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

Optimal. Leaf size=628 \[ -\frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right ) \left (3 a^{3/2} B \sqrt{c} e^2+a e (A c e-2 b B e+2 B c d)-\sqrt{a} c^{3/2} d (2 A e+B d)+A c^2 d^2\right )}{2 a^{3/4} c^{7/4} \left (b-2 \sqrt{a} \sqrt{c}\right ) \sqrt{a+b x^2+c x^4}}+\frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right ) \left (A c \left (a b e^2-4 a c d e+b c d^2\right )-2 a B \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )\right )}{a^{3/4} c^{7/4} \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}-\frac{x \sqrt{a+b x^2+c x^4} \left (A c \left (a b e^2-4 a c d e+b c d^2\right )-2 a B \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )\right )}{a c^{3/2} \left (b^2-4 a c\right ) \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{x \left (x^2 \left (-\left (A c \left (a b e^2-4 a c d e+b c d^2\right )-a B \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )\right )\right )-A c \left (-2 a b d e-2 a \left (c d^2-a e^2\right )+b^2 d^2\right )+a B \left (a b e^2-4 a c d e+b c d^2\right )\right )}{a c \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}} \]

[Out]

-((x*(a*B*(b*c*d^2 - 4*a*c*d*e + a*b*e^2) - A*c*(b^2*d^2 - 2*a*b*d*e - 2*a*(c*d^2 - a*e^2)) - (A*c*(b*c*d^2 -
4*a*c*d*e + a*b*e^2) - a*B*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e)))*x^2))/(a*c*(b^2 - 4*a*c)*Sqrt[a + b*x^2
+ c*x^4])) - ((A*c*(b*c*d^2 - 4*a*c*d*e + a*b*e^2) - 2*a*B*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e)))*x*Sqrt[a +
b*x^2 + c*x^4])/(a*c^(3/2)*(b^2 - 4*a*c)*(Sqrt[a] + Sqrt[c]*x^2)) + ((A*c*(b*c*d^2 - 4*a*c*d*e + a*b*e^2) - 2
*a*B*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e)))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt
[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(a^(3/4)*c^(7/4)*(b^2 - 4*a
*c)*Sqrt[a + b*x^2 + c*x^4]) - ((A*c^2*d^2 + 3*a^(3/2)*B*Sqrt[c]*e^2 - Sqrt[a]*c^(3/2)*d*(B*d + 2*A*e) + a*e*(
2*B*c*d - 2*b*B*e + A*c*e))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Ellipt
icF[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^(7/4)*Sq
rt[a + b*x^2 + c*x^4])

________________________________________________________________________________________

Rubi [A]  time = 0.584427, antiderivative size = 633, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {1678, 1197, 1103, 1195} \[ \frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right ) \left (A c \left (a b e^2-4 a c d e+b c d^2\right )-2 a B \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )\right )}{a^{3/4} c^{7/4} \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}-\frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right ) \left (3 a^{3/2} B \sqrt{c} e^2+a e (A c e-2 b B e+2 B c d)-\sqrt{a} c^{3/2} d (2 A e+B d)+A c^2 d^2\right )}{2 a^{3/4} c^{7/4} \left (b-2 \sqrt{a} \sqrt{c}\right ) \sqrt{a+b x^2+c x^4}}-\frac{x \sqrt{a+b x^2+c x^4} \left (A c \left (a b e^2-4 a c d e+b c d^2\right )-2 a B \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )\right )}{a c^{3/2} \left (b^2-4 a c\right ) \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{x \left (c \left (\frac{a B \left (a b e^2-4 a c d e+b c d^2\right )}{c}-A \left (-2 a b d e-2 a \left (c d^2-a e^2\right )+b^2 d^2\right )\right )-x^2 \left (A c \left (a b e^2-4 a c d e+b c d^2\right )-a B \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )\right )\right )}{a c \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}} \]

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*(c*((a*B*(b*c*d^2 - 4*a*c*d*e + a*b*e^2))/c - A*(b^2*d^2 - 2*a*b*d*e - 2*a*(c*d^2 - a*e^2))) - (A*c*(b*c*
d^2 - 4*a*c*d*e + a*b*e^2) - a*B*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e)))*x^2))/(a*c*(b^2 - 4*a*c)*Sqrt[a +
b*x^2 + c*x^4])) - ((A*c*(b*c*d^2 - 4*a*c*d*e + a*b*e^2) - 2*a*B*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e)))*x*Sq
rt[a + b*x^2 + c*x^4])/(a*c^(3/2)*(b^2 - 4*a*c)*(Sqrt[a] + Sqrt[c]*x^2)) + ((A*c*(b*c*d^2 - 4*a*c*d*e + a*b*e^
2) - 2*a*B*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e)))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a]
+ Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(a^(3/4)*c^(7/4)*(b^2
- 4*a*c)*Sqrt[a + b*x^2 + c*x^4]) - ((A*c^2*d^2 + 3*a^(3/2)*B*Sqrt[c]*e^2 - Sqrt[a]*c^(3/2)*d*(B*d + 2*A*e) +
a*e*(2*B*c*d - 2*b*B*e + A*c*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^(3/4)*(b - 2*Sqrt[a]*Sqrt[c])*c^(7
/4)*Sqrt[a + b*x^2 + c*x^4])

Rule 1678

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Pq, a +
b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[Pq, a + b*x^2 + c*x^4, x], x, 2]}, Simp[(x*(a + b*x^2
+ c*x^4)^(p + 1)*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*
a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuot
ient[Pq, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4*p + 7)*(b*d - 2*a*e)*x^2,
x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rule 1197

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 1103

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)]*EllipticF[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(2*q*Sqrt[a + b*x^2 + c
*x^4]), x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1195

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

Rubi steps

\begin{align*} \int \frac{\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx &=-\frac{x \left (c \left (\frac{a B \left (b c d^2-4 a c d e+a b e^2\right )}{c}-A \left (b^2 d^2-2 a b d e-2 a \left (c d^2-a e^2\right )\right )\right )-\left (A c \left (b c d^2-4 a c d e+a b e^2\right )-a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )\right ) x^2\right )}{a c \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}-\frac{\int \frac{\frac{a (a e (4 B c d-b B e+2 A c e)+c d (2 A c d-b (B d+2 A e)))}{c}+\frac{\left (A c \left (b c d^2-4 a c d e+a b e^2\right )-2 a B \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )\right ) x^2}{c}}{\sqrt{a+b x^2+c x^4}} \, dx}{a \left (b^2-4 a c\right )}\\ &=-\frac{x \left (c \left (\frac{a B \left (b c d^2-4 a c d e+a b e^2\right )}{c}-A \left (b^2 d^2-2 a b d e-2 a \left (c d^2-a e^2\right )\right )\right )-\left (A c \left (b c d^2-4 a c d e+a b e^2\right )-a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )\right ) x^2\right )}{a c \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}-\frac{\left (A c^2 d^2+3 a^{3/2} B \sqrt{c} e^2-\sqrt{a} c^{3/2} d (B d+2 A e)+a e (2 B c d-2 b B e+A c e)\right ) \int \frac{1}{\sqrt{a+b x^2+c x^4}} \, dx}{\sqrt{a} \left (b-2 \sqrt{a} \sqrt{c}\right ) c^{3/2}}+\frac{\left (A c \left (b c d^2-4 a c d e+a b e^2\right )-2 a B \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+b x^2+c x^4}} \, dx}{\sqrt{a} c^{3/2} \left (b^2-4 a c\right )}\\ &=-\frac{x \left (c \left (\frac{a B \left (b c d^2-4 a c d e+a b e^2\right )}{c}-A \left (b^2 d^2-2 a b d e-2 a \left (c d^2-a e^2\right )\right )\right )-\left (A c \left (b c d^2-4 a c d e+a b e^2\right )-a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )\right ) x^2\right )}{a c \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}-\frac{\left (A c \left (b c d^2-4 a c d e+a b e^2\right )-2 a B \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )\right ) x \sqrt{a+b x^2+c x^4}}{a c^{3/2} \left (b^2-4 a c\right ) \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{\left (A c \left (b c d^2-4 a c d e+a b e^2\right )-2 a B \left (c^2 d^2+b^2 e^2-c e (b d+3 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}} 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} c^{7/4} \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}-\frac{\left (A c^2 d^2+3 a^{3/2} B \sqrt{c} e^2-\sqrt{a} c^{3/2} d (B d+2 A e)+a e (2 B c d-2 b B e+A c 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}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{2 a^{3/4} \left (b-2 \sqrt{a} \sqrt{c}\right ) c^{7/4} \sqrt{a+b x^2+c x^4}}\\ \end{align*}

