3.126 \(\int \frac{x^4 (d+e x^2+f x^4+g x^6)}{(a+b x^2+c x^4)^2} \, dx\)

Optimal. Leaf size=594 \[ \frac{x \left (x^2 \left (-b^2 c (c e-4 a g)+b c^2 (c d-3 a f)+2 a c^2 (c e-a g)+b^3 c f+b^4 (-g)\right )+a \left (-c^2 (2 a f+b e)+b c (3 a g+b f)+b^3 (-g)+2 c^3 d\right )\right )}{2 c^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (-\frac{-b^2 c^2 (19 a f+c d)-b^3 c (c e-34 a g)+4 a b c^2 (2 c e-13 a g)-4 a c^3 (c d-5 a f)+3 b^4 c f-5 b^5 g}{\sqrt{b^2-4 a c}}-b^2 c (c e-24 a g)-b c^2 (13 a f+c d)+2 a c^2 (3 c e-7 a g)+3 b^3 c f-5 b^4 g\right )}{2 \sqrt{2} c^{7/2} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (\frac{-b^2 c^2 (19 a f+c d)-b^3 c (c e-34 a g)+4 a b c^2 (2 c e-13 a g)-4 a c^3 (c d-5 a f)+3 b^4 c f-5 b^5 g}{\sqrt{b^2-4 a c}}-b^2 c (c e-24 a g)-b c^2 (13 a f+c d)+2 a c^2 (3 c e-7 a g)+3 b^3 c f-5 b^4 g\right )}{2 \sqrt{2} c^{7/2} \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x (c f-2 b g)}{c^3}+\frac{g x^3}{3 c^2} \]

[Out]

((c*f - 2*b*g)*x)/c^3 + (g*x^3)/(3*c^2) + (x*(a*(2*c^3*d - c^2*(b*e + 2*a*f) - b^3*g + b*c*(b*f + 3*a*g)) + (b
^3*c*f + b*c^2*(c*d - 3*a*f) - b^4*g - b^2*c*(c*e - 4*a*g) + 2*a*c^2*(c*e - a*g))*x^2))/(2*c^3*(b^2 - 4*a*c)*(
a + b*x^2 + c*x^4)) - ((3*b^3*c*f - b*c^2*(c*d + 13*a*f) - 5*b^4*g - b^2*c*(c*e - 24*a*g) + 2*a*c^2*(3*c*e - 7
*a*g) - (3*b^4*c*f - 4*a*c^3*(c*d - 5*a*f) - b^2*c^2*(c*d + 19*a*f) - 5*b^5*g - b^3*c*(c*e - 34*a*g) + 4*a*b*c
^2*(2*c*e - 13*a*g))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*c^
(7/2)*(b^2 - 4*a*c)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - ((3*b^3*c*f - b*c^2*(c*d + 13*a*f) - 5*b^4*g - b^2*c*(c*e -
24*a*g) + 2*a*c^2*(3*c*e - 7*a*g) + (3*b^4*c*f - 4*a*c^3*(c*d - 5*a*f) - b^2*c^2*(c*d + 19*a*f) - 5*b^5*g - b
^3*c*(c*e - 34*a*g) + 4*a*b*c^2*(2*c*e - 13*a*g))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[
b^2 - 4*a*c]]])/(2*Sqrt[2]*c^(7/2)*(b^2 - 4*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

________________________________________________________________________________________

Rubi [A]  time = 14.1129, antiderivative size = 594, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {1668, 1676, 1166, 205} \[ \frac{x \left (x^2 \left (-b^2 c (c e-4 a g)+b c^2 (c d-3 a f)+2 a c^2 (c e-a g)+b^3 c f+b^4 (-g)\right )+a \left (-c^2 (2 a f+b e)+b c (3 a g+b f)+b^3 (-g)+2 c^3 d\right )\right )}{2 c^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (-\frac{-b^2 c^2 (19 a f+c d)-b^3 c (c e-34 a g)+4 a b c^2 (2 c e-13 a g)-4 a c^3 (c d-5 a f)+3 b^4 c f-5 b^5 g}{\sqrt{b^2-4 a c}}-b^2 c (c e-24 a g)-b c^2 (13 a f+c d)+2 a c^2 (3 c e-7 a g)+3 b^3 c f-5 b^4 g\right )}{2 \sqrt{2} c^{7/2} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (\frac{-b^2 c^2 (19 a f+c d)-b^3 c (c e-34 a g)+4 a b c^2 (2 c e-13 a g)-4 a c^3 (c d-5 a f)+3 b^4 c f-5 b^5 g}{\sqrt{b^2-4 a c}}-b^2 c (c e-24 a g)-b c^2 (13 a f+c d)+2 a c^2 (3 c e-7 a g)+3 b^3 c f-5 b^4 g\right )}{2 \sqrt{2} c^{7/2} \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x (c f-2 b g)}{c^3}+\frac{g x^3}{3 c^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(d + e*x^2 + f*x^4 + g*x^6))/(a + b*x^2 + c*x^4)^2,x]

[Out]

((c*f - 2*b*g)*x)/c^3 + (g*x^3)/(3*c^2) + (x*(a*(2*c^3*d - c^2*(b*e + 2*a*f) - b^3*g + b*c*(b*f + 3*a*g)) + (b
^3*c*f + b*c^2*(c*d - 3*a*f) - b^4*g - b^2*c*(c*e - 4*a*g) + 2*a*c^2*(c*e - a*g))*x^2))/(2*c^3*(b^2 - 4*a*c)*(
a + b*x^2 + c*x^4)) - ((3*b^3*c*f - b*c^2*(c*d + 13*a*f) - 5*b^4*g - b^2*c*(c*e - 24*a*g) + 2*a*c^2*(3*c*e - 7
*a*g) - (3*b^4*c*f - 4*a*c^3*(c*d - 5*a*f) - b^2*c^2*(c*d + 19*a*f) - 5*b^5*g - b^3*c*(c*e - 34*a*g) + 4*a*b*c
^2*(2*c*e - 13*a*g))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*c^
(7/2)*(b^2 - 4*a*c)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - ((3*b^3*c*f - b*c^2*(c*d + 13*a*f) - 5*b^4*g - b^2*c*(c*e -
24*a*g) + 2*a*c^2*(3*c*e - 7*a*g) + (3*b^4*c*f - 4*a*c^3*(c*d - 5*a*f) - b^2*c^2*(c*d + 19*a*f) - 5*b^5*g - b
^3*c*(c*e - 34*a*g) + 4*a*b*c^2*(2*c*e - 13*a*g))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[
b^2 - 4*a*c]]])/(2*Sqrt[2]*c^(7/2)*(b^2 - 4*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

Rule 1668

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainde
r[x^m*Pq, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 2]}, S
imp[(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)*PolynomialQuotient[x^m*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] && GtQ[Expon[Pq, x^2], 1] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[m/2, 0]

Rule 1676

Int[(Pq_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^2 + c*x^4), x], x
] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1

Rule 1166

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

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{x^4 \left (d+e x^2+f x^4+g x^6\right )}{\left (a+b x^2+c x^4\right )^2} \, dx &=\frac{x \left (a \left (2 c^3 d-c^2 (b e+2 a f)-b^3 g+b c (b f+3 a g)\right )+\left (b^3 c f+b c^2 (c d-3 a f)-b^4 g-b^2 c (c e-4 a g)+2 a c^2 (c e-a g)\right ) x^2\right )}{2 c^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\int \frac{\frac{a^2 \left (2 c^3 d-c^2 (b e+2 a f)-b^3 g+b c (b f+3 a g)\right )}{c^3}+\frac{a \left (b^3 c f-b c^2 (c d+5 a f)-b^4 g-b^2 c (c e-6 a g)+6 a c^2 (c e-a g)\right ) x^2}{c^3}-\frac{2 a \left (b^2-4 a c\right ) (c f-b g) x^4}{c^2}+2 a \left (4 a-\frac{b^2}{c}\right ) g x^6}{a+b x^2+c x^4} \, dx}{2 a \left (b^2-4 a c\right )}\\ &=\frac{x \left (a \left (2 c^3 d-c^2 (b e+2 a f)-b^3 g+b c (b f+3 a g)\right )+\left (b^3 c f+b c^2 (c d-3 a f)-b^4 g-b^2 c (c e-4 a g)+2 a c^2 (c e-a g)\right ) x^2\right )}{2 c^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\int \left (-\frac{2 a \left (b^2-4 a c\right ) (c f-2 b g)}{c^3}-\frac{2 a \left (b^2-4 a c\right ) g x^2}{c^2}+\frac{a^2 \left (2 c^3 d-c^2 (b e+10 a f)-5 b^3 g+b c (3 b f+19 a g)\right )+a \left (3 b^3 c f-b c^2 (c d+13 a f)-5 b^4 g-b^2 c (c e-24 a g)+2 a c^2 (3 c e-7 a g)\right ) x^2}{c^3 \left (a+b x^2+c x^4\right )}\right ) \, dx}{2 a \left (b^2-4 a c\right )}\\ &=\frac{(c f-2 b g) x}{c^3}+\frac{g x^3}{3 c^2}+\frac{x \left (a \left (2 c^3 d-c^2 (b e+2 a f)-b^3 g+b c (b f+3 a g)\right )+\left (b^3 c f+b c^2 (c d-3 a f)-b^4 g-b^2 c (c e-4 a g)+2 a c^2 (c e-a g)\right ) x^2\right )}{2 c^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\int \frac{a^2 \left (2 c^3 d-c^2 (b e+10 a f)-5 b^3 g+b c (3 b f+19 a g)\right )+a \left (3 b^3 c f-b c^2 (c d+13 a f)-5 b^4 g-b^2 c (c e-24 a g)+2 a c^2 (3 c e-7 a g)\right ) x^2}{a+b x^2+c x^4} \, dx}{2 a c^3 \left (b^2-4 a c\right )}\\ &=\frac{(c f-2 b g) x}{c^3}+\frac{g x^3}{3 c^2}+\frac{x \left (a \left (2 c^3 d-c^2 (b e+2 a f)-b^3 g+b c (b f+3 a g)\right )+\left (b^3 c f+b c^2 (c d-3 a f)-b^4 g-b^2 c (c e-4 a g)+2 a c^2 (c e-a g)\right ) x^2\right )}{2 c^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\left (3 b^3 c f-b c^2 (c d+13 a f)-5 b^4 g-b^2 c (c e-24 a g)+2 a c^2 (3 c e-7 a g)-\frac{3 b^4 c f-4 a c^3 (c d-5 a f)-b^2 c^2 (c d+19 a f)-5 b^5 g-b^3 c (c e-34 a g)+4 a b c^2 (2 c e-13 a g)}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{4 c^3 \left (b^2-4 a c\right )}-\frac{\left (3 b^3 c f-b c^2 (c d+13 a f)-5 b^4 g-b^2 c (c e-24 a g)+2 a c^2 (3 c e-7 a g)+\frac{3 b^4 c f-4 a c^3 (c d-5 a f)-b^2 c^2 (c d+19 a f)-5 b^5 g-b^3 c (c e-34 a g)+4 a b c^2 (2 c e-13 a g)}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{4 c^3 \left (b^2-4 a c\right )}\\ &=\frac{(c f-2 b g) x}{c^3}+\frac{g x^3}{3 c^2}+\frac{x \left (a \left (2 c^3 d-c^2 (b e+2 a f)-b^3 g+b c (b f+3 a g)\right )+\left (b^3 c f+b c^2 (c d-3 a f)-b^4 g-b^2 c (c e-4 a g)+2 a c^2 (c e-a g)\right ) x^2\right )}{2 c^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\left (3 b^3 c f-b c^2 (c d+13 a f)-5 b^4 g-b^2 c (c e-24 a g)+2 a c^2 (3 c e-7 a g)-\frac{3 b^4 c f-4 a c^3 (c d-5 a f)-b^2 c^2 (c d+19 a f)-5 b^5 g-b^3 c (c e-34 a g)+4 a b c^2 (2 c e-13 a g)}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} c^{7/2} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left (3 b^3 c f-b c^2 (c d+13 a f)-5 b^4 g-b^2 c (c e-24 a g)+2 a c^2 (3 c e-7 a g)+\frac{3 b^4 c f-4 a c^3 (c d-5 a f)-b^2 c^2 (c d+19 a f)-5 b^5 g-b^3 c (c e-34 a g)+4 a b c^2 (2 c e-13 a g)}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} c^{7/2} \left (b^2-4 a c\right ) \sqrt{b+\sqrt{b^2-4 a c}}}\\ \end{align*}

Mathematica [A]  time = 2.84651, size = 721, normalized size = 1.21 \[ \frac{\frac{6 \sqrt{c} x \left (a^2 c \left (3 b g-2 c \left (f+g x^2\right )\right )+a \left (b^2 c \left (f+4 g x^2\right )+b^3 (-g)-b c^2 \left (e+3 f x^2\right )+2 c^3 \left (d+e x^2\right )\right )+b x^2 \left (b^2 c f+b^3 (-g)-b c^2 e+c^3 d\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{3 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (-b^2 c \left (-c e \sqrt{b^2-4 a c}+24 a g \sqrt{b^2-4 a c}+19 a c f+c^2 d\right )+b c^2 \left (c \left (d \sqrt{b^2-4 a c}+8 a e\right )+13 a \left (f \sqrt{b^2-4 a c}-4 a g\right )\right )+2 a c^2 \left (-3 c e \sqrt{b^2-4 a c}+7 a g \sqrt{b^2-4 a c}+10 a c f-2 c^2 d\right )-b^3 c \left (3 f \sqrt{b^2-4 a c}-34 a g+c e\right )+b^4 \left (5 g \sqrt{b^2-4 a c}+3 c f\right )-5 b^5 g\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (b^2 c \left (c e \sqrt{b^2-4 a c}-24 a g \sqrt{b^2-4 a c}+19 a c f+c^2 d\right )+b c^2 \left (c \left (d \sqrt{b^2-4 a c}-8 a e\right )+13 a \left (f \sqrt{b^2-4 a c}+4 a g\right )\right )+2 a c^2 \left (-3 c e \sqrt{b^2-4 a c}+7 a g \sqrt{b^2-4 a c}-10 a c f+2 c^2 d\right )+b^3 c \left (-3 f \sqrt{b^2-4 a c}-34 a g+c e\right )+b^4 \left (5 g \sqrt{b^2-4 a c}-3 c f\right )+5 b^5 g\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+12 \sqrt{c} x (c f-2 b g)+4 c^{3/2} g x^3}{12 c^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(d + e*x^2 + f*x^4 + g*x^6))/(a + b*x^2 + c*x^4)^2,x]

[Out]

(12*Sqrt[c]*(c*f - 2*b*g)*x + 4*c^(3/2)*g*x^3 + (6*Sqrt[c]*x*(b*(c^3*d - b*c^2*e + b^2*c*f - b^3*g)*x^2 + a^2*
c*(3*b*g - 2*c*(f + g*x^2)) + a*(-(b^3*g) + 2*c^3*(d + e*x^2) - b*c^2*(e + 3*f*x^2) + b^2*c*(f + 4*g*x^2))))/(
(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (3*Sqrt[2]*(-5*b^5*g - b^3*c*(c*e + 3*Sqrt[b^2 - 4*a*c]*f - 34*a*g) + b^4
*(3*c*f + 5*Sqrt[b^2 - 4*a*c]*g) + 2*a*c^2*(-2*c^2*d - 3*c*Sqrt[b^2 - 4*a*c]*e + 10*a*c*f + 7*a*Sqrt[b^2 - 4*a
*c]*g) - b^2*c*(c^2*d - c*Sqrt[b^2 - 4*a*c]*e + 19*a*c*f + 24*a*Sqrt[b^2 - 4*a*c]*g) + b*c^2*(c*(Sqrt[b^2 - 4*
a*c]*d + 8*a*e) + 13*a*(Sqrt[b^2 - 4*a*c]*f - 4*a*g)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]]
)/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[2]*(5*b^5*g + b^3*c*(c*e - 3*Sqrt[b^2 - 4*a*c]*f
- 34*a*g) + b^4*(-3*c*f + 5*Sqrt[b^2 - 4*a*c]*g) + b^2*c*(c^2*d + c*Sqrt[b^2 - 4*a*c]*e + 19*a*c*f - 24*a*Sqr
t[b^2 - 4*a*c]*g) + 2*a*c^2*(2*c^2*d - 3*c*Sqrt[b^2 - 4*a*c]*e - 10*a*c*f + 7*a*Sqrt[b^2 - 4*a*c]*g) + b*c^2*(
c*(Sqrt[b^2 - 4*a*c]*d - 8*a*e) + 13*a*(Sqrt[b^2 - 4*a*c]*f + 4*a*g)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqr
t[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(12*c^(7/2))

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Maple [B]  time = 0.059, size = 3028, normalized size = 5.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(g*x^6+f*x^4+e*x^2+d)/(c*x^4+b*x^2+a)^2,x)

[Out]

-3/4/c^2/(4*a*c-b^2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(
1/2))*b^3*f+1/4/c/(4*a*c-b^2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2
)-b)*c)^(1/2))*b^2*e+5/(4*a*c-b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(
1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*a^2*f-1/4/(4*a*c-b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b
)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*b^2*d+5/(4*a*c-b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2
)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a^2*f-1/4/(4*a*c-b^2)/
(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2
))*b^2*d+3/2/c/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^3*a*b*f+1/2/c/(c*x^4+b*x^2+a)*a/(4*a*c-b^2)*x*b*e+1/c/(c*x^4+b*x^
2+a)/(4*a*c-b^2)*x^3*a^2*g+1/2/c^3/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^3*b^4*g-1/2/c^2/(c*x^4+b*x^2+a)*a/(4*a*c-b^2)
*x*b^2*f+3/4/c^2/(4*a*c-b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2
))*c)^(1/2))*b^3*f-1/4/c/(4*a*c-b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b
^2)^(1/2))*c)^(1/2))*b^2*e-13/c/(4*a*c-b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan
(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a^2*b*g+17/2/c^2/(4*a*c-b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4
*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*b^3*g-13/c/(4*a*c-b^2)/(-4*a*
c+b^2)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*a^
2*b*g+17/2/c^2/(4*a*c-b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((
-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*a*b^3*g-19/4/c/(4*a*c-b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c
)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*a*b^2*f-19/4/c/(4*a*c-b^2)/(-4*a*c+b^2)^(1/2)*2^
(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*b^2*f-1/2/c^2/(c
*x^4+b*x^2+a)/(4*a*c-b^2)*x^3*b^3*f+1/2/c/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^3*b^2*e+1/c/(c*x^4+b*x^2+a)*a^2/(4*a*c
-b^2)*x*f+1/4/(4*a*c-b^2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)
*c)^(1/2))*b*d+3/2/(4*a*c-b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1
/2))*c)^(1/2))*a*e-1/4/(4*a*c-b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2
)^(1/2))*c)^(1/2))*b*d-3/2/(4*a*c-b^2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+
b^2)^(1/2)-b)*c)^(1/2))*a*e-2/c^3*x*b*g-1/(c*x^4+b*x^2+a)*a/(4*a*c-b^2)*x*d-1/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^3*
a*e-1/2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^3*b*d+f*x/c^2+1/2/c^3/(c*x^4+b*x^2+a)*a/(4*a*c-b^2)*x*b^3*g-2/c^2/(c*x^4
+b*x^2+a)/(4*a*c-b^2)*x^3*a*b^2*g-3/2/c^2/(c*x^4+b*x^2+a)*a^2/(4*a*c-b^2)*x*b*g+7/2/c/(4*a*c-b^2)*2^(1/2)/(((-
4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*a^2*g+5/4/c^3/(4*a*c-b^2)*2
^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*b^4*g-7/2/c/(4*a
*c-b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a^2*g-5/
4/c^3/(4*a*c-b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)
)*b^4*g+1/3*g*x^3/c^2-5/4/c^3/(4*a*c-b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(
c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*b^5*g+6/c^2/(4*a*c-b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)
*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*b^2*g-5/4/c^3/(4*a*c-b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2)/(
(b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^5*g-6/c^2/(4*a*c-b^2)*2
^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*a*b^2*g+3/4/c^2/
(4*a*c-b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/
2)-b)*c)^(1/2))*b^4*f-1/4/c/(4*a*c-b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*
x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*b^3*e-13/4/c/(4*a*c-b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*
arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*b*f-c/(4*a*c-b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*
c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*d+3/4/c^2/(4*a*c-b^2)/(-4*a*c+b^
2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^4*f-1
/4/c/(4*a*c-b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^
2)^(1/2))*c)^(1/2))*b^3*e+2/(4*a*c-b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*
x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*a*b*e+2/(4*a*c-b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1
/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*b*e+13/4/c/(4*a*c-b^2)*2^(1/2)/(((-4*a*c+
b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*a*b*f-c/(4*a*c-b^2)/(-4*a*c+b^2)^
(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*a*d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (b c^{3} d -{\left (b^{2} c^{2} - 2 \, a c^{3}\right )} e +{\left (b^{3} c - 3 \, a b c^{2}\right )} f -{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} g\right )} x^{3} +{\left (2 \, a c^{3} d - a b c^{2} e +{\left (a b^{2} c - 2 \, a^{2} c^{2}\right )} f -{\left (a b^{3} - 3 \, a^{2} b c\right )} g\right )} x}{2 \,{\left (a b^{2} c^{3} - 4 \, a^{2} c^{4} +{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} +{\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{2}\right )}} + \frac{-\int \frac{2 \, a c^{3} d - a b c^{2} e -{\left (b c^{3} d +{\left (b^{2} c^{2} - 6 \, a c^{3}\right )} e -{\left (3 \, b^{3} c - 13 \, a b c^{2}\right )} f +{\left (5 \, b^{4} - 24 \, a b^{2} c + 14 \, a^{2} c^{2}\right )} g\right )} x^{2} +{\left (3 \, a b^{2} c - 10 \, a^{2} c^{2}\right )} f -{\left (5 \, a b^{3} - 19 \, a^{2} b c\right )} g}{c x^{4} + b x^{2} + a}\,{d x}}{2 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )}} + \frac{c g x^{3} + 3 \,{\left (c f - 2 \, b g\right )} x}{3 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(g*x^6+f*x^4+e*x^2+d)/(c*x^4+b*x^2+a)^2,x, algorithm="maxima")

[Out]

1/2*((b*c^3*d - (b^2*c^2 - 2*a*c^3)*e + (b^3*c - 3*a*b*c^2)*f - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*g)*x^3 + (2*a*c^
3*d - a*b*c^2*e + (a*b^2*c - 2*a^2*c^2)*f - (a*b^3 - 3*a^2*b*c)*g)*x)/(a*b^2*c^3 - 4*a^2*c^4 + (b^2*c^4 - 4*a*
c^5)*x^4 + (b^3*c^3 - 4*a*b*c^4)*x^2) + 1/2*integrate(-(2*a*c^3*d - a*b*c^2*e - (b*c^3*d + (b^2*c^2 - 6*a*c^3)
*e - (3*b^3*c - 13*a*b*c^2)*f + (5*b^4 - 24*a*b^2*c + 14*a^2*c^2)*g)*x^2 + (3*a*b^2*c - 10*a^2*c^2)*f - (5*a*b
^3 - 19*a^2*b*c)*g)/(c*x^4 + b*x^2 + a), x)/(b^2*c^3 - 4*a*c^4) + 1/3*(c*g*x^3 + 3*(c*f - 2*b*g)*x)/c^3

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Fricas [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(x^4*(g*x^6+f*x^4+e*x^2+d)/(c*x^4+b*x^2+a)^2,x, algorithm="fricas")

[Out]

Timed out

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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(x**4*(g*x**6+f*x**4+e*x**2+d)/(c*x**4+b*x**2+a)**2,x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(g*x^6+f*x^4+e*x^2+d)/(c*x^4+b*x^2+a)^2,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError