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

Optimal. Leaf size=859 \[ \frac{B x \sqrt{c x^4+b x^2+a} e^3}{3 c^2}-\frac{\left (a B \left (6 c^3 d^3-9 c^2 e (b d+6 a e) d-8 b^3 e^3+b c e^2 (18 b d+29 a e)\right )+3 A c \left (2 a b^2 e^3+6 a c \left (c d^2-a e^2\right ) e-b c d \left (c d^2+3 a e^2\right )\right )\right ) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+b x^2+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{3 a^{3/4} c^{11/4} \left (b^2-4 a c\right ) \sqrt{c x^4+b x^2+a}}-\frac{\left (3 A c^3 d^3-3 \sqrt{a} c^{5/2} (B d+3 A e) d^2-5 a^2 B c e^3+a e (3 c d-2 b e) (3 B c d-4 b B e+3 A c e)+3 a^{3/2} \sqrt{c} e^2 (9 B c d-4 b B e+3 A c e)\right ) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+b x^2+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{6 a^{3/4} \left (b-2 \sqrt{a} \sqrt{c}\right ) c^{11/4} \sqrt{c x^4+b x^2+a}}+\frac{\left (a B \left (6 c^3 d^3-9 c^2 e (b d+6 a e) d-8 b^3 e^3+b c e^2 (18 b d+29 a e)\right )+3 A c \left (2 a b^2 e^3+6 a c \left (c d^2-a e^2\right ) e-b c d \left (c d^2+3 a e^2\right )\right )\right ) x \sqrt{c x^4+b x^2+a}}{3 a c^{5/2} \left (b^2-4 a c\right ) \left (\sqrt{c} x^2+\sqrt{a}\right )}+\frac{x \left (-\left (a B (2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )+A c \left (a b^2 e^3+2 a c \left (3 c d^2-a e^2\right ) e-b c d \left (c d^2+3 a e^2\right )\right )\right ) x^2+A c \left (b^2 c d^3-2 a c \left (c d^2-3 a e^2\right ) d-a b e \left (3 c d^2+a e^2\right )\right )+a B \left (a b^2 e^3+2 a c \left (3 c d^2-a e^2\right ) e-b c d \left (c d^2+3 a e^2\right )\right )\right )}{a c^2 \left (b^2-4 a c\right ) \sqrt{c x^4+b x^2+a}} \]

[Out]

(x*(A*c*(b^2*c*d^3 - 2*a*c*d*(c*d^2 - 3*a*e^2) - a*b*e*(3*c*d^2 + a*e^2)) + a*B*(a*b^2*e^3 + 2*a*c*e*(3*c*d^2
- a*e^2) - b*c*d*(c*d^2 + 3*a*e^2)) - (a*B*(2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e)) + A*c*(a*b^2*
e^3 + 2*a*c*e*(3*c*d^2 - a*e^2) - b*c*d*(c*d^2 + 3*a*e^2)))*x^2))/(a*c^2*(b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4]
) + (B*e^3*x*Sqrt[a + b*x^2 + c*x^4])/(3*c^2) + ((a*B*(6*c^3*d^3 - 8*b^3*e^3 - 9*c^2*d*e*(b*d + 6*a*e) + b*c*e
^2*(18*b*d + 29*a*e)) + 3*A*c*(2*a*b^2*e^3 + 6*a*c*e*(c*d^2 - a*e^2) - b*c*d*(c*d^2 + 3*a*e^2)))*x*Sqrt[a + b*
x^2 + c*x^4])/(3*a*c^(5/2)*(b^2 - 4*a*c)*(Sqrt[a] + Sqrt[c]*x^2)) - ((a*B*(6*c^3*d^3 - 8*b^3*e^3 - 9*c^2*d*e*(
b*d + 6*a*e) + b*c*e^2*(18*b*d + 29*a*e)) + 3*A*c*(2*a*b^2*e^3 + 6*a*c*e*(c*d^2 - a*e^2) - b*c*d*(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]*EllipticE[2*ArcTan[(c^(1/4)
*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(3*a^(3/4)*c^(11/4)*(b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4]) - ((3*A
*c^3*d^3 - 5*a^2*B*c*e^3 - 3*Sqrt[a]*c^(5/2)*d^2*(B*d + 3*A*e) + a*e*(3*c*d - 2*b*e)*(3*B*c*d - 4*b*B*e + 3*A*
c*e) + 3*a^(3/2)*Sqrt[c]*e^2*(9*B*c*d - 4*b*B*e + 3*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])/(6*a^(3/4)*(b
- 2*Sqrt[a]*Sqrt[c])*c^(11/4)*Sqrt[a + b*x^2 + c*x^4])

________________________________________________________________________________________

Rubi [A]  time = 1.37963, antiderivative size = 859, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {1678, 1679, 1197, 1103, 1195} \[ \frac{B x \sqrt{c x^4+b x^2+a} e^3}{3 c^2}-\frac{\left (a B \left (6 c^3 d^3-9 c^2 e (b d+6 a e) d-8 b^3 e^3+b c e^2 (18 b d+29 a e)\right )+3 A c \left (2 a b^2 e^3+6 a c \left (c d^2-a e^2\right ) e-b c d \left (c d^2+3 a e^2\right )\right )\right ) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+b x^2+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{3 a^{3/4} c^{11/4} \left (b^2-4 a c\right ) \sqrt{c x^4+b x^2+a}}-\frac{\left (3 A c^3 d^3-3 \sqrt{a} c^{5/2} (B d+3 A e) d^2-5 a^2 B c e^3+a e (3 c d-2 b e) (3 B c d-4 b B e+3 A c e)+3 a^{3/2} \sqrt{c} e^2 (9 B c d-4 b B e+3 A c e)\right ) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+b x^2+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{6 a^{3/4} \left (b-2 \sqrt{a} \sqrt{c}\right ) c^{11/4} \sqrt{c x^4+b x^2+a}}+\frac{\left (a B \left (6 c^3 d^3-9 c^2 e (b d+6 a e) d-8 b^3 e^3+b c e^2 (18 b d+29 a e)\right )+3 A c \left (2 a b^2 e^3+6 a c \left (c d^2-a e^2\right ) e-b c d \left (c d^2+3 a e^2\right )\right )\right ) x \sqrt{c x^4+b x^2+a}}{3 a c^{5/2} \left (b^2-4 a c\right ) \left (\sqrt{c} x^2+\sqrt{a}\right )}+\frac{x \left (-\left (a B (2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )+A c \left (a b^2 e^3+2 a c \left (3 c d^2-a e^2\right ) e-b c d \left (c d^2+3 a e^2\right )\right )\right ) x^2+A c \left (b^2 c d^3-2 a c \left (c d^2-3 a e^2\right ) d-a b e \left (3 c d^2+a e^2\right )\right )+a B \left (a b^2 e^3+2 a c \left (3 c d^2-a e^2\right ) e-b c d \left (c d^2+3 a e^2\right )\right )\right )}{a c^2 \left (b^2-4 a c\right ) \sqrt{c x^4+b x^2+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(A*c*(b^2*c*d^3 - 2*a*c*d*(c*d^2 - 3*a*e^2) - a*b*e*(3*c*d^2 + a*e^2)) + a*B*(a*b^2*e^3 + 2*a*c*e*(3*c*d^2
- a*e^2) - b*c*d*(c*d^2 + 3*a*e^2)) - (a*B*(2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e)) + A*c*(a*b^2*
e^3 + 2*a*c*e*(3*c*d^2 - a*e^2) - b*c*d*(c*d^2 + 3*a*e^2)))*x^2))/(a*c^2*(b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4]
) + (B*e^3*x*Sqrt[a + b*x^2 + c*x^4])/(3*c^2) + ((a*B*(6*c^3*d^3 - 8*b^3*e^3 - 9*c^2*d*e*(b*d + 6*a*e) + b*c*e
^2*(18*b*d + 29*a*e)) + 3*A*c*(2*a*b^2*e^3 + 6*a*c*e*(c*d^2 - a*e^2) - b*c*d*(c*d^2 + 3*a*e^2)))*x*Sqrt[a + b*
x^2 + c*x^4])/(3*a*c^(5/2)*(b^2 - 4*a*c)*(Sqrt[a] + Sqrt[c]*x^2)) - ((a*B*(6*c^3*d^3 - 8*b^3*e^3 - 9*c^2*d*e*(
b*d + 6*a*e) + b*c*e^2*(18*b*d + 29*a*e)) + 3*A*c*(2*a*b^2*e^3 + 6*a*c*e*(c*d^2 - a*e^2) - b*c*d*(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]*EllipticE[2*ArcTan[(c^(1/4)
*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(3*a^(3/4)*c^(11/4)*(b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4]) - ((3*A
*c^3*d^3 - 5*a^2*B*c*e^3 - 3*Sqrt[a]*c^(5/2)*d^2*(B*d + 3*A*e) + a*e*(3*c*d - 2*b*e)*(3*B*c*d - 4*b*B*e + 3*A*
c*e) + 3*a^(3/2)*Sqrt[c]*e^2*(9*B*c*d - 4*b*B*e + 3*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])/(6*a^(3/4)*(b
- 2*Sqrt[a]*Sqrt[c])*c^(11/4)*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 1679

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{q = Expon[Pq, x^2], e = Coeff[Pq, x^2,
Expon[Pq, x^2]]}, Simp[(e*x^(2*q - 3)*(a + b*x^2 + c*x^4)^(p + 1))/(c*(2*q + 4*p + 1)), x] + Dist[1/(c*(2*q +
4*p + 1)), Int[(a + b*x^2 + c*x^4)^p*ExpandToSum[c*(2*q + 4*p + 1)*Pq - a*e*(2*q - 3)*x^(2*q - 4) - b*e*(2*q
+ 2*p - 1)*x^(2*q - 2) - c*e*(2*q + 4*p + 1)*x^(2*q), x], x], x]] /; FreeQ[{a, b, c, p}, 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 )^3}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\frac{x \left (A c \left (b^2 c d^3-2 a c d \left (c d^2-3 a e^2\right )-a b e \left (3 c d^2+a e^2\right )\right )+a B \left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )-\left (a B (2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )+A c \left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )\right ) x^2\right )}{a c^2 \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}-\frac{\int \frac{\frac{a \left (a b^2 B e^3-b c \left (B c d^3+3 A c d^2 e+3 a B d e^2+a A e^3\right )+2 c \left (a B e \left (3 c d^2-a e^2\right )+A c d \left (c d^2+3 a e^2\right )\right )\right )}{c^2}-\frac{\left (a B \left (2 c^3 d^3-2 b^3 e^3-3 c^2 d e (b d+6 a e)+b c e^2 (6 b d+7 a e)\right )+A c \left (2 a b^2 e^3+6 a c e \left (c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )\right ) x^2}{c^2}+a B \left (4 a-\frac{b^2}{c}\right ) e^3 x^4}{\sqrt{a+b x^2+c x^4}} \, dx}{a \left (b^2-4 a c\right )}\\ &=\frac{x \left (A c \left (b^2 c d^3-2 a c d \left (c d^2-3 a e^2\right )-a b e \left (3 c d^2+a e^2\right )\right )+a B \left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )-\left (a B (2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )+A c \left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )\right ) x^2\right )}{a c^2 \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}+\frac{B e^3 x \sqrt{a+b x^2+c x^4}}{3 c^2}-\frac{\int \frac{\frac{a \left (4 a b^2 B e^3-3 b c \left (B c d^3+3 A c d^2 e+3 a B d e^2+a A e^3\right )+2 c \left (a B e \left (9 c d^2-5 a e^2\right )+3 A c d \left (c d^2+3 a e^2\right )\right )\right )}{c}-\frac{\left (a B \left (6 c^3 d^3-8 b^3 e^3-9 c^2 d e (b d+6 a e)+b c e^2 (18 b d+29 a e)\right )+3 A c \left (2 a b^2 e^3+6 a c e \left (c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )\right ) x^2}{c}}{\sqrt{a+b x^2+c x^4}} \, dx}{3 a c \left (b^2-4 a c\right )}\\ &=\frac{x \left (A c \left (b^2 c d^3-2 a c d \left (c d^2-3 a e^2\right )-a b e \left (3 c d^2+a e^2\right )\right )+a B \left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )-\left (a B (2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )+A c \left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )\right ) x^2\right )}{a c^2 \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}+\frac{B e^3 x \sqrt{a+b x^2+c x^4}}{3 c^2}-\frac{\left (3 A c^3 d^3-5 a^2 B c e^3-3 \sqrt{a} c^{5/2} d^2 (B d+3 A e)+a e (3 c d-2 b e) (3 B c d-4 b B e+3 A c e)+3 a^{3/2} \sqrt{c} e^2 (9 B c d-4 b B e+3 A c e)\right ) \int \frac{1}{\sqrt{a+b x^2+c x^4}} \, dx}{3 \sqrt{a} \left (b-2 \sqrt{a} \sqrt{c}\right ) c^{5/2}}-\frac{\left (a B \left (6 c^3 d^3-8 b^3 e^3-9 c^2 d e (b d+6 a e)+b c e^2 (18 b d+29 a e)\right )+3 A c \left (2 a b^2 e^3+6 a c e \left (c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+b x^2+c x^4}} \, dx}{3 \sqrt{a} c^{5/2} \left (b^2-4 a c\right )}\\ &=\frac{x \left (A c \left (b^2 c d^3-2 a c d \left (c d^2-3 a e^2\right )-a b e \left (3 c d^2+a e^2\right )\right )+a B \left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )-\left (a B (2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )+A c \left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )\right ) x^2\right )}{a c^2 \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}+\frac{B e^3 x \sqrt{a+b x^2+c x^4}}{3 c^2}+\frac{\left (a B \left (6 c^3 d^3-8 b^3 e^3-9 c^2 d e (b d+6 a e)+b c e^2 (18 b d+29 a e)\right )+3 A c \left (2 a b^2 e^3+6 a c e \left (c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )\right ) x \sqrt{a+b x^2+c x^4}}{3 a c^{5/2} \left (b^2-4 a c\right ) \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\left (a B \left (6 c^3 d^3-8 b^3 e^3-9 c^2 d e (b d+6 a e)+b c e^2 (18 b d+29 a e)\right )+3 A c \left (2 a b^2 e^3+6 a c e \left (c d^2-a e^2\right )-b c d \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 )}{3 a^{3/4} c^{11/4} \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}-\frac{\left (3 A c^3 d^3-5 a^2 B c e^3-3 \sqrt{a} c^{5/2} d^2 (B d+3 A e)+a e (3 c d-2 b e) (3 B c d-4 b B e+3 A c e)+3 a^{3/2} \sqrt{c} e^2 (9 B c d-4 b B e+3 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 )}{6 a^{3/4} \left (b-2 \sqrt{a} \sqrt{c}\right ) c^{11/4} \sqrt{a+b x^2+c x^4}}\\ \end{align*}

