^2-b d e+a e^2\right )^2 \left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}}\right ) \, dx\\ &=\frac {\int \frac {a B e (2 c d-b e)+A \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right )-c \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) x^2}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx}{\left (c d^2-b d e+a e^2\right )^2}-\frac {(e (B d-A e)) \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a+b x^2+c x^4}} \, dx}{c d^2-b d e+a e^2}+\frac {\left (e \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right )\right ) \int \frac {1}{\left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx}{\left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {x \left (a b c \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right )+\left (b^2-2 a c\right ) \left (a B e (2 c d-b e)+A \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right )\right )-c \left (a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )+A \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right )\right ) x^2\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}}-\frac {e^3 (B d-A e) x \sqrt {a+b x^2+c x^4}}{2 d \left (c d^2-b d e+a e^2\right )^2 \left (d+e x^2\right )}-\frac {\int \frac {a c \left (A b^2 e^2+2 c \left (A c d^2+2 a B d e-a A e^2\right )-b \left (B c d^2+2 A c d e+a B e^2\right )\right )-c \left (a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )+A \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right )\right ) x^2}{\sqrt {a+b x^2+c x^4}} \, dx}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2}+\frac {(e (B d-A e)) \int \frac {-2 c d^2+e (2 b d-a e)+2 c d e x^2+c e^2 x^4}{\left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx}{2 d \left (c d^2-b d e+a e^2\right )^2}+\frac {\left (\sqrt {c} e \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right )\right ) \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx}{\left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b d e+a e^2\right )^2}-\frac {\left (\sqrt {a} e^2 \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right )\right ) \int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx}{\left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {x \left (a b c \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right )+\left (b^2-2 a c\right ) \left (a B e (2 c d-b e)+A \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right )\right )-c \left (a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )+A \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right )\right ) x^2\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}}-\frac {e^3 (B d-A e) x \sqrt {a+b x^2+c x^4}}{2 d \left (c d^2-b d e+a e^2\right )^2 \left (d+e x^2\right )}+\frac {e^{3/2} \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) \tan ^{-1}\left (\frac {\sqrt {c d^2-b d e+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {d} \left (c d^2-b d e+a e^2\right )^{5/2}}+\frac {\sqrt [4]{c} e \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+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}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{a} e \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+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}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{4 \sqrt [4]{c} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}}+\frac {(B d-A e) \int \frac {\sqrt {a} c^{3/2} d e^2+c e \left (-2 c d^2+e (2 b d-a e)\right )+\left (2 c^2 d e^2-c e^2 \left (c d-\sqrt {a} \sqrt {c} e\right )\right ) x^2}{\left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx}{2 c d \left (c d^2-b d e+a e^2\right )^2}-\frac {\left (\sqrt {a} \sqrt {c} e^2 (B d-A e)\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx}{2 d \left (c d^2-b d e+a e^2\right )^2}-\frac {\left (\sqrt {c} \left (a^{3/2} B \sqrt {c} e^2+A (c d-b e)^2+a e (2 B c d-b B e-A c e)-\sqrt {a} \sqrt {c} \left (B c d^2-A e (2 c d-b e)\right )\right )\right ) \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx}{\sqrt {a} \left (b-2 \sqrt {a} \sqrt {c}\right ) \left (c d^2-b d e+a e^2\right )^2}-\frac {\left (\sqrt {c} \left (a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )+A \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right )\right )\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx}{\sqrt {a} \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {x \left (a b c \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right )+\left (b^2-2 a c\right ) \left (a B e (2 c d-b e)+A \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right )\right )-c \left (a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )+A \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right )\right ) x^2\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}}+\frac {\sqrt {c} e^2 (B d-A e) x \sqrt {a+b x^2+c x^4}}{2 d \left (c d^2-b d e+a e^2\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {\sqrt {c} \left (a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )+A \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right )\right ) x \sqrt {a+b x^2+c x^4}}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {e^3 (B d-A e) x \sqrt {a+b x^2+c x^4}}{2 d \left (c d^2-b d e+a e^2\right )^2 \left (d+e x^2\right )}+\frac {e^{3/2} \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) \tan ^{-1}\left (\frac {\sqrt {c d^2-b d e+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {d} \left (c d^2-b d e+a e^2\right )^{5/2}}-\frac {\sqrt [4]{a} \sqrt [4]{c} e^2 (B d-A e) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+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}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 d \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{c} \left (a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )+A \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+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}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{a^{3/4} \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{c} e \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+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}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{c} \left (a^{3/2} B \sqrt {c} e^2+A (c d-b e)^2+a e (2 B c d-b B e-A c e)-\sqrt {a} \sqrt {c} \left (B c d^2-A e (2 c d-b e)\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+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}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 a^{3/4} \left (b-2 \sqrt {a} \sqrt {c}\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{a} e \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+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}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{4 \sqrt [4]{c} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}}-\frac {\left (\sqrt {c} e (B d-A e)\right ) \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx}{d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b d e+a e^2\right )}+\frac {\left (\sqrt {a} e^2 (B d-A e) \left (3 c d^2-e (2 b d-a e)\right )\right ) \int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx}{2 d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {x \left (a b c \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right )+\left (b^2-2 a c\right ) \left (a B e (2 c d-b e)+A \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right )\right )-c \left (a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )+A \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right )\right ) x^2\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}}+\frac {\sqrt {c} e^2 (B d-A e) x \sqrt {a+b x^2+c x^4}}{2 d \left (c d^2-b d e+a e^2\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {\sqrt {c} \left (a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )+A \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right )\right ) x \sqrt {a+b x^2+c x^4}}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {e^3 (B d-A e) x \sqrt {a+b x^2+c x^4}}{2 d \left (c d^2-b d e+a e^2\right )^2 \left (d+e x^2\right )}-\frac {e^{3/2} (B d-A e) \left (3 c d^2-e (2 b d-a e)\right ) \tan ^{-1}\left (\frac {\sqrt {c d^2-b d e+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+c x^4}}\right )}{4 d^{3/2} \left (c d^2-b d e+a e^2\right )^{5/2}}+\frac {e^{3/2} \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) \tan ^{-1}\left (\frac {\sqrt {c d^2-b d e+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {d} \left (c d^2-b d e+a e^2\right )^{5/2}}-\frac {\sqrt [4]{a} \sqrt [4]{c} e^2 (B d-A e) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+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}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 d \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{c} \left (a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )+A \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+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}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{a^{3/4} \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{c} e (B d-A e) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+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}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{c} e \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+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}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{c} \left (a^{3/2} B \sqrt {c} e^2+A (c d-b e)^2+a e (2 B c d-b B e-A c e)-\sqrt {a} \sqrt {c} \left (B c d^2-A e (2 c d-b e)\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+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}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 a^{3/4} \left (b-2 \sqrt {a} \sqrt {c}\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}}+\frac {e \left (\sqrt {c} d+\sqrt {a} e\right ) (B d-A e) \left (3 c d^2-e (2 b d-a e)\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+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}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{a} e \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+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}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{4 \sqrt [4]{c} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 14.94, size = 1116, normalized size = 0.86 \begin {gather*} \frac {4 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} d \left (-a \left (b^2-4 a c\right ) e^3 (B d-A e) x \left (a+b x^2+c x^4\right )+2 d x \left (d+e x^2\right ) \left (a B \left (-b^3 e^2+b^2 c e \left (2 d-e x^2\right )+b c \left (3 a e^2-c d \left (d-2 e x^2\right )\right )-2 c^2 \left (c d^2 x^2+a e \left (2 d-e x^2\right )\right )\right )+A \left (b^4 e^2+b^3 c e \left (-2 d+e x^2\right )+2 a c^2 \left (a e^2-c d \left (d-2 e x^2\right )\right )+b^2 c \left (-4 a e^2+c d \left (d-2 e x^2\right )\right )+b c^2 \left (c d^2 x^2-3 a e \left (-2 d+e x^2\right )\right )\right )\right )\right )-i \sqrt {2} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \left (d+e x^2\right ) \left (\left (-b+\sqrt {b^2-4 a c}\right ) d \left (a B d \left (-4 c^2 d^2-3 b^2 e^2+4 c e (b d+2 a e)\right )+A \left (2 b^3 d e^2+2 b c d \left (c d^2-3 a e^2\right )-4 a c e \left (-2 c d^2+a e^2\right )+b^2 \left (-4 c d^2 e+a e^3\right )\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-d \left (a B d \left (3 b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^2-4 c^2 d \left (\sqrt {b^2-4 a c} d-6 a e\right )+2 c \left (-3 b+2 \sqrt {b^2-4 a c}\right ) e (b d+2 a e)\right )+A \left (-2 b^4 d e^2+b^2 \left (-2 c^2 d^3+a \sqrt {b^2-4 a c} e^3-4 c d e \left (\sqrt {b^2-4 a c} d-3 a e\right )\right )-4 a c \left (-2 c^2 d^3+a \sqrt {b^2-4 a c} e^3-2 c d e \left (\sqrt {b^2-4 a c} d-2 a e\right )\right )+b^3 e \left (4 c d^2+e \left (2 \sqrt {b^2-4 a c} d-a e\right )\right )+2 b c \left (c d^2 \left (\sqrt {b^2-4 a c} d-8 a e\right )+a e^2 \left (-3 \sqrt {b^2-4 a c} d+2 a e\right )\right )\right )\right ) F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-2 a \left (-b^2+4 a c\right ) e \left (A e \left (7 c d^2+e (-4 b d+a e)\right )+B \left (-5 c d^3+d e (2 b d+a e)\right )\right ) \Pi \left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{8 a \left (b^2-4 a c\right ) \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \left (c d^3+d e (-b d+a e)\right )^2 \left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*(-(a*(b^2 - 4*a*c)*e^3*(B*d - A*e)*x*(a + b*x^2 + c*x^4)) + 2*d*x*(d + e*
x^2)*(a*B*(-(b^3*e^2) + b^2*c*e*(2*d - e*x^2) + b*c*(3*a*e^2 - c*d*(d - 2*e*x^2)) - 2*c^2*(c*d^2*x^2 + a*e*(2*
d - e*x^2))) + A*(b^4*e^2 + b^3*c*e*(-2*d + e*x^2) + 2*a*c^2*(a*e^2 - c*d*(d - 2*e*x^2)) + b^2*c*(-4*a*e^2 + c
*d*(d - 2*e*x^2)) + b*c^2*(c*d^2*x^2 - 3*a*e*(-2*d + e*x^2))))) - I*Sqrt[2]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*
x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(d + e*x^2)*((-b + Sqrt[b^2 - 4*a*c]
)*d*(a*B*d*(-4*c^2*d^2 - 3*b^2*e^2 + 4*c*e*(b*d + 2*a*e)) + A*(2*b^3*d*e^2 + 2*b*c*d*(c*d^2 - 3*a*e^2) - 4*a*c
*e*(-2*c*d^2 + a*e^2) + b^2*(-4*c*d^2*e + a*e^3)))*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]
*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - d*(a*B*d*(3*b^2*(b - Sqrt[b^2 - 4*a*c])*e^2 - 4*c^2*d*
(Sqrt[b^2 - 4*a*c]*d - 6*a*e) + 2*c*(-3*b + 2*Sqrt[b^2 - 4*a*c])*e*(b*d + 2*a*e)) + A*(-2*b^4*d*e^2 + b^2*(-2*
c^2*d^3 + a*Sqrt[b^2 - 4*a*c]*e^3 - 4*c*d*e*(Sqrt[b^2 - 4*a*c]*d - 3*a*e)) - 4*a*c*(-2*c^2*d^3 + a*Sqrt[b^2 -
4*a*c]*e^3 - 2*c*d*e*(Sqrt[b^2 - 4*a*c]*d - 2*a*e)) + b^3*e*(4*c*d^2 + e*(2*Sqrt[b^2 - 4*a*c]*d - a*e)) + 2*b*
c*(c*d^2*(Sqrt[b^2 - 4*a*c]*d - 8*a*e) + a*e^2*(-3*Sqrt[b^2 - 4*a*c]*d + 2*a*e))))*EllipticF[I*ArcSinh[Sqrt[2]
*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - 2*a*(-b^2 + 4*a*c)*e*(
A*e*(7*c*d^2 + e*(-4*b*d + a*e)) + B*(-5*c*d^3 + d*e*(2*b*d + a*e)))*EllipticPi[((b + Sqrt[b^2 - 4*a*c])*e)/(2
*c*d), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])
)/(8*a*(b^2 - 4*a*c)*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*(c*d^3 + d*e*(-(b*d) + a*e))^2*(d + e*x^2)*Sqrt[a + b*x^2
+ c*x^4])

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(8275\) vs. \(2(1252)=2504\).
time = 0.20, size = 8276, normalized size = 6.36

method	result	size
default	\(\text {Expression too large to display}\)	\(8276\)
elliptic	\(\text {Expression too large to display}\)	\(9725\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

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)^2/(c*x^4+b*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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)^2/(c*x^4+b*x^2+a)^(3/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________