Mathematica [C]  time = 4.76351, size = 766, normalized size = 1.22 \[ \frac{i \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}}\right ),\frac{\sqrt{b^2-4 a c}+b}{b-\sqrt{b^2-4 a c}}\right ) \left (A c \left (b^2 \left (a e^2+c d^2\right )-b \sqrt{b^2-4 a c} \left (a e^2+c d^2\right )-4 a c \left (-d e \sqrt{b^2-4 a c}+a e^2+c d^2\right )\right )+2 a B \left (c^2 d \left (d \sqrt{b^2-4 a c}-4 a e\right )+c e \left (-b d \sqrt{b^2-4 a c}-3 a e \sqrt{b^2-4 a c}+4 a b e+b^2 d\right )+b^2 e^2 \left (\sqrt{b^2-4 a c}-b\right )\right )\right )-4 c x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left (A c \left (2 a^2 e^2+a b e \left (e x^2-2 d\right )-2 a c d \left (d+2 e x^2\right )+b^2 d^2+b c d^2 x^2\right )-a B \left (a b e^2-2 a c e \left (2 d+e x^2\right )+b^2 e^2 x^2+b c d \left (d-2 e x^2\right )+2 c^2 d^2 x^2\right )\right )-i \left (\sqrt{b^2-4 a c}-b\right ) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right ) \left (2 a B \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )-A c \left (a b e^2-4 a c d e+b c d^2\right )\right )}{4 a c^2 \left (4 a c-b^2\right ) \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}} \]

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*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(-(a*B*(a*b*e^2 + 2*c^2*d^2*x^2 + b^2*e^2*x^2 + b*c*d*(d - 2*e*x^2) -
2*a*c*e*(2*d + e*x^2))) + A*c*(b^2*d^2 + 2*a^2*e^2 + b*c*d^2*x^2 + a*b*e*(-2*d + e*x^2) - 2*a*c*d*(d + 2*e*x^2
))) - I*(-b + Sqrt[b^2 - 4*a*c])*(-(A*c*(b*c*d^2 - 4*a*c*d*e + a*b*e^2)) + 2*a*B*(c^2*d^2 + b^2*e^2 - c*e*(b*d
+ 3*a*e)))*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] +
4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*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])] + I*(2*a*B*(b^2*(-b + Sqrt[b^2 - 4*a*c])*e^2 + c^2*d*(Sqrt[b^2 - 4*a*c]*d
- 4*a*e) + c*e*(b^2*d - b*Sqrt[b^2 - 4*a*c]*d + 4*a*b*e - 3*a*Sqrt[b^2 - 4*a*c]*e)) + A*c*(b^2*(c*d^2 + a*e^2
) - b*Sqrt[b^2 - 4*a*c]*(c*d^2 + a*e^2) - 4*a*c*(c*d^2 - Sqrt[b^2 - 4*a*c]*d*e + a*e^2)))*Sqrt[(b + Sqrt[b^2 -
4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])
]*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])])/(4*a*c^2*(-b^2 + 4*a*c)*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*x^2 + c*x^4])

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Maple [B]  time = 0.008, size = 1891, normalized size = 3. \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)^2/(c*x^4+b*x^2+a)^(3/2),x)

[Out]

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

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

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B e^{2} x^{6} +{\left (2 \, B d e + A e^{2}\right )} x^{4} + A d^{2} +{\left (B d^{2} + 2 \, A d e\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2} + a}}{c^{2} x^{8} + 2 \, b c x^{6} +{\left (b^{2} + 2 \, a c\right )} x^{4} + 2 \, a b x^{2} + a^{2}}, x\right ) \end{align*}

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]

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

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

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]

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

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

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