Mathematica [C]  time = 6.67421, size = 5432, normalized size = 6.32 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

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Maple [B]  time = 0.034, size = 2445, normalized size = 2.9 \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)^3/(c*x^4+b*x^2+a)^(3/2),x)

[Out]

B*e^3*(-2*c*(-1/2*b/c^3*(3*a*c-b^2)/(4*a*c-b^2)*x^3-1/2*(2*a*c-b^2)*a/c^3/(4*a*c-b^2)*x)/((x^4+x^2*b/c+a/c)*c)
^(1/2)+1/3/c^2*x*(c*x^4+b*x^2+a)^(1/2)+1/4*(-1/c^2*(2*a*c-b^2)*a/(4*a*c-b^2)-1/3*a/c^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*(-5/3*b/c^2-b/c^2*(3*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))-E
llipticE(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^
3+3*B*d*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))))+(3*A*d*e^2+3*B*d^2*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))))+(3*A*d^2*e+B*d^3)*(-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^3*(-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 )}^{3}}{{\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)^3/(c*x^4+b*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)*(e*x^2 + d)^3/(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^{3} x^{8} +{\left (3 \, B d e^{2} + A e^{3}\right )} x^{6} + 3 \,{\left (B d^{2} e + A d e^{2}\right )} x^{4} + A d^{3} +{\left (B d^{3} + 3 \, A d^{2} 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)^3/(c*x^4+b*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

integral((B*e^3*x^8 + (3*B*d*e^2 + A*e^3)*x^6 + 3*(B*d^2*e + A*d*e^2)*x^4 + A*d^3 + (B*d^3 + 3*A*d^2*e)*x^2)*s
qrt(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(-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)**3/(c*x**4+b*x**2+a)**(3/2),x)

[Out]

Timed out

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

[Out]

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