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

Optimal. Leaf size=542 \[ \frac{x \left (a^2 \left (\frac{b^4 d}{a^2}-\frac{b^2 (b e+4 c d)}{a}-a (b g+2 c f)+b^2 f+3 b c e+2 c^2 d\right )+c x^2 \left (2 a^2 (c e-a g)-a b^2 e-a b (3 c d-a f)+b^3 d\right )\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (\frac{4 a^2 b (a g+4 c e)+4 a^2 c (7 c d-3 a f)-a b^2 (29 c d-a f)-3 a b^3 e+5 b^4 d}{\sqrt{b^2-4 a c}}+2 a^2 (5 c e-a g)-3 a b^2 e-a b (19 c d-a f)+5 b^3 d\right )}{2 \sqrt{2} a^3 \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (-\frac{4 a^2 b (a g+4 c e)+4 a^2 c (7 c d-3 a f)-a b^2 (29 c d-a f)-3 a b^3 e+5 b^4 d}{\sqrt{b^2-4 a c}}+2 a^2 (5 c e-a g)-3 a b^2 e-a b (19 c d-a f)+5 b^3 d\right )}{2 \sqrt{2} a^3 \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 b d-a e}{a^3 x}-\frac{d}{3 a^2 x^3} \]

[Out]

-d/(3*a^2*x^3) + (2*b*d - a*e)/(a^3*x) + (x*(a^2*((b^4*d)/a^2 + 2*c^2*d + 3*b*c*e - (b^2*(4*c*d + b*e))/a + b^
2*f - a*(2*c*f + b*g)) + c*(b^3*d - a*b^2*e - a*b*(3*c*d - a*f) + 2*a^2*(c*e - a*g))*x^2))/(2*a^3*(b^2 - 4*a*c
)*(a + b*x^2 + c*x^4)) + (Sqrt[c]*(5*b^3*d - 3*a*b^2*e - a*b*(19*c*d - a*f) + 2*a^2*(5*c*e - a*g) + (5*b^4*d -
3*a*b^3*e + 4*a^2*c*(7*c*d - 3*a*f) - a*b^2*(29*c*d - a*f) + 4*a^2*b*(4*c*e + 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]*a^3*(b^2 - 4*a*c)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) +
(Sqrt[c]*(5*b^3*d - 3*a*b^2*e - a*b*(19*c*d - a*f) + 2*a^2*(5*c*e - a*g) - (5*b^4*d - 3*a*b^3*e + 4*a^2*c*(7*
c*d - 3*a*f) - a*b^2*(29*c*d - a*f) + 4*a^2*b*(4*c*e + a*g))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqr
t[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a^3*(b^2 - 4*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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Rubi [A]  time = 7.26496, antiderivative size = 542, 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 = {1669, 1664, 1166, 205} \[ \frac{x \left (a^2 \left (\frac{b^4 d}{a^2}-\frac{b^2 (b e+4 c d)}{a}-a (b g+2 c f)+b^2 f+3 b c e+2 c^2 d\right )+c x^2 \left (2 a^2 (c e-a g)-a b^2 e-a b (3 c d-a f)+b^3 d\right )\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (\frac{4 a^2 b (a g+4 c e)+4 a^2 c (7 c d-3 a f)-a b^2 (29 c d-a f)-3 a b^3 e+5 b^4 d}{\sqrt{b^2-4 a c}}+2 a^2 (5 c e-a g)-3 a b^2 e-a b (19 c d-a f)+5 b^3 d\right )}{2 \sqrt{2} a^3 \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (-\frac{4 a^2 b (a g+4 c e)+4 a^2 c (7 c d-3 a f)-a b^2 (29 c d-a f)-3 a b^3 e+5 b^4 d}{\sqrt{b^2-4 a c}}+2 a^2 (5 c e-a g)-3 a b^2 e-a b (19 c d-a f)+5 b^3 d\right )}{2 \sqrt{2} a^3 \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 b d-a e}{a^3 x}-\frac{d}{3 a^2 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-d/(3*a^2*x^3) + (2*b*d - a*e)/(a^3*x) + (x*(a^2*((b^4*d)/a^2 + 2*c^2*d + 3*b*c*e - (b^2*(4*c*d + b*e))/a + b^
2*f - a*(2*c*f + b*g)) + c*(b^3*d - a*b^2*e - a*b*(3*c*d - a*f) + 2*a^2*(c*e - a*g))*x^2))/(2*a^3*(b^2 - 4*a*c
)*(a + b*x^2 + c*x^4)) + (Sqrt[c]*(5*b^3*d - 3*a*b^2*e - a*b*(19*c*d - a*f) + 2*a^2*(5*c*e - a*g) + (5*b^4*d -
3*a*b^3*e + 4*a^2*c*(7*c*d - 3*a*f) - a*b^2*(29*c*d - a*f) + 4*a^2*b*(4*c*e + 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]*a^3*(b^2 - 4*a*c)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) +
(Sqrt[c]*(5*b^3*d - 3*a*b^2*e - a*b*(19*c*d - a*f) + 2*a^2*(5*c*e - a*g) - (5*b^4*d - 3*a*b^3*e + 4*a^2*c*(7*
c*d - 3*a*f) - a*b^2*(29*c*d - a*f) + 4*a^2*b*(4*c*e + a*g))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqr
t[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a^3*(b^2 - 4*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

Rule 1669

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[x^m*(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])/x^m + (b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e)
/x^m + c*(4*p + 7)*(b*d - 2*a*e)*x^(2 - m), 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] && ILtQ[m/2, 0]

Rule 1664

Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x
)^m*Pq*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && PolyQ[Pq, x^2] && IGtQ[p, -2]

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}{x^4 \left (a+b x^2+c x^4\right )^2} \, dx &=\frac{x \left (a^2 \left (\frac{b^4 d}{a^2}+2 c^2 d+3 b c e-\frac{b^2 (4 c d+b e)}{a}+b^2 f-a (2 c f+b g)\right )+c \left (b^3 d-a b^2 e-a b (3 c d-a f)+2 a^2 (c e-a g)\right ) x^2\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\int \frac{-2 \left (b^2-4 a c\right ) d+\frac{2 \left (b^2-4 a c\right ) (b d-a e) x^2}{a}-\frac{\left (b^4 d-a b^3 e+6 a^2 c (c d-a f)-a b^2 (6 c d-a f)+a^2 b (5 c e+a g)\right ) x^4}{a^2}-c \left (\frac{b^3 d}{a^2}+2 c e-\frac{b (3 c d+b e)}{a}+b f-2 a g\right ) x^6}{x^4 \left (a+b x^2+c x^4\right )} \, dx}{2 a \left (b^2-4 a c\right )}\\ &=\frac{x \left (a^2 \left (\frac{b^4 d}{a^2}+2 c^2 d+3 b c e-\frac{b^2 (4 c d+b e)}{a}+b^2 f-a (2 c f+b g)\right )+c \left (b^3 d-a b^2 e-a b (3 c d-a f)+2 a^2 (c e-a g)\right ) x^2\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\int \left (\frac{2 \left (-b^2+4 a c\right ) d}{a x^4}+\frac{2 \left (-b^2+4 a c\right ) (-2 b d+a e)}{a^2 x^2}+\frac{-5 b^4 d+3 a b^3 e-2 a^2 c (7 c d-3 a f)+a b^2 (24 c d-a f)-a^2 b (13 c e+a g)-c \left (5 b^3 d-3 a b^2 e-a b (19 c d-a f)+2 a^2 (5 c e-a g)\right ) x^2}{a^2 \left (a+b x^2+c x^4\right )}\right ) \, dx}{2 a \left (b^2-4 a c\right )}\\ &=-\frac{d}{3 a^2 x^3}+\frac{2 b d-a e}{a^3 x}+\frac{x \left (a^2 \left (\frac{b^4 d}{a^2}+2 c^2 d+3 b c e-\frac{b^2 (4 c d+b e)}{a}+b^2 f-a (2 c f+b g)\right )+c \left (b^3 d-a b^2 e-a b (3 c d-a f)+2 a^2 (c e-a g)\right ) x^2\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\int \frac{-5 b^4 d+3 a b^3 e-2 a^2 c (7 c d-3 a f)+a b^2 (24 c d-a f)-a^2 b (13 c e+a g)-c \left (5 b^3 d-3 a b^2 e-a b (19 c d-a f)+2 a^2 (5 c e-a g)\right ) x^2}{a+b x^2+c x^4} \, dx}{2 a^3 \left (b^2-4 a c\right )}\\ &=-\frac{d}{3 a^2 x^3}+\frac{2 b d-a e}{a^3 x}+\frac{x \left (a^2 \left (\frac{b^4 d}{a^2}+2 c^2 d+3 b c e-\frac{b^2 (4 c d+b e)}{a}+b^2 f-a (2 c f+b g)\right )+c \left (b^3 d-a b^2 e-a b (3 c d-a f)+2 a^2 (c e-a g)\right ) x^2\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (c \left (5 b^3 d-3 a b^2 e-a b (19 c d-a f)+2 a^2 (5 c e-a g)-\frac{5 b^4 d-3 a b^3 e+4 a^2 c (7 c d-3 a f)-a b^2 (29 c d-a f)+4 a^2 b (4 c e+a g)}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{4 a^3 \left (b^2-4 a c\right )}+\frac{\left (c \left (5 b^3 d-3 a b^2 e-a b (19 c d-a f)+2 a^2 (5 c e-a g)+\frac{5 b^4 d-3 a b^3 e+4 a^2 c (7 c d-3 a f)-a b^2 (29 c d-a f)+4 a^2 b (4 c e+a g)}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{4 a^3 \left (b^2-4 a c\right )}\\ &=-\frac{d}{3 a^2 x^3}+\frac{2 b d-a e}{a^3 x}+\frac{x \left (a^2 \left (\frac{b^4 d}{a^2}+2 c^2 d+3 b c e-\frac{b^2 (4 c d+b e)}{a}+b^2 f-a (2 c f+b g)\right )+c \left (b^3 d-a b^2 e-a b (3 c d-a f)+2 a^2 (c e-a g)\right ) x^2\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \left (5 b^3 d-3 a b^2 e-a b (19 c d-a f)+2 a^2 (5 c e-a g)+\frac{5 b^4 d-3 a b^3 e+4 a^2 c (7 c d-3 a f)-a b^2 (29 c d-a f)+4 a^2 b (4 c e+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} a^3 \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left (5 b^3 d-3 a b^2 e-a b (19 c d-a f)+2 a^2 (5 c e-a g)-\frac{5 b^4 d-3 a b^3 e+4 a^2 c (7 c d-3 a f)-a b^2 (29 c d-a f)+4 a^2 b (4 c e+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} a^3 \left (b^2-4 a c\right ) \sqrt{b+\sqrt{b^2-4 a c}}}\\ \end{align*}

Mathematica [A]  time = 2.33592, size = 612, normalized size = 1.13 \[ \frac{\frac{6 x \left (a b \left (a^2 (-g)+a c \left (3 e+f x^2\right )-3 c^2 d x^2\right )+2 a^2 c \left (c \left (d+e x^2\right )-a \left (f+g x^2\right )\right )+a b^2 \left (a f-c \left (4 d+e x^2\right )\right )+b^3 \left (c d x^2-a e\right )+b^4 d\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{3 \sqrt{2} \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (-2 a^2 \left (-5 c e \sqrt{b^2-4 a c}+a g \sqrt{b^2-4 a c}+6 a c f-14 c^2 d\right )+a b \left (4 a^2 g-19 c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}+16 a c e\right )+a b^2 \left (-3 e \sqrt{b^2-4 a c}+a f-29 c d\right )+b^3 \left (5 d \sqrt{b^2-4 a c}-3 a e\right )+5 b^4 d\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{2} \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (2 a^2 \left (-5 c e \sqrt{b^2-4 a c}+a g \sqrt{b^2-4 a c}-6 a c f+14 c^2 d\right )+a b \left (4 a^2 g+19 c d \sqrt{b^2-4 a c}-a f \sqrt{b^2-4 a c}+16 a c e\right )+a b^2 \left (3 e \sqrt{b^2-4 a c}+a f-29 c d\right )-b^3 \left (5 d \sqrt{b^2-4 a c}+3 a e\right )+5 b^4 d\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{24 b d-12 a e}{x}-\frac{4 a d}{x^3}}{12 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*a*d)/x^3 + (24*b*d - 12*a*e)/x + (6*x*(b^4*d + b^3*(-(a*e) + c*d*x^2) + a*b^2*(a*f - c*(4*d + e*x^2)) + a
*b*(-(a^2*g) - 3*c^2*d*x^2 + a*c*(3*e + f*x^2)) + 2*a^2*c*(c*(d + e*x^2) - a*(f + g*x^2))))/((b^2 - 4*a*c)*(a
+ b*x^2 + c*x^4)) + (3*Sqrt[2]*Sqrt[c]*(5*b^4*d + b^3*(5*Sqrt[b^2 - 4*a*c]*d - 3*a*e) + a*b^2*(-29*c*d - 3*Sqr
t[b^2 - 4*a*c]*e + a*f) + a*b*(-19*c*Sqrt[b^2 - 4*a*c]*d + 16*a*c*e + a*Sqrt[b^2 - 4*a*c]*f + 4*a^2*g) - 2*a^2
*(-14*c^2*d - 5*c*Sqrt[b^2 - 4*a*c]*e + 6*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]]) - (3*Sqrt[2]*Sqrt[c]*(5*b^4*d - b^3*(5*
Sqrt[b^2 - 4*a*c]*d + 3*a*e) + a*b^2*(-29*c*d + 3*Sqrt[b^2 - 4*a*c]*e + a*f) + a*b*(19*c*Sqrt[b^2 - 4*a*c]*d +
16*a*c*e - a*Sqrt[b^2 - 4*a*c]*f + 4*a^2*g) + 2*a^2*(14*c^2*d - 5*c*Sqrt[b^2 - 4*a*c]*e - 6*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]]))/(12*a^3)

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Maple [B]  time = 0.048, size = 2503, normalized size = 4.6 \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)/x^4/(c*x^4+b*x^2+a)^2,x)

[Out]

1/2*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)
)*g-1/2*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))*g-1/a/(c*x^4+b*x^2+a)*c^2/(4*a*c-b^2)*x^3*e-1/2/a/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x*b^2*f-1/a/(c*x^4+b*x^2+
a)/(4*a*c-b^2)*x*c^2*d+1/2/a^2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x*b^3*e-1/2/a^3/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x*b^4*d
-3/4/a^2*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-29/4/a^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))*b^2*d+5/4/a^3*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^4*d+5/4/a^3*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^4*d-29/4/a^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)*arctan
h(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*b^2*d+1/4/a*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^2*f+1/4/a*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^2
*f+4/a*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*e-3/4/a^2*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+4/a*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*e+1/(c*x^4+b*x^2+a)/(4
*a*c-b^2)*x*c*f-1/a^2/x*e+1/2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x*b*g+1/(c*x^4+b*x^2+a)*c/(4*a*c-b^2)*x^3*g+2/a^3/x*
b*d-1/3*d/a^2/x^3+7/a*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))*d-1/4/a*c/(4*a*c-b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arcta
n(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b*f+1/4/a*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*f-3/4/a^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))*b^2*e-19/4/a^2*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*d+3/4/a^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))*b^2*e+19/4/a^2*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
*d+7/a*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))*d+5/4/a^3*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^3*d-5/4/a^3*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^3*d+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*g+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*g-1/2/a^3/(c*x^4
+b*x^2+a)*c/(4*a*c-b^2)*x^3*b^3*d-1/2/a/(c*x^4+b*x^2+a)*c/(4*a*c-b^2)*x^3*b*f+1/2/a^2/(c*x^4+b*x^2+a)*c/(4*a*c
-b^2)*x^3*b^2*e+2/a^2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x*b^2*c*d+3/2/a^2/(c*x^4+b*x^2+a)*c^2/(4*a*c-b^2)*x^3*b*d+5/
2/a*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))*e-5/2/a*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))*e-3*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))*f-3*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))*f-3/2/a/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x*b*c*e

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{3 \,{\left (a^{2} b c f - 2 \, a^{3} c g +{\left (5 \, b^{3} c - 19 \, a b c^{2}\right )} d -{\left (3 \, a b^{2} c - 10 \, a^{2} c^{2}\right )} e\right )} x^{6} -{\left (3 \, a^{3} b g -{\left (15 \, b^{4} - 62 \, a b^{2} c + 14 \, a^{2} c^{2}\right )} d + 3 \,{\left (3 \, a b^{3} - 11 \, a^{2} b c\right )} e - 3 \,{\left (a^{2} b^{2} - 2 \, a^{3} c\right )} f\right )} x^{4} + 2 \,{\left (5 \,{\left (a b^{3} - 4 \, a^{2} b c\right )} d - 3 \,{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} e\right )} x^{2} - 2 \,{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d}{6 \,{\left ({\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} x^{7} +{\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x^{5} +{\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{3}\right )}} - \frac{-\int \frac{a^{3} b g +{\left (a^{2} b c f - 2 \, a^{3} c g +{\left (5 \, b^{3} c - 19 \, a b c^{2}\right )} d -{\left (3 \, a b^{2} c - 10 \, a^{2} c^{2}\right )} e\right )} x^{2} +{\left (5 \, b^{4} - 24 \, a b^{2} c + 14 \, a^{2} c^{2}\right )} d -{\left (3 \, a b^{3} - 13 \, a^{2} b c\right )} e +{\left (a^{2} b^{2} - 6 \, a^{3} c\right )} f}{c x^{4} + b x^{2} + a}\,{d x}}{2 \,{\left (a^{3} b^{2} - 4 \, a^{4} c\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)/x^4/(c*x^4+b*x^2+a)^2,x, algorithm="maxima")

[Out]

1/6*(3*(a^2*b*c*f - 2*a^3*c*g + (5*b^3*c - 19*a*b*c^2)*d - (3*a*b^2*c - 10*a^2*c^2)*e)*x^6 - (3*a^3*b*g - (15*
b^4 - 62*a*b^2*c + 14*a^2*c^2)*d + 3*(3*a*b^3 - 11*a^2*b*c)*e - 3*(a^2*b^2 - 2*a^3*c)*f)*x^4 + 2*(5*(a*b^3 - 4
*a^2*b*c)*d - 3*(a^2*b^2 - 4*a^3*c)*e)*x^2 - 2*(a^2*b^2 - 4*a^3*c)*d)/((a^3*b^2*c - 4*a^4*c^2)*x^7 + (a^3*b^3
- 4*a^4*b*c)*x^5 + (a^4*b^2 - 4*a^5*c)*x^3) - 1/2*integrate(-(a^3*b*g + (a^2*b*c*f - 2*a^3*c*g + (5*b^3*c - 19
*a*b*c^2)*d - (3*a*b^2*c - 10*a^2*c^2)*e)*x^2 + (5*b^4 - 24*a*b^2*c + 14*a^2*c^2)*d - (3*a*b^3 - 13*a^2*b*c)*e
+ (a^2*b^2 - 6*a^3*c)*f)/(c*x^4 + b*x^2 + a), x)/(a^3*b^2 - 4*a^4*c)

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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)/x^4/(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)/x**4/(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)/x^4/(c*x^4+b*x^2+a)^2,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError