### 3.128 $$\int \frac{d+e x^2+f x^4+g x^6}{(a+b x^2+c x^4)^2} \, dx$$

Optimal. Leaf size=449 $\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (\frac{b^2 c (c d-a f)-a b^3 g+4 a b c (2 a g+c e)-4 a c^2 (a f+3 c d)}{c \sqrt{b^2-4 a c}}+\frac{a b^2 g}{c}+b (a f+c d)-2 a (3 a g+c e)\right )}{2 \sqrt{2} a \sqrt{c} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (-\frac{b^2 c (c d-a f)-a b^3 g+4 a b c (2 a g+c e)-4 a c^2 (a f+3 c d)}{c \sqrt{b^2-4 a c}}+\frac{a b^2 g}{c}+b (a f+c d)-2 a (3 a g+c e)\right )}{2 \sqrt{2} a \sqrt{c} \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x \left (x^2 \left (-a b^2 g+b c (a f+c d)-2 a c (c e-a g)\right )+c \left (-\frac{a b (a g+c e)}{c}-2 a (c d-a f)+b^2 d\right )\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}$

[Out]

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

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Rubi [A]  time = 2.86695, antiderivative size = 449, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 32, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.094, Rules used = {1678, 1166, 205} $\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (\frac{b^2 c (c d-a f)-a b^3 g+4 a b c (2 a g+c e)-4 a c^2 (a f+3 c d)}{c \sqrt{b^2-4 a c}}+\frac{a b^2 g}{c}+b (a f+c d)-2 a (3 a g+c e)\right )}{2 \sqrt{2} a \sqrt{c} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (-\frac{b^2 c (c d-a f)-a b^3 g+4 a b c (2 a g+c e)-4 a c^2 (a f+3 c d)}{c \sqrt{b^2-4 a c}}+\frac{a b^2 g}{c}+b (a f+c d)-2 a (3 a g+c e)\right )}{2 \sqrt{2} a \sqrt{c} \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x \left (x^2 \left (-a b^2 g+b c (a f+c d)-2 a c (c e-a g)\right )+c \left (-\frac{a b (a g+c e)}{c}-2 a (c d-a f)+b^2 d\right )\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

Mathematica [A]  time = 1.83305, size = 512, normalized size = 1.14 $\frac{\frac{2 \sqrt{c} x \left (b \left (a^2 (-g)-a c e+a c f x^2+c^2 d x^2\right )+b^2 \left (c d-a g x^2\right )+2 a c \left (a \left (f+g x^2\right )-c \left (d+e x^2\right )\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (b c \left (8 a^2 g+c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}+4 a c e\right )-2 a c \left (c e \sqrt{b^2-4 a c}+3 a g \sqrt{b^2-4 a c}+2 a c f+6 c^2 d\right )+b^2 \left (a g \sqrt{b^2-4 a c}-a c f+c^2 d\right )-a b^3 g\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (b c \left (-8 a^2 g+c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}-4 a c e\right )+2 a c \left (-c e \sqrt{b^2-4 a c}-3 a g \sqrt{b^2-4 a c}+2 a c f+6 c^2 d\right )+b^2 \left (a g \sqrt{b^2-4 a c}+a c f+c^2 (-d)\right )+a b^3 g\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}}{4 a c^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((2*Sqrt[c]*x*(b*(-(a*c*e) - a^2*g + c^2*d*x^2 + a*c*f*x^2) + b^2*(c*d - a*g*x^2) + 2*a*c*(-(c*(d + e*x^2)) +
a*(f + g*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(-(a*b^3*g) + b*c*(c*Sqrt[b^2 - 4*a*c]*d + 4*a
*c*e + a*Sqrt[b^2 - 4*a*c]*f + 8*a^2*g) + b^2*(c^2*d - a*c*f + a*Sqrt[b^2 - 4*a*c]*g) - 2*a*c*(6*c^2*d + c*Sqr
t[b^2 - 4*a*c]*e + 2*a*c*f + 3*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]*(a*b^3*g + b*c*(c*Sqrt[b^2 - 4*a*c]*d - 4*a*c*e
+ a*Sqrt[b^2 - 4*a*c]*f - 8*a^2*g) + 2*a*c*(6*c^2*d - c*Sqrt[b^2 - 4*a*c]*e + 2*a*c*f - 3*a*Sqrt[b^2 - 4*a*c]
*g) + b^2*(-(c^2*d) + a*c*f + 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*a*c^(3/2))

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Maple [B]  time = 0.042, size = 1760, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)
)*e+1/4/a/(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^2*d+1/4/a/(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^2*d+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*f-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))*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*f+(-1/2/a*(2*a^2*c*g-a*b^2*g+a*b*c*
f-2*a*c^2*e+b*c^2*d)/(4*a*c-b^2)/c*x^3+1/2*(a^2*b*g-2*a^2*c*f+a*b*c*e+2*a*c^2*d-b^2*c*d)/a/c/(4*a*c-b^2)*x)/(c
*x^4+b*x^2+a)-3/2*a/(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))*g+3/2*a/(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))*g+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*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*g-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*f-3/(4*a*c-b^2)*c^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))*d-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*f-3/(4*a*c-b^2)*c^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))*d+1/4/a/(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*d-a/(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))*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*e-1/4/a/(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*d-a/(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))*
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*e+2*a/(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*g-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*g+2*a/(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*g-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*g

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((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^2*d - 2*a*c^2*e + a*b*c*f - (a*b^2 - 2*a^2*c)*g)*x^3 - (a*b*c*e - 2*a^2*c*f + a^2*b*g - (b^2*c - 2*a
*c^2)*d)*x)/(a^2*b^2*c - 4*a^3*c^2 + (a*b^2*c^2 - 4*a^2*c^3)*x^4 + (a*b^3*c - 4*a^2*b*c^2)*x^2) - 1/2*integrat
e(-(a*b*c*e - 2*a^2*c*f + a^2*b*g + (b*c^2*d - 2*a*c^2*e + a*b*c*f + (a*b^2 - 6*a^2*c)*g)*x^2 + (b^2*c - 6*a*c
^2)*d)/(c*x^4 + b*x^2 + a), x)/(a*b^2*c - 4*a^2*c^2)

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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((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((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((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