### 3.127 $$\int \frac{x^2 (d+e x^2+f x^4+g x^6)}{(a+b x^2+c x^4)^2} \, dx$$

Optimal. Leaf size=471 $-\frac{x \left (x^2 \left (-c^2 (2 a f+b e)+b c (3 a g+b f)+b^3 (-g)+2 c^3 d\right )-a b^2 g+b c (a f+c d)-2 a c (c e-a g)\right )}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (\frac{b^2 c (19 a g+c e)-4 b c^2 (2 a f+c d)+4 a c^2 (c e-5 a g)+b^3 c f-3 b^4 g}{\sqrt{b^2-4 a c}}-c^2 (b e-6 a f)-b c (13 a g+b f)+3 b^3 g+2 c^3 d\right )}{2 \sqrt{2} c^{5/2} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (-\frac{b^2 c (19 a g+c e)-4 b c^2 (2 a f+c d)+4 a c^2 (c e-5 a g)+b^3 c f-3 b^4 g}{\sqrt{b^2-4 a c}}-c^2 (b e-6 a f)-b c (13 a g+b f)+3 b^3 g+2 c^3 d\right )}{2 \sqrt{2} c^{5/2} \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{g x}{c^2}$

[Out]

(g*x)/c^2 - (x*(b*c*(c*d + a*f) - a*b^2*g - 2*a*c*(c*e - a*g) + (2*c^3*d - c^2*(b*e + 2*a*f) - b^3*g + b*c*(b*
f + 3*a*g))*x^2))/(2*c^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((2*c^3*d - c^2*(b*e - 6*a*f) + 3*b^3*g - b*c*(b
*f + 13*a*g) + (b^3*c*f - 4*b*c^2*(c*d + 2*a*f) - 3*b^4*g + 4*a*c^2*(c*e - 5*a*g) + b^2*c*(c*e + 19*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^(5/2)*(b^2 - 4*a*c)*Sqrt[
b - Sqrt[b^2 - 4*a*c]]) - ((2*c^3*d - c^2*(b*e - 6*a*f) + 3*b^3*g - b*c*(b*f + 13*a*g) - (b^3*c*f - 4*b*c^2*(c
*d + 2*a*f) - 3*b^4*g + 4*a*c^2*(c*e - 5*a*g) + b^2*c*(c*e + 19*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^(5/2)*(b^2 - 4*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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Rubi [A]  time = 6.66183, antiderivative size = 471, 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} $-\frac{x \left (x^2 \left (-c^2 (2 a f+b e)+b c (3 a g+b f)+b^3 (-g)+2 c^3 d\right )-a b^2 g+b c (a f+c d)-2 a c (c e-a g)\right )}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (\frac{b^2 c (19 a g+c e)-4 b c^2 (2 a f+c d)+4 a c^2 (c e-5 a g)+b^3 c f-3 b^4 g}{\sqrt{b^2-4 a c}}-c^2 (b e-6 a f)-b c (13 a g+b f)+3 b^3 g+2 c^3 d\right )}{2 \sqrt{2} c^{5/2} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (-\frac{b^2 c (19 a g+c e)-4 b c^2 (2 a f+c d)+4 a c^2 (c e-5 a g)+b^3 c f-3 b^4 g}{\sqrt{b^2-4 a c}}-c^2 (b e-6 a f)-b c (13 a g+b f)+3 b^3 g+2 c^3 d\right )}{2 \sqrt{2} c^{5/2} \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{g x}{c^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(g*x)/c^2 - (x*(b*c*(c*d + a*f) - a*b^2*g - 2*a*c*(c*e - a*g) + (2*c^3*d - c^2*(b*e + 2*a*f) - b^3*g + b*c*(b*
f + 3*a*g))*x^2))/(2*c^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((2*c^3*d - c^2*(b*e - 6*a*f) + 3*b^3*g - b*c*(b
*f + 13*a*g) + (b^3*c*f - 4*b*c^2*(c*d + 2*a*f) - 3*b^4*g + 4*a*c^2*(c*e - 5*a*g) + b^2*c*(c*e + 19*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^(5/2)*(b^2 - 4*a*c)*Sqrt[
b - Sqrt[b^2 - 4*a*c]]) - ((2*c^3*d - c^2*(b*e - 6*a*f) + 3*b^3*g - b*c*(b*f + 13*a*g) - (b^3*c*f - 4*b*c^2*(c
*d + 2*a*f) - 3*b^4*g + 4*a*c^2*(c*e - 5*a*g) + b^2*c*(c*e + 19*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^(5/2)*(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^2 \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 (b c (c d+a f)-a b^2 g-2 a c (c e-a g)+\left (2 c^3 d-c^2 (b e+2 a f)-b^3 g+b c (b f+3 a g)\right ) x^2\right )}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\int \frac{-\frac{a \left (b c (c d+a f)-a b^2 g-2 a c (c e-a g)\right )}{c^2}+\frac{a \left (2 c^3 d-c^2 (b e-6 a f)+b^3 g-b c (b f+5 a g)\right ) x^2}{c^2}+2 a \left (4 a-\frac{b^2}{c}\right ) g x^4}{a+b x^2+c x^4} \, dx}{2 a \left (b^2-4 a c\right )}\\ &=-\frac{x \left (b c (c d+a f)-a b^2 g-2 a c (c e-a g)+\left (2 c^3 d-c^2 (b e+2 a f)-b^3 g+b c (b f+3 a g)\right ) x^2\right )}{2 c^2 \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 ) g}{c^2}-\frac{a \left (b c (c d+a f)-3 a b^2 g-2 a c (c e-5 a g)\right )-a \left (2 c^3 d-c^2 (b e-6 a f)+3 b^3 g-b c (b f+13 a g)\right ) x^2}{c^2 \left (a+b x^2+c x^4\right )}\right ) \, dx}{2 a \left (b^2-4 a c\right )}\\ &=\frac{g x}{c^2}-\frac{x \left (b c (c d+a f)-a b^2 g-2 a c (c e-a g)+\left (2 c^3 d-c^2 (b e+2 a f)-b^3 g+b c (b f+3 a g)\right ) x^2\right )}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\int \frac{a \left (b c (c d+a f)-3 a b^2 g-2 a c (c e-5 a g)\right )-a \left (2 c^3 d-c^2 (b e-6 a f)+3 b^3 g-b c (b f+13 a g)\right ) x^2}{a+b x^2+c x^4} \, dx}{2 a c^2 \left (b^2-4 a c\right )}\\ &=\frac{g x}{c^2}-\frac{x \left (b c (c d+a f)-a b^2 g-2 a c (c e-a g)+\left (2 c^3 d-c^2 (b e+2 a f)-b^3 g+b c (b f+3 a g)\right ) x^2\right )}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\left (2 c^3 d-c^2 (b e-6 a f)+3 b^3 g-b c (b f+13 a g)-\frac{b^3 c f-4 b c^2 (c d+2 a f)-3 b^4 g+4 a c^2 (c e-5 a g)+b^2 c (c e+19 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^2 \left (b^2-4 a c\right )}-\frac{\left (2 c^3 d-c^2 (b e-6 a f)+3 b^3 g-b c (b f+13 a g)+\frac{b^3 c f-4 b c^2 (c d+2 a f)-3 b^4 g+4 a c^2 (c e-5 a g)+b^2 c (c e+19 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^2 \left (b^2-4 a c\right )}\\ &=\frac{g x}{c^2}-\frac{x \left (b c (c d+a f)-a b^2 g-2 a c (c e-a g)+\left (2 c^3 d-c^2 (b e+2 a f)-b^3 g+b c (b f+3 a g)\right ) x^2\right )}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\left (2 c^3 d-c^2 (b e-6 a f)+3 b^3 g-b c (b f+13 a g)+\frac{b^3 c f-4 b c^2 (c d+2 a f)-3 b^4 g+4 a c^2 (c e-5 a g)+b^2 c (c e+19 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^{5/2} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left (2 c^3 d-c^2 (b e-6 a f)+3 b^3 g-b c (b f+13 a g)-\frac{b^3 c f-4 b c^2 (c d+2 a f)-3 b^4 g+4 a c^2 (c e-5 a g)+b^2 c (c e+19 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^{5/2} \left (b^2-4 a c\right ) \sqrt{b+\sqrt{b^2-4 a c}}}\\ \end{align*}

Mathematica [A]  time = 2.1155, size = 575, normalized size = 1.22 $\frac{-\frac{2 \sqrt{c} x \left (2 c \left (a^2 g-a c \left (e+f x^2\right )+c^2 d x^2\right )+b^2 \left (c f x^2-a g\right )+b c \left (a \left (f+3 g x^2\right )+c \left (d-e x^2\right )\right )+b^3 (-g) x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (2 c^2 \left (-10 a^2 g+c d \sqrt{b^2-4 a c}+3 a f \sqrt{b^2-4 a c}+2 a c e\right )-b c \left (c e \sqrt{b^2-4 a c}+13 a g \sqrt{b^2-4 a c}+8 a c f+4 c^2 d\right )+b^2 c \left (-f \sqrt{b^2-4 a c}+19 a g+c e\right )+b^3 \left (3 g \sqrt{b^2-4 a c}+c f\right )-3 b^4 g\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (2 c^2 \left (10 a^2 g+c d \sqrt{b^2-4 a c}+3 a f \sqrt{b^2-4 a c}-2 a c e\right )+b c \left (-c e \sqrt{b^2-4 a c}-13 a g \sqrt{b^2-4 a c}+8 a c f+4 c^2 d\right )-b^2 c \left (f \sqrt{b^2-4 a c}+19 a g+c e\right )+b^3 \left (3 g \sqrt{b^2-4 a c}-c f\right )+3 b^4 g\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+4 \sqrt{c} g x}{4 c^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(4*Sqrt[c]*g*x - (2*Sqrt[c]*x*(-(b^3*g*x^2) + b^2*(-(a*g) + c*f*x^2) + 2*c*(a^2*g + c^2*d*x^2 - a*c*(e + f*x^2
)) + b*c*(c*(d - e*x^2) + a*(f + 3*g*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (Sqrt[2]*(-3*b^4*g + b^2*c*
(c*e - Sqrt[b^2 - 4*a*c]*f + 19*a*g) + 2*c^2*(c*Sqrt[b^2 - 4*a*c]*d + 2*a*c*e + 3*a*Sqrt[b^2 - 4*a*c]*f - 10*a
^2*g) + b^3*(c*f + 3*Sqrt[b^2 - 4*a*c]*g) - b*c*(4*c^2*d + c*Sqrt[b^2 - 4*a*c]*e + 8*a*c*f + 13*a*Sqrt[b^2 - 4
*a*c]*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]]) - (Sqrt[2]*(3*b^4*g - b^2*c*(c*e + Sqrt[b^2 - 4*a*c]*f + 19*a*g) + 2*c^2*(c*Sqrt[b^2 - 4*a*c]*d - 2*a*c
*e + 3*a*Sqrt[b^2 - 4*a*c]*f + 10*a^2*g) + b^3*(-(c*f) + 3*Sqrt[b^2 - 4*a*c]*g) + b*c*(4*c^2*d - c*Sqrt[b^2 -
4*a*c]*e + 8*a*c*f - 13*a*Sqrt[b^2 - 4*a*c]*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]]))/(4*c^(5/2))

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Maple [B]  time = 0.05, size = 2300, normalized size = 4.9 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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*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*g-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*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*e-1/2/(4*a*c-b^2)*c*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))*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*f-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*e+1/2/(4*a*c-b^2)*c*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))*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*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*g-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*g+13/4/c/(4*a*c-b^2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arc
tanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*a*b*g-1/2/c^2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^3*b^3*g+1/2/c/(
c*x^4+b*x^2+a)/(4*a*c-b^2)*x^3*b^2*f+1/c/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x*a^2*g-1/4/(4*a*c-b^2)/c*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*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*e+
c/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^3*d-1/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^3*a*f-1/2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^3*
b*e-1/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x*a*e+1/2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x*b*d+1/4/(4*a*c-b^2)/c*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*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
*e+g*x/c^2-1/(4*a*c-b^2)*c/(-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*e-1/4/(4*a*c-b^2)/c/(-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*f+1/(4*a*c-b^2)*c/(-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*d+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*f-1/
(4*a*c-b^2)*c/(-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*e-1/4/(4*a*c-b^2)/c/(-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*f+1/(4*a*c-b^2)*c/(-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*d+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*f-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*g+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*g-1/2/c^2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x*a*b^2*g+3/2/c/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^3*a*b*g+1/2/c/(c*x^4
+b*x^2+a)/(4*a*c-b^2)*x*a*b*f+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)*arctan
h(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*a^2*g+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*g

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

[Out]

Timed out

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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^2*(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**2*(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^2*(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