3.1.24 \(\int \frac {(A+B x^2) (d+e x^2)^3}{(a+b x^2+c x^4)^{3/2}} \, dx\) [24]

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

[Out]

x*(A*c*(b^2*c*d^3-2*a*c*d*(-3*a*e^2+c*d^2)-a*b*e*(a*e^2+3*c*d^2))+a*B*(a*b^2*e^3+2*a*c*e*(-a*e^2+3*c*d^2)-b*c*
d*(3*a*e^2+c*d^2))-(a*B*(-b*e+2*c*d)*(c^2*d^2+b^2*e^2-c*e*(3*a*e+b*d))+A*c*(a*b^2*e^3+2*a*c*e*(-a*e^2+3*c*d^2)
-b*c*d*(3*a*e^2+c*d^2)))*x^2)/a/c^2/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^(1/2)+1/3*B*e^3*x*(c*x^4+b*x^2+a)^(1/2)/c^2+1
/3*(a*B*(6*c^3*d^3-8*b^3*e^3-9*c^2*d*e*(6*a*e+b*d)+b*c*e^2*(29*a*e+18*b*d))+3*A*c*(2*a*b^2*e^3+6*a*c*e*(-a*e^2
+c*d^2)-b*c*d*(3*a*e^2+c*d^2)))*x*(c*x^4+b*x^2+a)^(1/2)/a/c^(5/2)/(-4*a*c+b^2)/(a^(1/2)+x^2*c^(1/2))-1/3*(a*B*
(6*c^3*d^3-8*b^3*e^3-9*c^2*d*e*(6*a*e+b*d)+b*c*e^2*(29*a*e+18*b*d))+3*A*c*(2*a*b^2*e^3+6*a*c*e*(-a*e^2+c*d^2)-
b*c*d*(3*a*e^2+c*d^2)))*(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)/c^(11/4)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^(1/2)-1/6*(cos(2*arctan(c^(1/4)*x/a^(1
/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^
(1/2))^(1/2))*(3*A*c^3*d^3-5*a^2*B*c*e^3+a*e*(-2*b*e+3*c*d)*(3*A*c*e-4*B*b*e+3*B*c*d)-3*c^(5/2)*d^2*(3*A*e+B*d
)*a^(1/2)+3*a^(3/2)*e^2*(3*A*c*e-4*B*b*e+9*B*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)/c^(11/4)/(b-2*a^(1/2)*c^(1/2))/(c*x^4+b*x^2+a)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.89, antiderivative size = 859, normalized size of antiderivative = 1.00, 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 = {1692, 1693, 1211, 1117, 1209} \begin {gather*} \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 \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 )}{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 \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 )}{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 (\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\right )+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}} \end {gather*}

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 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 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 1692

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 1693

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]

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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 14.74, size = 1058, normalized size = 1.23 \begin {gather*} \frac {-4 c \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x \left (3 A c \left (b^2 \left (c d^3-a e^3 x^2\right )+b \left (-a^2 e^3+c^2 d^3 x^2-3 a c d e \left (d-e x^2\right )\right )+2 a c \left (a e^2 \left (3 d+e x^2\right )-c d^2 \left (d+3 e x^2\right )\right )\right )+a B \left (4 b^3 e^3 x^2+b^2 e^2 \left (4 a e-9 c d x^2+c e x^4\right )-b c \left (3 c d^2 \left (d-3 e x^2\right )+a e^2 \left (9 d+13 e x^2\right )\right )-2 c \left (5 a^2 e^3+3 c^2 d^3 x^2+a c e \left (-9 d^2-9 d e x^2+2 e^2 x^4\right )\right )\right )\right )+i \left (-b+\sqrt {b^2-4 a c}\right ) \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 ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+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 )-i \left (a B \left (8 b^3 \left (-b+\sqrt {b^2-4 a c}\right ) e^3-6 c^3 d^2 \left (\sqrt {b^2-4 a c} d-6 a e\right )+b c e^2 \left (18 b^2 d-18 b \sqrt {b^2-4 a c} d+37 a b e-29 a \sqrt {b^2-4 a c} e\right )+c^2 e \left (-9 b^2 d^2+2 a e \left (27 \sqrt {b^2-4 a c} d-10 a e\right )+9 b d \left (\sqrt {b^2-4 a c} d-8 a e\right )\right )\right )+3 A c \left (2 a b^3 e^3-b^2 \left (c^2 d^3+3 a c d e^2+2 a \sqrt {b^2-4 a c} e^3\right )+b c \left (c \sqrt {b^2-4 a c} d^3+a e^2 \left (3 \sqrt {b^2-4 a c} d-8 a e\right )\right )+2 a c \left (2 c^2 d^3+3 a \sqrt {b^2-4 a c} e^3-3 c d e \left (\sqrt {b^2-4 a c} d-2 a e\right )\right )\right )\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} 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 )}{12 a c^3 \left (-b^2+4 a c\right ) \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {a+b x^2+c x^4}} \end {gather*}

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]

(-4*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(3*A*c*(b^2*(c*d^3 - a*e^3*x^2) + b*(-(a^2*e^3) + c^2*d^3*x^2 - 3*a*c*
d*e*(d - e*x^2)) + 2*a*c*(a*e^2*(3*d + e*x^2) - c*d^2*(d + 3*e*x^2))) + a*B*(4*b^3*e^3*x^2 + b^2*e^2*(4*a*e -
9*c*d*x^2 + c*e*x^4) - b*c*(3*c*d^2*(d - 3*e*x^2) + a*e^2*(9*d + 13*e*x^2)) - 2*c*(5*a^2*e^3 + 3*c^2*d^3*x^2 +
a*c*e*(-9*d^2 - 9*d*e*x^2 + 2*e^2*x^4)))) + I*(-b + Sqrt[b^2 - 4*a*c])*(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[(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*(a*B*(8*b^3*(-b + Sqrt[b^2 - 4*a*c])*e^3 - 6*c^3*d^2*(Sqrt[b^2 - 4
*a*c]*d - 6*a*e) + b*c*e^2*(18*b^2*d - 18*b*Sqrt[b^2 - 4*a*c]*d + 37*a*b*e - 29*a*Sqrt[b^2 - 4*a*c]*e) + c^2*e
*(-9*b^2*d^2 + 2*a*e*(27*Sqrt[b^2 - 4*a*c]*d - 10*a*e) + 9*b*d*(Sqrt[b^2 - 4*a*c]*d - 8*a*e))) + 3*A*c*(2*a*b^
3*e^3 - b^2*(c^2*d^3 + 3*a*c*d*e^2 + 2*a*Sqrt[b^2 - 4*a*c]*e^3) + b*c*(c*Sqrt[b^2 - 4*a*c]*d^3 + a*e^2*(3*Sqrt
[b^2 - 4*a*c]*d - 8*a*e)) + 2*a*c*(2*c^2*d^3 + 3*a*Sqrt[b^2 - 4*a*c]*e^3 - 3*c*d*e*(Sqrt[b^2 - 4*a*c]*d - 2*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])]*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])])/(12*a*c^3*(-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] Leaf count of result is larger than twice the leaf count of optimal. \(2444\) vs. \(2(841)=1682\).
time = 0.16, size = 2445, normalized size = 2.85

method	result	size
elliptic	\(-\frac {2 c \left (\frac {\left (2 A \,a^{2} c^{2} e^{3}-A a \,b^{2} c \,e^{3}+3 A a b \,c^{2} d \,e^{2}-6 A a \,c^{3} d^{2} e +A b \,c^{3} d^{3}-3 B \,a^{2} b c \,e^{3}+6 B \,a^{2} c^{2} d \,e^{2}+B a \,b^{3} e^{3}-3 B a \,b^{2} c d \,e^{2}+3 B a b \,c^{2} d^{2} e -2 B a \,c^{3} d^{3}\right ) x^{3}}{2 c^{3} a \left (4 a c -b^{2}\right )}-\frac {\left (A \,a^{2} b c \,e^{3}-6 A \,a^{2} c^{2} d \,e^{2}+3 A a b \,c^{2} d^{2} e +2 A a \,c^{3} d^{3}-A \,b^{2} c^{2} d^{3}+2 B \,a^{3} c \,e^{3}-B \,a^{2} b^{2} e^{3}+3 B \,a^{2} b c d \,e^{2}-6 B \,a^{2} c^{2} d^{2} e +B a b \,c^{2} d^{3}\right ) x}{2 c^{3} \left (4 a c -b^{2}\right ) a}\right )}{\sqrt {\left (x^{4}+\frac {b \,x^{2}}{c}+\frac {a}{c}\right ) c}}+\frac {B \,e^{3} x \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 c^{2}}+\frac {\left (-\frac {e \left (A b c \,e^{2}-3 c^{2} d e A +B a c \,e^{2}-B \,b^{2} e^{2}+3 B b c d e -3 B \,c^{2} d^{2}\right )}{c^{3}}+\frac {A a b c \,e^{3}-3 A a \,c^{2} d \,e^{2}+A \,c^{3} d^{3}+B \,a^{2} c \,e^{3}-B a \,b^{2} e^{3}+3 B a b c d \,e^{2}-3 B a \,c^{2} d^{2} e}{c^{3} a}-\frac {A \,a^{2} b c \,e^{3}-6 A \,a^{2} c^{2} d \,e^{2}+3 A a b \,c^{2} d^{2} e +2 A a \,c^{3} d^{3}-A \,b^{2} c^{2} d^{3}+2 B \,a^{3} c \,e^{3}-B \,a^{2} b^{2} e^{3}+3 B \,a^{2} b c d \,e^{2}-6 B \,a^{2} c^{2} d^{2} e +B a b \,c^{2} d^{3}}{a \,c^{2} \left (4 a c -b^{2}\right )}-\frac {a B \,e^{3}}{3 c^{2}}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (\frac {e^{2} \left (A c e -B b e +3 B c d \right )}{c^{2}}+\frac {2 A \,a^{2} c^{2} e^{3}-A a \,b^{2} c \,e^{3}+3 A a b \,c^{2} d \,e^{2}-6 A a \,c^{3} d^{2} e +A b \,c^{3} d^{3}-3 B \,a^{2} b c \,e^{3}+6 B \,a^{2} c^{2} d \,e^{2}+B a \,b^{3} e^{3}-3 B a \,b^{2} c d \,e^{2}+3 B a b \,c^{2} d^{2} e -2 B a \,c^{3} d^{3}}{a \,c^{2} \left (4 a c -b^{2}\right )}-\frac {2 e^{3} B b}{3 c^{2}}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\EllipticE \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\)	\(1141\)
default	\(\text {Expression too large to display}\)	\(2445\)
risch	\(\text {Expression too large to display}\)	\(2482\)

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,method=_RETURNVERBOSE)

[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+b/c*x^2+a/c)*c)
^(1/2)+1/3/c^2*x*(c*x^4+b*x^2+a)^(1/2)+1/4*(-a*(2*a*c-b^2)/(4*a*c-b^2)/c^2-1/3*a/c^2)*2^(1/2)/((-b+(-4*a*c+b^2
)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/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)*((-b+(-4*a*c+b^2)^(1/2))/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*(3*a*c-b^2)/c^2/(4*a*c-b^2))*a*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-
4*a*c+b^2)^(1/2))/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)*((-b+(-4*a*c+b^2)^(1/2))/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)*((-b+(-4*a*c+b^2)^(1/2))/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+b/c*x^2+a/c)*c)^
(1/2)-1/4*a*b/c/(4*a*c-b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/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)*((-b+(-4*a*c+b^2)^(1/
2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))-1/2*(1/c+(2*a*c-b^2)/c/(4*a*c-b^2))*a*2^(1/2)/((-b
+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/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)*((-b+(-4*a*c+b^2)^(1/2))/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)*((-b+(-4*a*c+b^2)^(1/2))/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/c/(4*a*c-b^2)*x^3+a/c/(4*a*c-b^2)*
x)/((x^4+b/c*x^2+a/c)*c)^(1/2)+1/2*a/(4*a*c-b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^
2)^(1/2))/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)*
((-b+(-4*a*c+b^2)^(1/2))/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)/(
(-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/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)*((-b+(-4*a*c+b^2)^(1/2))/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)*((-b+(-4*a*c+b^2)^(1/2))/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+b/c*x^2+a/c)*c)^(1/2)-1/4*b/(4*a*c-b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)
^(1/2))/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)*((
-b+(-4*a*c+b^2)^(1/2))/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)/((-b+(-
4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/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)*((-b+(-4*a*c+b^2)^(1/2))/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)*((-b+(-4*a*c+b^2)^(1/2))/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+b/c*x^2+a/c)*c)^(1/2)+1/4*(1/a-(2*a*c-b^2)/a/(4*a*c-b^2))*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2
*(-b+(-4*a*c+b^2)^(1/2))/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)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))-1/2*b*c/(4*a*c
-b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/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)*((-b+(-4*a*c+b^2)^(
1/2))/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)*((-b+(-4*a*c+b^2)^(1/2))
/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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> Symbolic function elliptic_ec takes exactly 1 arguments (2 given)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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