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

Optimal. Leaf size=1150 \[ \text{result too large to display} \]

[Out]

-(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)/(4*c^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (
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))/(
4*a*c^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + ((b^3*j)/c + 2*b*(3*c*e + a*j) -
16*a^2*l - (b^4*l)/c^2 - b^2*(3*g - (5*a*l)/c) + 2*(6*c^2*e - 3*b*c*g + b^2*j +
2*a*c*j - 3*a*b*l)*x^2)/(4*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (x*(4*a^2*b*c^
2*(2*c*f + a*k) + a*b^3*c*(c*f + 2*a*k) - a*b^2*c*(25*c^2*d + 7*a*c*h - 11*a^2*m
) + 4*a^2*c^2*(7*c^2*d + a*c*h - 9*a^2*m) + b^4*(3*c^2*d - 2*a^2*m) + c*(a*b^2*c
*(c*f + 3*a*k) + 4*a^2*c^2*(5*c*f + 3*a*k) + b^3*(3*c^2*d + a^2*m) - 4*a*b*c*(6*
c^2*d + 3*a*c*h + 4*a^2*m))*x^2))/(8*a^2*c^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)
) + ((a*b^2*c*(c*f + 3*a*k) + 4*a^2*c^2*(5*c*f + 3*a*k) + b^3*(3*c^2*d + a^2*m)
- 4*a*b*c*(6*c^2*d + 3*a*c*h + 4*a^2*m) + (a*b^3*c*(c*f - 3*a*k) - 4*a^2*b*c^2*(
13*c*f + 9*a*k) - 6*a*b^2*c*(5*c^2*d - 3*a*c*h - 3*a^2*m) + b^4*(3*c^2*d - a^2*m
) + 8*a^2*c^2*(21*c^2*d + 3*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]]])/(8*Sqrt[2]*a^2*c^(3/2)*(b^2 - 4*a*c)^2
*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((a*b^2*c*(c*f + 3*a*k) + 4*a^2*c^2*(5*c*f + 3*a
*k) + b^3*(3*c^2*d + a^2*m) - 4*a*b*c*(6*c^2*d + 3*a*c*h + 4*a^2*m) - (a*b^3*c*(
c*f - 3*a*k) - 4*a^2*b*c^2*(13*c*f + 9*a*k) - 6*a*b^2*c*(5*c^2*d - 3*a*c*h - 3*a
^2*m) + b^4*(3*c^2*d - a^2*m) + 8*a^2*c^2*(21*c^2*d + 3*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]]])/(8*Sqrt[2]
*a^2*c^(3/2)*(b^2 - 4*a*c)^2*Sqrt[b + Sqrt[b^2 - 4*a*c]]) - ((6*c^2*e - 3*b*c*g
+ b^2*j + 2*a*c*j - 3*a*b*l)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*
a*c)^(5/2)

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Rubi [A]  time = 24.6691, antiderivative size = 1144, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 9, integrand size = 55, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.164 \[ \frac{-\frac{l b^4}{c^2}+\frac{j b^3}{c}-\left (3 g-\frac{5 a l}{c}\right ) b^2+2 (3 c e+a j) b+2 \left (j b^2-3 c g b-3 a l b+6 c^2 e+2 a c j\right ) x^2-16 a^2 l}{4 \left (b^2-4 a c\right )^2 \left (c x^4+b x^2+a\right )}+\frac{\left (\left (\frac{m a^2}{c}+3 c d\right ) b^3+a (c f+3 a k) b^2-4 a \left (4 m a^2+3 c h a+6 c^2 d\right ) b+4 a^2 c (5 c f+3 a k)+\frac{\left (3 c^2 d-a^2 m\right ) b^4+a c (c f-3 a k) b^3-6 a c \left (-3 m a^2-3 c h a+5 c^2 d\right ) b^2-4 a^2 c^2 (13 c f+9 a k) b+8 a^2 c^2 \left (5 m a^2+3 c h a+21 c^2 d\right )}{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 )}{8 \sqrt{2} a^2 \sqrt{c} \left (b^2-4 a c\right )^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (\left (\frac{m a^2}{c}+3 c d\right ) b^3+a (c f+3 a k) b^2-4 a \left (4 m a^2+3 c h a+6 c^2 d\right ) b+4 a^2 c (5 c f+3 a k)-\frac{\left (3 c^2 d-a^2 m\right ) b^4+a c (c f-3 a k) b^3-6 a c \left (-3 m a^2-3 c h a+5 c^2 d\right ) b^2-4 a^2 c^2 (13 c f+9 a k) b+8 a^2 c^2 \left (5 m a^2+3 c h a+21 c^2 d\right )}{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 )}{8 \sqrt{2} a^2 \sqrt{c} \left (b^2-4 a c\right )^2 \sqrt{b+\sqrt{b^2-4 a c}}}-\frac{\left (j b^2-3 c g b-3 a l b+6 c^2 e+2 a c j\right ) \tanh ^{-1}\left (\frac{2 c x^2+b}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{x \left (\left (3 c d-\frac{2 a^2 m}{c}\right ) b^4+a (c f+2 a k) b^3-a \left (-11 m a^2+7 c h a+25 c^2 d\right ) b^2+4 a^2 c (2 c f+a k) b+\left (\left (m a^2+3 c^2 d\right ) b^3+a c (c f+3 a k) b^2-4 a c \left (4 m a^2+3 c h a+6 c^2 d\right ) b+4 a^2 c^2 (5 c f+3 a k)\right ) x^2+4 a^2 c \left (-9 m a^2+c h a+7 c^2 d\right )\right )}{8 a^2 c \left (b^2-4 a c\right )^2 \left (c x^4+b x^2+a\right )}-\frac{-a l b^2+c (c e+a j) b+\left (-l b^3+c (b j+3 a l) b+2 c^3 e-c^2 (b g+2 a j)\right ) x^2-2 a c (c g-a l)}{4 c^2 \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )^2}-\frac{x \left (-\left (m a^2+c^2 d\right ) b^2+a c (c f+a k) b+\left (-a m b^3+a c k b^2-c \left (-3 m a^2+c h a+c^2 d\right ) b+2 a c^2 (c f-a k)\right ) x^2+2 a c \left (m a^2-c h a+c^2 d\right )\right )}{4 a c^2 \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )^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)^3,x]

[Out]

-(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)/(4*c^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (
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))/(
4*a*c^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + ((b^3*j)/c + 2*b*(3*c*e + a*j) -
16*a^2*l - (b^4*l)/c^2 - b^2*(3*g - (5*a*l)/c) + 2*(6*c^2*e - 3*b*c*g + b^2*j +
2*a*c*j - 3*a*b*l)*x^2)/(4*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (x*(4*a^2*b*c*
(2*c*f + a*k) + a*b^3*(c*f + 2*a*k) - a*b^2*(25*c^2*d + 7*a*c*h - 11*a^2*m) + 4*
a^2*c*(7*c^2*d + a*c*h - 9*a^2*m) + b^4*(3*c*d - (2*a^2*m)/c) + (a*b^2*c*(c*f +
3*a*k) + 4*a^2*c^2*(5*c*f + 3*a*k) + b^3*(3*c^2*d + a^2*m) - 4*a*b*c*(6*c^2*d +
3*a*c*h + 4*a^2*m))*x^2))/(8*a^2*c*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + ((a*b^
2*(c*f + 3*a*k) + 4*a^2*c*(5*c*f + 3*a*k) - 4*a*b*(6*c^2*d + 3*a*c*h + 4*a^2*m)
+ b^3*(3*c*d + (a^2*m)/c) + (a*b^3*c*(c*f - 3*a*k) - 4*a^2*b*c^2*(13*c*f + 9*a*k
) - 6*a*b^2*c*(5*c^2*d - 3*a*c*h - 3*a^2*m) + b^4*(3*c^2*d - a^2*m) + 8*a^2*c^2*
(21*c^2*d + 3*a*c*h + 5*a^2*m))/(c*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*x
)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*Sqrt[c]*(b^2 - 4*a*c)^2*Sqrt[b -
Sqrt[b^2 - 4*a*c]]) + ((a*b^2*(c*f + 3*a*k) + 4*a^2*c*(5*c*f + 3*a*k) - 4*a*b*(6
*c^2*d + 3*a*c*h + 4*a^2*m) + b^3*(3*c*d + (a^2*m)/c) - (a*b^3*c*(c*f - 3*a*k) -
4*a^2*b*c^2*(13*c*f + 9*a*k) - 6*a*b^2*c*(5*c^2*d - 3*a*c*h - 3*a^2*m) + b^4*(3
*c^2*d - a^2*m) + 8*a^2*c^2*(21*c^2*d + 3*a*c*h + 5*a^2*m))/(c*Sqrt[b^2 - 4*a*c]
))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*Sqrt[
c]*(b^2 - 4*a*c)^2*Sqrt[b + Sqrt[b^2 - 4*a*c]]) - ((6*c^2*e - 3*b*c*g + b^2*j +
2*a*c*j - 3*a*b*l)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(5/2)

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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)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 9.31706, size = 1590, normalized size = 1.38 \[ \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}{4 a c^2 \left (4 a c-b^2\right ) \left (c x^4+b x^2+a\right )^2}+\frac{\left (40 c^2 m a^4+24 c^3 h a^3-36 b c^2 k a^3+12 c^2 \sqrt{b^2-4 a c} k a^3+18 b^2 c m a^3-16 b c \sqrt{b^2-4 a c} m a^3+168 c^4 d a^2-52 b c^3 f a^2+20 c^3 \sqrt{b^2-4 a c} f a^2+18 b^2 c^2 h a^2-12 b c^2 \sqrt{b^2-4 a c} h a^2-3 b^3 c k a^2+3 b^2 c \sqrt{b^2-4 a c} k a^2-b^4 m a^2+b^3 \sqrt{b^2-4 a c} m a^2-30 b^2 c^3 d a-24 b c^3 \sqrt{b^2-4 a c} d a+b^3 c^2 f a+b^2 c^2 \sqrt{b^2-4 a c} f a+3 b^4 c^2 d+3 b^3 c^2 \sqrt{b^2-4 a c} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (-40 c^2 m a^4-24 c^3 h a^3+36 b c^2 k a^3+12 c^2 \sqrt{b^2-4 a c} k a^3-18 b^2 c m a^3-16 b c \sqrt{b^2-4 a c} m a^3-168 c^4 d a^2+52 b c^3 f a^2+20 c^3 \sqrt{b^2-4 a c} f a^2-18 b^2 c^2 h a^2-12 b c^2 \sqrt{b^2-4 a c} h a^2+3 b^3 c k a^2+3 b^2 c \sqrt{b^2-4 a c} k a^2+b^4 m a^2+b^3 \sqrt{b^2-4 a c} m a^2+30 b^2 c^3 d a-24 b c^3 \sqrt{b^2-4 a c} d a-b^3 c^2 f a+b^2 c^2 \sqrt{b^2-4 a c} f a-3 b^4 c^2 d+3 b^3 c^2 \sqrt{b^2-4 a c} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt{b+\sqrt{b^2-4 a c}}}+\frac{\left (j b^2-3 c g b-3 a l b+6 c^2 e+2 a c j\right ) \log \left (-2 c x^2-b+\sqrt{b^2-4 a c}\right )}{2 \left (b^2-4 a c\right )^{5/2}}+\frac{\left (-j b^2+3 c g b+3 a l b-6 c^2 e-2 a c j\right ) \log \left (2 c x^2+b+\sqrt{b^2-4 a c}\right )}{2 \left (b^2-4 a c\right )^{5/2}}+\frac{-32 c^2 l a^4-36 c^2 m x a^4+12 c^3 k x^3 a^3-16 b c^2 m x^3 a^3+8 c^3 j x^2 a^3-12 b c^2 l x^2 a^3+4 b c^2 j a^3+10 b^2 c l a^3+4 c^3 h x a^3+4 b c^2 k x a^3+11 b^2 c m x a^3+20 c^4 f x^3 a^2-12 b c^3 h x^3 a^2+3 b^2 c^2 k x^3 a^2+b^3 c m x^3 a^2+24 c^4 e x^2 a^2-12 b c^3 g x^2 a^2+4 b^2 c^2 j x^2 a^2+12 b c^3 e a^2-6 b^2 c^2 g a^2+2 b^3 c j a^2-2 b^4 l a^2+28 c^4 d x a^2+8 b c^3 f x a^2-7 b^2 c^2 h x a^2+2 b^3 c k x a^2-2 b^4 m x a^2-24 b c^4 d x^3 a+b^2 c^3 f x^3 a-25 b^2 c^3 d x a+b^3 c^2 f x a+3 b^3 c^3 d x^3+3 b^4 c^2 d x}{8 a^2 c^2 \left (4 a c-b^2\right )^2 \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)^3,x]

[Out]

(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)/(4*a*c^2*(-b^2 + 4*a*c)*(a +
b*x^2 + c*x^4)^2) + (12*a^2*b*c^3*e - 6*a^2*b^2*c^2*g + 2*a^2*b^3*c*j + 4*a^3*b
*c^2*j - 2*a^2*b^4*l + 10*a^3*b^2*c*l - 32*a^4*c^2*l + 3*b^4*c^2*d*x - 25*a*b^2*
c^3*d*x + 28*a^2*c^4*d*x + a*b^3*c^2*f*x + 8*a^2*b*c^3*f*x - 7*a^2*b^2*c^2*h*x +
4*a^3*c^3*h*x + 2*a^2*b^3*c*k*x + 4*a^3*b*c^2*k*x - 2*a^2*b^4*m*x + 11*a^3*b^2*
c*m*x - 36*a^4*c^2*m*x + 24*a^2*c^4*e*x^2 - 12*a^2*b*c^3*g*x^2 + 4*a^2*b^2*c^2*j
*x^2 + 8*a^3*c^3*j*x^2 - 12*a^3*b*c^2*l*x^2 + 3*b^3*c^3*d*x^3 - 24*a*b*c^4*d*x^3
+ a*b^2*c^3*f*x^3 + 20*a^2*c^4*f*x^3 - 12*a^2*b*c^3*h*x^3 + 3*a^2*b^2*c^2*k*x^3
+ 12*a^3*c^3*k*x^3 + a^2*b^3*c*m*x^3 - 16*a^3*b*c^2*m*x^3)/(8*a^2*c^2*(-b^2 + 4
*a*c)^2*(a + b*x^2 + c*x^4)) + ((3*b^4*c^2*d - 30*a*b^2*c^3*d + 168*a^2*c^4*d +
3*b^3*c^2*Sqrt[b^2 - 4*a*c]*d - 24*a*b*c^3*Sqrt[b^2 - 4*a*c]*d + a*b^3*c^2*f - 5
2*a^2*b*c^3*f + a*b^2*c^2*Sqrt[b^2 - 4*a*c]*f + 20*a^2*c^3*Sqrt[b^2 - 4*a*c]*f +
18*a^2*b^2*c^2*h + 24*a^3*c^3*h - 12*a^2*b*c^2*Sqrt[b^2 - 4*a*c]*h - 3*a^2*b^3*
c*k - 36*a^3*b*c^2*k + 3*a^2*b^2*c*Sqrt[b^2 - 4*a*c]*k + 12*a^3*c^2*Sqrt[b^2 - 4
*a*c]*k - a^2*b^4*m + 18*a^3*b^2*c*m + 40*a^4*c^2*m + a^2*b^3*Sqrt[b^2 - 4*a*c]*
m - 16*a^3*b*c*Sqrt[b^2 - 4*a*c]*m)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2
- 4*a*c]]])/(8*Sqrt[2]*a^2*c^(3/2)*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*
c]]) + ((-3*b^4*c^2*d + 30*a*b^2*c^3*d - 168*a^2*c^4*d + 3*b^3*c^2*Sqrt[b^2 - 4*
a*c]*d - 24*a*b*c^3*Sqrt[b^2 - 4*a*c]*d - a*b^3*c^2*f + 52*a^2*b*c^3*f + a*b^2*c
^2*Sqrt[b^2 - 4*a*c]*f + 20*a^2*c^3*Sqrt[b^2 - 4*a*c]*f - 18*a^2*b^2*c^2*h - 24*
a^3*c^3*h - 12*a^2*b*c^2*Sqrt[b^2 - 4*a*c]*h + 3*a^2*b^3*c*k + 36*a^3*b*c^2*k +
3*a^2*b^2*c*Sqrt[b^2 - 4*a*c]*k + 12*a^3*c^2*Sqrt[b^2 - 4*a*c]*k + a^2*b^4*m - 1
8*a^3*b^2*c*m - 40*a^4*c^2*m + a^2*b^3*Sqrt[b^2 - 4*a*c]*m - 16*a^3*b*c*Sqrt[b^2
- 4*a*c]*m)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]
*a^2*c^(3/2)*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + ((6*c^2*e - 3*b*
c*g + b^2*j + 2*a*c*j - 3*a*b*l)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(2*(b^2
- 4*a*c)^(5/2)) + ((-6*c^2*e + 3*b*c*g - b^2*j - 2*a*c*j + 3*a*b*l)*Log[b + Sqrt
[b^2 - 4*a*c] + 2*c*x^2])/(2*(b^2 - 4*a*c)^(5/2))

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Maple [B]  time = 0.344, size = 36326, normalized size = 31.6 \[ \text{output too large to display} \]

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)^3,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{result too large to display} \]

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)^3,x, algorithm="maxima")

[Out]

-1/8*((12*a^2*b*c^3*h - 3*(b^3*c^3 - 8*a*b*c^4)*d - (a*b^2*c^3 + 20*a^2*c^4)*f -
3*(a^2*b^2*c^2 + 4*a^3*c^3)*k - (a^2*b^3*c - 16*a^3*b*c^2)*m)*x^7 - 12*a^4*b*c*
j - 4*(6*a^2*c^4*e - 3*a^2*b*c^3*g - 3*a^3*b*c^2*l + (a^2*b^2*c^2 + 2*a^3*c^3)*j
)*x^6 - ((6*b^4*c^2 - 49*a*b^2*c^3 + 28*a^2*c^4)*d + 2*(a*b^3*c^2 + 14*a^2*b*c^3
)*f - (19*a^2*b^2*c^2 - 4*a^3*c^3)*h + (5*a^2*b^3*c + 16*a^3*b*c^2)*k - (a^2*b^4
+ 5*a^3*b^2*c + 36*a^4*c^2)*m)*x^5 - 2*(18*a^2*b*c^3*e - 9*a^2*b^2*c^2*g + 3*(a
^2*b^3*c + 2*a^3*b*c^2)*j - (a^2*b^4 + a^3*b^2*c + 16*a^4*c^2)*l)*x^4 - ((3*b^5*
c - 20*a*b^3*c^2 - 4*a^2*b*c^3)*d + (a*b^4*c + 5*a^2*b^2*c^2 + 36*a^3*c^3)*f - (
5*a^2*b^3*c + 16*a^3*b*c^2)*h + (19*a^3*b^2*c - 4*a^4*c^2)*k - 2*(a^3*b^3 + 14*a
^4*b*c)*m)*x^3 - 4*(2*(a^2*b^2*c^2 + 5*a^3*c^3)*e - (a^2*b^3*c + 5*a^3*b*c^2)*g
+ (5*a^3*b^2*c - 2*a^4*c^2)*j - (a^3*b^3 + 5*a^4*b*c)*l)*x^2 + 2*(a^2*b^3*c - 10
*a^3*b*c^2)*e + 2*(a^3*b^2*c + 8*a^4*c^2)*g + 2*(a^4*b^2 + 8*a^5*c)*l - (12*a^4*
b*c*k + (5*a*b^4*c - 37*a^2*b^2*c^2 + 44*a^3*c^3)*d - (a^2*b^3*c - 16*a^3*b*c^2)
*f - 3*(a^3*b^2*c + 4*a^4*c^2)*h - (a^4*b^2 + 20*a^5*c)*m)*x)/(a^4*b^4*c - 8*a^5
*b^2*c^2 + 16*a^6*c^3 + (a^2*b^4*c^3 - 8*a^3*b^2*c^4 + 16*a^4*c^5)*x^8 + 2*(a^2*
b^5*c^2 - 8*a^3*b^3*c^3 + 16*a^4*b*c^4)*x^6 + (a^2*b^6*c - 6*a^3*b^4*c^2 + 32*a^
5*c^4)*x^4 + 2*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*x^2) - 1/8*integrate((
12*a^3*b*c*k + (12*a^2*b*c^2*h - 3*(b^3*c^2 - 8*a*b*c^3)*d - (a*b^2*c^2 + 20*a^2
*c^3)*f - 3*(a^2*b^2*c + 4*a^3*c^2)*k - (a^2*b^3 - 16*a^3*b*c)*m)*x^2 - 3*(b^4*c
- 9*a*b^2*c^2 + 28*a^2*c^3)*d - (a*b^3*c - 16*a^2*b*c^2)*f - 3*(a^2*b^2*c + 4*a
^3*c^2)*h - (a^3*b^2 + 20*a^4*c)*m - 8*(6*a^2*c^3*e - 3*a^2*b*c^2*g - 3*a^3*b*c*
l + (a^2*b^2*c + 2*a^3*c^2)*j)*x)/(c*x^4 + b*x^2 + a), x)/(a^2*b^4*c - 8*a^3*b^2
*c^2 + 16*a^4*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)^3,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)**3,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)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError