3.41 \(\int \frac{d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{\left (a+b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=770 \[ -\frac{x \left (b^2 \left (-\left (a^2 m+c^2 d\right )\right )+x^2 \left (-b c \left (-3 a^2 m+a c h+c^2 d\right )-a b^3 m+a b^2 c k+2 a c^2 (c f-a k)\right )+2 a c \left (a^2 m-a c h+c^2 d\right )+a b c (a k+c f)\right )}{2 a c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (-\frac{-b^2 c \left (-19 a^2 m-a c h+c^2 d\right )+4 a c^2 \left (-5 a^2 m+a c h+3 c^2 d\right )-3 a b^4 m+a b^3 c k-4 a b c^2 (2 a k+c f)}{\sqrt{b^2-4 a c}}+b c \left (13 a^2 m+a c h+c^2 d\right )-3 a b^3 m+a b^2 c k-2 a c^2 (3 a k+c f)\right )}{2 \sqrt{2} a c^{5/2} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (\frac{-b^2 c \left (-19 a^2 m-a c h+c^2 d\right )+4 a c^2 \left (-5 a^2 m+a c h+3 c^2 d\right )-3 a b^4 m+a b^3 c k-4 a b c^2 (2 a k+c f)}{\sqrt{b^2-4 a c}}+b c \left (13 a^2 m+a c h+c^2 d\right )-3 a b^3 m+a b^2 c k-2 a c^2 (3 a k+c f)\right )}{2 \sqrt{2} a c^{5/2} \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (-c^2 (2 b g-4 a j)-6 a b c l+b^3 l+4 c^3 e\right )}{2 c^2 \left (b^2-4 a c\right )^{3/2}}-\frac{x^2 \left (-c^2 (2 a j+b g)+b c (3 a l+b j)+b^3 (-l)+2 c^3 e\right )-a b^2 l+b c (a j+c e)-2 a c (c g-a l)}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{l \log \left (a+b x^2+c x^4\right )}{4 c^2}+\frac{m x}{c^2} \]

[Out]

(m*x)/c^2 - (b*c*(c*e + a*j) - a*b^2*l - 2*a*c*(c*g - a*l) + (2*c^3*e - c^2*(b*g
+ 2*a*j) - b^3*l + b*c*(b*j + 3*a*l))*x^2)/(2*c^2*(b^2 - 4*a*c)*(a + b*x^2 + c*
x^4)) - (x*(a*b*c*(c*f + a*k) - b^2*(c^2*d + a^2*m) + 2*a*c*(c^2*d - a*c*h + a^2
*m) + (a*b^2*c*k + 2*a*c^2*(c*f - a*k) - a*b^3*m - b*c*(c^2*d + a*c*h - 3*a^2*m)
)*x^2))/(2*a*c^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + ((a*b^2*c*k - 2*a*c^2*(c*f
+ 3*a*k) - 3*a*b^3*m + b*c*(c^2*d + a*c*h + 13*a^2*m) - (a*b^3*c*k - 4*a*b*c^2*
(c*f + 2*a*k) - 3*a*b^4*m - b^2*c*(c^2*d - a*c*h - 19*a^2*m) + 4*a*c^2*(3*c^2*d
+ a*c*h - 5*a^2*m))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[
b^2 - 4*a*c]]])/(2*Sqrt[2]*a*c^(5/2)*(b^2 - 4*a*c)*Sqrt[b - Sqrt[b^2 - 4*a*c]])
+ ((a*b^2*c*k - 2*a*c^2*(c*f + 3*a*k) - 3*a*b^3*m + b*c*(c^2*d + a*c*h + 13*a^2*
m) + (a*b^3*c*k - 4*a*b*c^2*(c*f + 2*a*k) - 3*a*b^4*m - b^2*c*(c^2*d - a*c*h - 1
9*a^2*m) + 4*a*c^2*(3*c^2*d + a*c*h - 5*a^2*m))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[
2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a*c^(5/2)*(b^2 - 4*a*c)*S
qrt[b + Sqrt[b^2 - 4*a*c]]) + ((4*c^3*e - c^2*(2*b*g - 4*a*j) + b^3*l - 6*a*b*c*
l)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^2*(b^2 - 4*a*c)^(3/2)) + (l*Lo
g[a + b*x^2 + c*x^4])/(4*c^2)

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Rubi [A]  time = 25.763, antiderivative size = 770, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 55, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{x \left (b^2 \left (-\left (a^2 m+c^2 d\right )\right )+x^2 \left (-b c \left (-3 a^2 m+a c h+c^2 d\right )-a b^3 m+a b^2 c k+2 a c^2 (c f-a k)\right )+2 a c \left (a^2 m-a c h+c^2 d\right )+a b c (a k+c f)\right )}{2 a c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (-\frac{-b^2 c \left (-19 a^2 m-a c h+c^2 d\right )+4 a c^2 \left (-5 a^2 m+a c h+3 c^2 d\right )-3 a b^4 m+a b^3 c k-4 a b c^2 (2 a k+c f)}{\sqrt{b^2-4 a c}}+b c \left (13 a^2 m+a c h+c^2 d\right )-3 a b^3 m+a b^2 c k-2 a c^2 (3 a k+c f)\right )}{2 \sqrt{2} a c^{5/2} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (\frac{-b^2 c \left (-19 a^2 m-a c h+c^2 d\right )+4 a c^2 \left (-5 a^2 m+a c h+3 c^2 d\right )-3 a b^4 m+a b^3 c k-4 a b c^2 (2 a k+c f)}{\sqrt{b^2-4 a c}}+b c \left (13 a^2 m+a c h+c^2 d\right )-3 a b^3 m+a b^2 c k-2 a c^2 (3 a k+c f)\right )}{2 \sqrt{2} a c^{5/2} \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (-c^2 (2 b g-4 a j)-6 a b c l+b^3 l+4 c^3 e\right )}{2 c^2 \left (b^2-4 a c\right )^{3/2}}-\frac{x^2 \left (-c^2 (2 a j+b g)+b c (3 a l+b j)+b^3 (-l)+2 c^3 e\right )-a b^2 l+b c (a j+c e)-2 a c (c g-a l)}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{l \log \left (a+b x^2+c x^4\right )}{4 c^2}+\frac{m x}{c^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x + f*x^2 + g*x^3 + h*x^4 + j*x^5 + k*x^6 + l*x^7 + m*x^8)/(a + b*x^2 + c*x^4)^2,x]

[Out]

(m*x)/c^2 - (b*c*(c*e + a*j) - a*b^2*l - 2*a*c*(c*g - a*l) + (2*c^3*e - c^2*(b*g
+ 2*a*j) - b^3*l + b*c*(b*j + 3*a*l))*x^2)/(2*c^2*(b^2 - 4*a*c)*(a + b*x^2 + c*
x^4)) - (x*(a*b*c*(c*f + a*k) - b^2*(c^2*d + a^2*m) + 2*a*c*(c^2*d - a*c*h + a^2
*m) + (a*b^2*c*k + 2*a*c^2*(c*f - a*k) - a*b^3*m - b*c*(c^2*d + a*c*h - 3*a^2*m)
)*x^2))/(2*a*c^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + ((a*b^2*c*k - 2*a*c^2*(c*f
+ 3*a*k) - 3*a*b^3*m + b*c*(c^2*d + a*c*h + 13*a^2*m) - (a*b^3*c*k - 4*a*b*c^2*
(c*f + 2*a*k) - 3*a*b^4*m - b^2*c*(c^2*d - a*c*h - 19*a^2*m) + 4*a*c^2*(3*c^2*d
+ a*c*h - 5*a^2*m))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[
b^2 - 4*a*c]]])/(2*Sqrt[2]*a*c^(5/2)*(b^2 - 4*a*c)*Sqrt[b - Sqrt[b^2 - 4*a*c]])
+ ((a*b^2*c*k - 2*a*c^2*(c*f + 3*a*k) - 3*a*b^3*m + b*c*(c^2*d + a*c*h + 13*a^2*
m) + (a*b^3*c*k - 4*a*b*c^2*(c*f + 2*a*k) - 3*a*b^4*m - b^2*c*(c^2*d - a*c*h - 1
9*a^2*m) + 4*a*c^2*(3*c^2*d + a*c*h - 5*a^2*m))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[
2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a*c^(5/2)*(b^2 - 4*a*c)*S
qrt[b + Sqrt[b^2 - 4*a*c]]) + ((4*c^3*e - c^2*(2*b*g - 4*a*j) + b^3*l - 6*a*b*c*
l)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^2*(b^2 - 4*a*c)^(3/2)) + (l*Lo
g[a + b*x^2 + c*x^4])/(4*c^2)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((m*x**8+l*x**7+k*x**6+j*x**5+h*x**4+g*x**3+f*x**2+e*x+d)/(c*x**4+b*x**2+a)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 8.76673, size = 1109, normalized size = 1.44 \[ \frac{m x}{c^2}-\frac{\left (-3 a m b^4+a c k b^3+3 a \sqrt{b^2-4 a c} m b^3-c^3 d b^2+a c^2 h b^2-a c \sqrt{b^2-4 a c} k b^2+19 a^2 c m b^2-c^3 \sqrt{b^2-4 a c} d b-4 a c^3 f b-a c^2 \sqrt{b^2-4 a c} h b-8 a^2 c^2 k b-13 a^2 c \sqrt{b^2-4 a c} m b+12 a c^4 d+2 a c^3 \sqrt{b^2-4 a c} f+4 a^2 c^3 h+6 a^2 c^2 \sqrt{b^2-4 a c} k-20 a^3 c^2 m\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a c^{5/2} \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left (3 a m b^4-a c k b^3+3 a \sqrt{b^2-4 a c} m b^3+c^3 d b^2-a c^2 h b^2-a c \sqrt{b^2-4 a c} k b^2-19 a^2 c m b^2-c^3 \sqrt{b^2-4 a c} d b+4 a c^3 f b-a c^2 \sqrt{b^2-4 a c} h b+8 a^2 c^2 k b-13 a^2 c \sqrt{b^2-4 a c} m b-12 a c^4 d+2 a c^3 \sqrt{b^2-4 a c} f-4 a^2 c^3 h+6 a^2 c^2 \sqrt{b^2-4 a c} k+20 a^3 c^2 m\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a c^{5/2} \left (b^2-4 a c\right )^{3/2} \sqrt{b+\sqrt{b^2-4 a c}}}-\frac{\left (l b^3-\sqrt{b^2-4 a c} l b^2-2 c^2 g b-6 a c l b+4 c^3 e+4 a c^2 j+4 a c \sqrt{b^2-4 a c} l\right ) \log \left (-2 c x^2-b+\sqrt{b^2-4 a c}\right )}{4 c^2 \left (b^2-4 a c\right )^{3/2}}-\frac{\left (-l b^3-\sqrt{b^2-4 a c} l b^2+2 c^2 g b+6 a c l b-4 c^3 e-4 a c^2 j+4 a c \sqrt{b^2-4 a c} l\right ) \log \left (2 c x^2+b+\sqrt{b^2-4 a c}\right )}{4 c^2 \left (b^2-4 a c\right )^{3/2}}+\frac{2 c l a^3+2 c m x a^3-2 c^2 k x^3 a^2+3 b c m x^3 a^2-2 c^2 j x^2 a^2+3 b c l x^2 a^2-2 c^2 g a^2+b c j a^2-b^2 l a^2-2 c^2 h x a^2+b c k x a^2-b^2 m x a^2+2 c^3 f x^3 a-b c^2 h x^3 a+b^2 c k x^3 a-b^3 m x^3 a+2 c^3 e x^2 a-b c^2 g x^2 a+b^2 c j x^2 a-b^3 l x^2 a+b c^2 e a+2 c^3 d x a+b c^2 f x a-b c^3 d x^3-b^2 c^2 d x}{2 a c^2 \left (4 a c-b^2\right ) \left (c x^4+b x^2+a\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + j*x^5 + k*x^6 + l*x^7 + m*x^8)/(a + b*x^2 + c*x^4)^2,x]

[Out]

(m*x)/c^2 + (a*b*c^2*e - 2*a^2*c^2*g + a^2*b*c*j - a^2*b^2*l + 2*a^3*c*l - b^2*c
^2*d*x + 2*a*c^3*d*x + a*b*c^2*f*x - 2*a^2*c^2*h*x + a^2*b*c*k*x - a^2*b^2*m*x +
2*a^3*c*m*x + 2*a*c^3*e*x^2 - a*b*c^2*g*x^2 + a*b^2*c*j*x^2 - 2*a^2*c^2*j*x^2 -
a*b^3*l*x^2 + 3*a^2*b*c*l*x^2 - b*c^3*d*x^3 + 2*a*c^3*f*x^3 - a*b*c^2*h*x^3 + a
*b^2*c*k*x^3 - 2*a^2*c^2*k*x^3 - a*b^3*m*x^3 + 3*a^2*b*c*m*x^3)/(2*a*c^2*(-b^2 +
4*a*c)*(a + b*x^2 + c*x^4)) - ((-(b^2*c^3*d) + 12*a*c^4*d - b*c^3*Sqrt[b^2 - 4*
a*c]*d - 4*a*b*c^3*f + 2*a*c^3*Sqrt[b^2 - 4*a*c]*f + a*b^2*c^2*h + 4*a^2*c^3*h -
a*b*c^2*Sqrt[b^2 - 4*a*c]*h + a*b^3*c*k - 8*a^2*b*c^2*k - a*b^2*c*Sqrt[b^2 - 4*
a*c]*k + 6*a^2*c^2*Sqrt[b^2 - 4*a*c]*k - 3*a*b^4*m + 19*a^2*b^2*c*m - 20*a^3*c^2
*m + 3*a*b^3*Sqrt[b^2 - 4*a*c]*m - 13*a^2*b*c*Sqrt[b^2 - 4*a*c]*m)*ArcTan[(Sqrt[
2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a*c^(5/2)*(b^2 - 4*a*c)^(
3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - ((b^2*c^3*d - 12*a*c^4*d - b*c^3*Sqrt[b^2 -
4*a*c]*d + 4*a*b*c^3*f + 2*a*c^3*Sqrt[b^2 - 4*a*c]*f - a*b^2*c^2*h - 4*a^2*c^3*h
- a*b*c^2*Sqrt[b^2 - 4*a*c]*h - a*b^3*c*k + 8*a^2*b*c^2*k - a*b^2*c*Sqrt[b^2 -
4*a*c]*k + 6*a^2*c^2*Sqrt[b^2 - 4*a*c]*k + 3*a*b^4*m - 19*a^2*b^2*c*m + 20*a^3*c
^2*m + 3*a*b^3*Sqrt[b^2 - 4*a*c]*m - 13*a^2*b*c*Sqrt[b^2 - 4*a*c]*m)*ArcTan[(Sqr
t[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a*c^(5/2)*(b^2 - 4*a*c)
^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) - ((4*c^3*e - 2*b*c^2*g + 4*a*c^2*j + b^3*l
- 6*a*b*c*l - b^2*Sqrt[b^2 - 4*a*c]*l + 4*a*c*Sqrt[b^2 - 4*a*c]*l)*Log[-b + Sqrt
[b^2 - 4*a*c] - 2*c*x^2])/(4*c^2*(b^2 - 4*a*c)^(3/2)) - ((-4*c^3*e + 2*b*c^2*g -
4*a*c^2*j - b^3*l + 6*a*b*c*l - b^2*Sqrt[b^2 - 4*a*c]*l + 4*a*c*Sqrt[b^2 - 4*a*
c]*l)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(4*c^2*(b^2 - 4*a*c)^(3/2))

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Maple [B]  time = 0.151, size = 16517, normalized size = 21.5 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((m*x^8+l*x^7+k*x^6+j*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^2,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{a b c^{2} e - 2 \, a^{2} c^{2} g + a^{2} b c j -{\left (b c^{3} d - 2 \, a c^{3} f + a b c^{2} h -{\left (a b^{2} c - 2 \, a^{2} c^{2}\right )} k +{\left (a b^{3} - 3 \, a^{2} b c\right )} m\right )} x^{3} +{\left (2 \, a c^{3} e - a b c^{2} g +{\left (a b^{2} c - 2 \, a^{2} c^{2}\right )} j -{\left (a b^{3} - 3 \, a^{2} b c\right )} l\right )} x^{2} -{\left (a^{2} b^{2} - 2 \, a^{3} c\right )} l +{\left (a b c^{2} f - 2 \, a^{2} c^{2} h + a^{2} b c k -{\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d -{\left (a^{2} b^{2} - 2 \, a^{3} c\right )} m\right )} x}{2 \,{\left (a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3} +{\left (a b^{2} c^{3} - 4 \, a^{2} c^{4}\right )} x^{4} +{\left (a b^{3} c^{2} - 4 \, a^{2} b c^{3}\right )} x^{2}\right )}} + \frac{m x}{c^{2}} - \frac{-\int \frac{a b c^{2} f - 2 \, a^{2} c^{2} h + a^{2} b c k + 2 \,{\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} l x^{3} +{\left (b c^{3} d - 2 \, a c^{3} f + a b c^{2} h +{\left (a b^{2} c - 6 \, a^{2} c^{2}\right )} k -{\left (3 \, a b^{3} - 13 \, a^{2} b c\right )} m\right )} x^{2} +{\left (b^{2} c^{2} - 6 \, a c^{3}\right )} d -{\left (3 \, a^{2} b^{2} - 10 \, a^{3} c\right )} m - 2 \,{\left (2 \, a c^{3} e - a b c^{2} g + 2 \, a^{2} c^{2} j - a^{2} b c l\right )} x}{c x^{4} + b x^{2} + a}\,{d x}}{2 \,{\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((m*x^8 + l*x^7 + k*x^6 + j*x^5 + h*x^4 + g*x^3 + f*x^2 + e*x + d)/(c*x^4 + b*x^2 + a)^2,x, algorithm="maxima")

[Out]

-1/2*(a*b*c^2*e - 2*a^2*c^2*g + a^2*b*c*j - (b*c^3*d - 2*a*c^3*f + a*b*c^2*h - (
a*b^2*c - 2*a^2*c^2)*k + (a*b^3 - 3*a^2*b*c)*m)*x^3 + (2*a*c^3*e - a*b*c^2*g + (
a*b^2*c - 2*a^2*c^2)*j - (a*b^3 - 3*a^2*b*c)*l)*x^2 - (a^2*b^2 - 2*a^3*c)*l + (a
*b*c^2*f - 2*a^2*c^2*h + a^2*b*c*k - (b^2*c^2 - 2*a*c^3)*d - (a^2*b^2 - 2*a^3*c)
*m)*x)/(a^2*b^2*c^2 - 4*a^3*c^3 + (a*b^2*c^3 - 4*a^2*c^4)*x^4 + (a*b^3*c^2 - 4*a
^2*b*c^3)*x^2) + m*x/c^2 - 1/2*integrate(-(a*b*c^2*f - 2*a^2*c^2*h + a^2*b*c*k +
2*(a*b^2*c - 4*a^2*c^2)*l*x^3 + (b*c^3*d - 2*a*c^3*f + a*b*c^2*h + (a*b^2*c - 6
*a^2*c^2)*k - (3*a*b^3 - 13*a^2*b*c)*m)*x^2 + (b^2*c^2 - 6*a*c^3)*d - (3*a^2*b^2
- 10*a^3*c)*m - 2*(2*a*c^3*e - a*b*c^2*g + 2*a^2*c^2*j - a^2*b*c*l)*x)/(c*x^4 +
b*x^2 + a), x)/(a*b^2*c^2 - 4*a^2*c^3)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((m*x^8 + l*x^7 + k*x^6 + j*x^5 + h*x^4 + g*x^3 + f*x^2 + e*x + 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. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((m*x**8+l*x**7+k*x**6+j*x**5+h*x**4+g*x**3+f*x**2+e*x+d)/(c*x**4+b*x**2+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((m*x^8 + l*x^7 + k*x^6 + j*x^5 + h*x^4 + g*x^3 + f*x^2 + e*x + d)/(c*x^4 + b*x^2 + a)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError