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

Optimal. Leaf size=755 \[ \frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right ) \left (\frac{\sqrt{c} \left (7 A c \left (4 a b e^3-15 a c d e^2+15 c^2 d^3\right )-a B e \left (-c e (25 a e+84 b d)+24 b^2 e^2+105 c^2 d^2\right )\right )}{\sqrt{a}}+7 A c e \left (-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right )+B \left (-21 c^2 d e (9 a e+10 b d)+8 b c e^2 (13 a e+21 b d)-48 b^3 e^3+105 c^3 d^3\right )\right )}{210 c^{15/4} \sqrt{a+b x^2+c x^4}}+\frac{e x \sqrt{a+b x^2+c x^4} \left (B \left (-c e (25 a e+84 b d)+24 b^2 e^2+105 c^2 d^2\right )+7 A c e (15 c d-4 b e)\right )}{105 c^3}+\frac{x \sqrt{a+b x^2+c x^4} \left (7 A c e \left (-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right )+B \left (-21 c^2 d e (9 a e+10 b d)+8 b c e^2 (13 a e+21 b d)-48 b^3 e^3+105 c^3 d^3\right )\right )}{105 c^{7/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right ) \left (7 A c e \left (-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right )+B \left (-21 c^2 d e (9 a e+10 b d)+8 b c e^2 (13 a e+21 b d)-48 b^3 e^3+105 c^3 d^3\right )\right )}{105 c^{15/4} \sqrt{a+b x^2+c x^4}}+\frac{e^2 x^3 \sqrt{a+b x^2+c x^4} (7 A c e-6 b B e+21 B c d)}{35 c^2}+\frac{B e^3 x^5 \sqrt{a+b x^2+c x^4}}{7 c} \]

[Out]

(e*(7*A*c*e*(15*c*d - 4*b*e) + B*(105*c^2*d^2 + 24*b^2*e^2 - c*e*(84*b*d + 25*a*e)))*x*Sqrt[a + b*x^2 + c*x^4]
)/(105*c^3) + (e^2*(21*B*c*d - 6*b*B*e + 7*A*c*e)*x^3*Sqrt[a + b*x^2 + c*x^4])/(35*c^2) + (B*e^3*x^5*Sqrt[a +
b*x^2 + c*x^4])/(7*c) + ((7*A*c*e*(45*c^2*d^2 + 8*b^2*e^2 - 3*c*e*(10*b*d + 3*a*e)) + B*(105*c^3*d^3 - 48*b^3*
e^3 - 21*c^2*d*e*(10*b*d + 9*a*e) + 8*b*c*e^2*(21*b*d + 13*a*e)))*x*Sqrt[a + b*x^2 + c*x^4])/(105*c^(7/2)*(Sqr
t[a] + Sqrt[c]*x^2)) - (a^(1/4)*(7*A*c*e*(45*c^2*d^2 + 8*b^2*e^2 - 3*c*e*(10*b*d + 3*a*e)) + B*(105*c^3*d^3 -
48*b^3*e^3 - 21*c^2*d*e*(10*b*d + 9*a*e) + 8*b*c*e^2*(21*b*d + 13*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])/
(105*c^(15/4)*Sqrt[a + b*x^2 + c*x^4]) + (a^(1/4)*(7*A*c*e*(45*c^2*d^2 + 8*b^2*e^2 - 3*c*e*(10*b*d + 3*a*e)) +
B*(105*c^3*d^3 - 48*b^3*e^3 - 21*c^2*d*e*(10*b*d + 9*a*e) + 8*b*c*e^2*(21*b*d + 13*a*e)) + (Sqrt[c]*(7*A*c*(1
5*c^2*d^3 - 15*a*c*d*e^2 + 4*a*b*e^3) - a*B*e*(105*c^2*d^2 + 24*b^2*e^2 - c*e*(84*b*d + 25*a*e))))/Sqrt[a])*(S
qrt[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])/(210*c^(15/4)*Sqrt[a + b*x^2 + c*x^4])

________________________________________________________________________________________

Rubi [A]  time = 1.35688, antiderivative size = 755, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {1679, 1197, 1103, 1195} \[ \frac{e x \sqrt{a+b x^2+c x^4} \left (B \left (-c e (25 a e+84 b d)+24 b^2 e^2+105 c^2 d^2\right )+7 A c e (15 c d-4 b e)\right )}{105 c^3}+\frac{x \sqrt{a+b x^2+c x^4} \left (7 A c e \left (-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right )+B \left (-21 c^2 d e (9 a e+10 b d)+8 b c e^2 (13 a e+21 b d)-48 b^3 e^3+105 c^3 d^3\right )\right )}{105 c^{7/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right ) \left (\frac{\sqrt{c} \left (7 A c \left (4 a b e^3-15 a c d e^2+15 c^2 d^3\right )-a B e \left (-c e (25 a e+84 b d)+24 b^2 e^2+105 c^2 d^2\right )\right )}{\sqrt{a}}+7 A c e \left (-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right )+B \left (-21 c^2 d e (9 a e+10 b d)+8 b c e^2 (13 a e+21 b d)-48 b^3 e^3+105 c^3 d^3\right )\right )}{210 c^{15/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right ) \left (7 A c e \left (-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right )+B \left (-21 c^2 d e (9 a e+10 b d)+8 b c e^2 (13 a e+21 b d)-48 b^3 e^3+105 c^3 d^3\right )\right )}{105 c^{15/4} \sqrt{a+b x^2+c x^4}}+\frac{e^2 x^3 \sqrt{a+b x^2+c x^4} (7 A c e-6 b B e+21 B c d)}{35 c^2}+\frac{B e^3 x^5 \sqrt{a+b x^2+c x^4}}{7 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*(7*A*c*e*(15*c*d - 4*b*e) + B*(105*c^2*d^2 + 24*b^2*e^2 - c*e*(84*b*d + 25*a*e)))*x*Sqrt[a + b*x^2 + c*x^4]
)/(105*c^3) + (e^2*(21*B*c*d - 6*b*B*e + 7*A*c*e)*x^3*Sqrt[a + b*x^2 + c*x^4])/(35*c^2) + (B*e^3*x^5*Sqrt[a +
b*x^2 + c*x^4])/(7*c) + ((7*A*c*e*(45*c^2*d^2 + 8*b^2*e^2 - 3*c*e*(10*b*d + 3*a*e)) + B*(105*c^3*d^3 - 48*b^3*
e^3 - 21*c^2*d*e*(10*b*d + 9*a*e) + 8*b*c*e^2*(21*b*d + 13*a*e)))*x*Sqrt[a + b*x^2 + c*x^4])/(105*c^(7/2)*(Sqr
t[a] + Sqrt[c]*x^2)) - (a^(1/4)*(7*A*c*e*(45*c^2*d^2 + 8*b^2*e^2 - 3*c*e*(10*b*d + 3*a*e)) + B*(105*c^3*d^3 -
48*b^3*e^3 - 21*c^2*d*e*(10*b*d + 9*a*e) + 8*b*c*e^2*(21*b*d + 13*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])/
(105*c^(15/4)*Sqrt[a + b*x^2 + c*x^4]) + (a^(1/4)*(7*A*c*e*(45*c^2*d^2 + 8*b^2*e^2 - 3*c*e*(10*b*d + 3*a*e)) +
B*(105*c^3*d^3 - 48*b^3*e^3 - 21*c^2*d*e*(10*b*d + 9*a*e) + 8*b*c*e^2*(21*b*d + 13*a*e)) + (Sqrt[c]*(7*A*c*(1
5*c^2*d^3 - 15*a*c*d*e^2 + 4*a*b*e^3) - a*B*e*(105*c^2*d^2 + 24*b^2*e^2 - c*e*(84*b*d + 25*a*e))))/Sqrt[a])*(S
qrt[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])/(210*c^(15/4)*Sqrt[a + b*x^2 + c*x^4])

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}{\sqrt{a+b x^2+c x^4}} \, dx &=\frac{B e^3 x^5 \sqrt{a+b x^2+c x^4}}{7 c}+\frac{\int \frac{7 A c d^3+7 c d^2 (B d+3 A e) x^2+e \left (21 B c d^2+21 A c d e-5 a B e^2\right ) x^4+e^2 (21 B c d-6 b B e+7 A c e) x^6}{\sqrt{a+b x^2+c x^4}} \, dx}{7 c}\\ &=\frac{e^2 (21 B c d-6 b B e+7 A c e) x^3 \sqrt{a+b x^2+c x^4}}{35 c^2}+\frac{B e^3 x^5 \sqrt{a+b x^2+c x^4}}{7 c}+\frac{\int \frac{35 A c^2 d^3+\left (21 A c e \left (5 c d^2-a e^2\right )+B \left (35 c^2 d^3-63 a c d e^2+18 a b e^3\right )\right ) x^2+e \left (7 A c e (15 c d-4 b e)+B \left (105 c^2 d^2+24 b^2 e^2-c e (84 b d+25 a e)\right )\right ) x^4}{\sqrt{a+b x^2+c x^4}} \, dx}{35 c^2}\\ &=\frac{e \left (7 A c e (15 c d-4 b e)+B \left (105 c^2 d^2+24 b^2 e^2-c e (84 b d+25 a e)\right )\right ) x \sqrt{a+b x^2+c x^4}}{105 c^3}+\frac{e^2 (21 B c d-6 b B e+7 A c e) x^3 \sqrt{a+b x^2+c x^4}}{35 c^2}+\frac{B e^3 x^5 \sqrt{a+b x^2+c x^4}}{7 c}+\frac{\int \frac{7 A c \left (15 c^2 d^3-15 a c d e^2+4 a b e^3\right )-a B e \left (105 c^2 d^2+24 b^2 e^2-c e (84 b d+25 a e)\right )+\left (7 A c e \left (45 c^2 d^2+8 b^2 e^2-3 c e (10 b d+3 a e)\right )+B \left (105 c^3 d^3-48 b^3 e^3-21 c^2 d e (10 b d+9 a e)+8 b c e^2 (21 b d+13 a e)\right )\right ) x^2}{\sqrt{a+b x^2+c x^4}} \, dx}{105 c^3}\\ &=\frac{e \left (7 A c e (15 c d-4 b e)+B \left (105 c^2 d^2+24 b^2 e^2-c e (84 b d+25 a e)\right )\right ) x \sqrt{a+b x^2+c x^4}}{105 c^3}+\frac{e^2 (21 B c d-6 b B e+7 A c e) x^3 \sqrt{a+b x^2+c x^4}}{35 c^2}+\frac{B e^3 x^5 \sqrt{a+b x^2+c x^4}}{7 c}-\frac{\left (\sqrt{a} \left (7 A c e \left (45 c^2 d^2+8 b^2 e^2-3 c e (10 b d+3 a e)\right )+B \left (105 c^3 d^3-48 b^3 e^3-21 c^2 d e (10 b d+9 a e)+8 b c e^2 (21 b d+13 a e)\right )\right )\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+b x^2+c x^4}} \, dx}{105 c^{7/2}}+\frac{\left (\sqrt{a} \left (7 A c e \left (45 c^2 d^2+8 b^2 e^2-3 c e (10 b d+3 a e)\right )+B \left (105 c^3 d^3-48 b^3 e^3-21 c^2 d e (10 b d+9 a e)+8 b c e^2 (21 b d+13 a e)\right )+\frac{\sqrt{c} \left (7 A c \left (15 c^2 d^3-15 a c d e^2+4 a b e^3\right )-a B e \left (105 c^2 d^2+24 b^2 e^2-c e (84 b d+25 a e)\right )\right )}{\sqrt{a}}\right )\right ) \int \frac{1}{\sqrt{a+b x^2+c x^4}} \, dx}{105 c^{7/2}}\\ &=\frac{e \left (7 A c e (15 c d-4 b e)+B \left (105 c^2 d^2+24 b^2 e^2-c e (84 b d+25 a e)\right )\right ) x \sqrt{a+b x^2+c x^4}}{105 c^3}+\frac{e^2 (21 B c d-6 b B e+7 A c e) x^3 \sqrt{a+b x^2+c x^4}}{35 c^2}+\frac{B e^3 x^5 \sqrt{a+b x^2+c x^4}}{7 c}+\frac{\left (7 A c e \left (45 c^2 d^2+8 b^2 e^2-3 c e (10 b d+3 a e)\right )+B \left (105 c^3 d^3-48 b^3 e^3-21 c^2 d e (10 b d+9 a e)+8 b c e^2 (21 b d+13 a e)\right )\right ) x \sqrt{a+b x^2+c x^4}}{105 c^{7/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\sqrt [4]{a} \left (7 A c e \left (45 c^2 d^2+8 b^2 e^2-3 c e (10 b d+3 a e)\right )+B \left (105 c^3 d^3-48 b^3 e^3-21 c^2 d e (10 b d+9 a e)+8 b c e^2 (21 b d+13 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 )}{105 c^{15/4} \sqrt{a+b x^2+c x^4}}+\frac{\sqrt [4]{a} \left (7 A c e \left (45 c^2 d^2+8 b^2 e^2-3 c e (10 b d+3 a e)\right )+B \left (105 c^3 d^3-48 b^3 e^3-21 c^2 d e (10 b d+9 a e)+8 b c e^2 (21 b d+13 a e)\right )+\frac{\sqrt{c} \left (7 A c \left (15 c^2 d^3-15 a c d e^2+4 a b e^3\right )-a B e \left (105 c^2 d^2+24 b^2 e^2-c e (84 b d+25 a e)\right )\right )}{\sqrt{a}}\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 )}{210 c^{15/4} \sqrt{a+b x^2+c x^4}}\\ \end{align*}

Mathematica [C]  time = 6.50581, size = 4473, normalized size = 5.92 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[a + b*x^2 + c*x^4]*(-(e*(-105*B*c^2*d^2 + 84*b*B*c*d*e - 105*A*c^2*d*e - 24*b^2*B*e^2 + 28*A*b*c*e^2 + 25
*a*B*c*e^2)*x)/(105*c^3) + (e^2*(21*B*c*d - 6*b*B*e + 7*A*c*e)*x^3)/(35*c^2) + (B*e^3*x^5)/(7*c)) + ((((105*I)
/2)*B*c^2*(-b + Sqrt[b^2 - 4*a*c])*d^3*Sqrt[1 - (2*c*x^2)/(-b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 - (2*c*x^2)/(-b + S
qrt[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])] - 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])]))/(Sqrt[2]*Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]
) - ((105*I)*b*B*c*(-b + Sqrt[b^2 - 4*a*c])*d^2*e*Sqrt[1 - (2*c*x^2)/(-b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 - (2*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])] - 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])]))/(Sqrt[2]*Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x
^2 + c*x^4]) + (((315*I)/2)*A*c^2*(-b + Sqrt[b^2 - 4*a*c])*d^2*e*Sqrt[1 - (2*c*x^2)/(-b - Sqrt[b^2 - 4*a*c])]*
Sqrt[1 - (2*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])] - 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])]))/(Sqrt[2]*Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c])
)]*Sqrt[a + b*x^2 + c*x^4]) + ((42*I)*Sqrt[2]*b^2*B*(-b + Sqrt[b^2 - 4*a*c])*d*e^2*Sqrt[1 - (2*c*x^2)/(-b - Sq
rt[b^2 - 4*a*c])]*Sqrt[1 - (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]*(EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[-(c/(-b - Sqr
t[b^2 - 4*a*c]))]*x], (-b - Sqrt[b^2 - 4*a*c])/(-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])]))/(Sqrt[-(c/(-b - Sqrt[b^2
- 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) - ((105*I)*A*b*c*(-b + Sqrt[b^2 - 4*a*c])*d*e^2*Sqrt[1 - (2*c*x^2)/(-b -
Sqrt[b^2 - 4*a*c])]*Sqrt[1 - (2*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])] - 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])]))/(Sqrt[2]*Sqrt[-(c/(-b
- Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) - (((189*I)/2)*a*B*c*(-b + Sqrt[b^2 - 4*a*c])*d*e^2*Sqrt[1 -
(2*c*x^2)/(-b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 - (2*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])] - EllipticF[I*ArcSi
nh[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])]))/(Sqrt[
2]*Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) + ((14*I)*Sqrt[2]*A*b^2*(-b + Sqrt[b^2 - 4*a*c
])*e^3*Sqrt[1 - (2*c*x^2)/(-b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 - (2*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])] - E
llipticF[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])]))/(Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) + ((26*I)*Sqrt[2]*a*b*B*(-b + Sqrt[b^2
- 4*a*c])*e^3*Sqrt[1 - (2*c*x^2)/(-b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 - (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]*(Elli
pticE[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])] - 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])]))/(Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) - ((12*I)*Sqrt[2]*b^3*B*(-b +
Sqrt[b^2 - 4*a*c])*e^3*Sqrt[1 - (2*c*x^2)/(-b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 - (2*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])] - 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])]))/(c*Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) - (((63*I)/2)*a*A*c*
(-b + Sqrt[b^2 - 4*a*c])*e^3*Sqrt[1 - (2*c*x^2)/(-b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 - (2*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 + S
qrt[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])]))/(Sqrt[2]*Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) - ((105*
I)*A*c^3*d^3*Sqrt[1 - (2*c*x^2)/(-b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 - (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]*Ellipti
cF[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])
])/(Sqrt[2]*Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) + ((105*I)*a*B*c^2*d^2*e*Sqrt[1 - (2*
c*x^2)/(-b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 - (2*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])])/(Sqrt[2]*Sqrt[-(c/(-b
- Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) - ((42*I)*Sqrt[2]*a*b*B*c*d*e^2*Sqrt[1 - (2*c*x^2)/(-b - Sqrt[
b^2 - 4*a*c])]*Sqrt[1 - (2*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])])/(Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*Sqr
t[a + b*x^2 + c*x^4]) + ((105*I)*a*A*c^2*d*e^2*Sqrt[1 - (2*c*x^2)/(-b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 - (2*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])])/(Sqrt[2]*Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) + (
(12*I)*Sqrt[2]*a*b^2*B*e^3*Sqrt[1 - (2*c*x^2)/(-b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 - (2*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])])/(Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) - ((14*I)*Sqrt[2]*a*A*b*c*e^3*S
qrt[1 - (2*c*x^2)/(-b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 - (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]*EllipticF[I*ArcSinh[S
qrt[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])])/(Sqrt[-(c/(
-b - Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) - ((25*I)*a^2*B*c*e^3*Sqrt[1 - (2*c*x^2)/(-b - Sqrt[b^2 - 4
*a*c])]*Sqrt[1 - (2*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])])/(Sqrt[2]*Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*Sq
rt[a + b*x^2 + c*x^4]))/(105*c^3)

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Maple [B]  time = 0.021, size = 1708, normalized size = 2.3 \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)^3/(c*x^4+b*x^2+a)^(1/2),x)

[Out]

B*e^3*(1/7/c*x^5*(c*x^4+b*x^2+a)^(1/2)-6/35*b/c^2*x^3*(c*x^4+b*x^2+a)^(1/2)+1/3*(-5/7*a/c+24/35*b^2/c^2)/c*x*(
c*x^4+b*x^2+a)^(1/2)-1/12*(-5/7*a/c+24/35*b^2/c^2)/c*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)*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*(18/35*b/c^2*a-2/3*(-5
/7*a/c+24/35*b^2/c^2)/c*b)*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^3+3*B*d*e^2)*(1/5/c*x^3*(c*
x^4+b*x^2+a)^(1/2)-4/15*b/c^2*x*(c*x^4+b*x^2+a)^(1/2)+1/15*b/c^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)*Elliptic
F(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*(-3/5*a/c+
8/15*b^2/c^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)*(1/3/c*x*(c*x^4+b*x^2+a
)^(1/2)-1/12*a/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)*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))+1/3*b/c*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))))-1/2*(
3*A*d^2*e+B*d^3)*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)))+1/4*A*d^3*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))

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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}}{\sqrt{c x^{4} + b x^{2} + a}}\,{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)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{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}}{\sqrt{c x^{4} + b x^{2} + a}}, 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)^(1/2),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), 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 )^{3}}{\sqrt{a + b x^{2} + c x^{4}}}\, dx \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)**(1/2),x)

[Out]

Integral((A + B*x**2)*(d + e*x**2)**3/sqrt(a + b*x**2 + c*x**4), 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 )}^{3}}{\sqrt{c x^{4} + b x^{2} + a}}\,{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)^(1/2),x, algorithm="giac")

[Out]

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