3.59 \(\int \frac{d+e x+f x^2+g x^3+h x^4+i x^5+j x^8+k x^{11}}{\left (a+b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=1177 \[ \text{result too large to display} \]

[Out]

-(x*(c^2*(a*b*f - b^2*(d + (a^2*j)/c^2) + 2*a*(c*d - a*h + (a^2*j)/c)) + (2*a*c^
3*f - a*b^3*j - b*c*(c^2*d + a*c*h - 3*a^2*j))*x^2))/(4*a*c^2*(b^2 - 4*a*c)*(a +
b*x^2 + c*x^4)^2) - (b*c^3*(c*e + a*i) - a*b^4*k + 4*a^2*b^2*c*k - 2*a*c^2*(c^2
*g + a^2*k) + (2*c^5*e + b^2*c^3*i - c^4*(b*g + 2*a*i) - b^5*k + 5*a*b^3*c*k - 5
*a^2*b*c^2*k)*x^2)/(4*c^4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (x*(c*(a*b^3*f
+ 8*a^2*b*c*f + 4*a^2*(7*c^2*d + a*c*h - 9*a^2*j) + b^4*(3*d - (2*a^2*j)/c^2) -
a*b^2*(25*c*d + 7*a*h - (11*a^2*j)/c)) + (a*b^2*c^2*f + 20*a^2*c^3*f + b^3*(3*c^
2*d + a^2*j) - 4*a*b*c*(6*c^2*d + 3*a*c*h + 4*a^2*j))*x^2))/(8*a^2*c*(b^2 - 4*a*
c)^2*(a + b*x^2 + c*x^4)) + (b^3*c^2*i + 2*b*c^3*(3*c*e + a*i) + 11*a*b^4*k - (b
^6*k)/c + 32*a^3*c^2*k - 3*b^2*(c^3*g + 13*a^2*c*k) + 2*(6*c^5*e + b^2*c^3*i - c
^4*(3*b*g - 2*a*i) + 2*b^5*k - 15*a*b^3*c*k + 25*a^2*b*c^2*k)*x^2)/(4*c^3*(b^2 -
4*a*c)^2*(a + b*x^2 + c*x^4)) + ((a*b^2*c^2*f + 20*a^2*c^3*f + b^3*(3*c^2*d + a
^2*j) - 4*a*b*c*(6*c^2*d + 3*a*c*h + 4*a^2*j) + (a*b^3*c^2*f - 52*a^2*b*c^3*f -
6*a*b^2*c*(5*c^2*d - 3*a*c*h - 3*a^2*j) + b^4*(3*c^2*d - a^2*j) + 8*a^2*c^2*(21*
c^2*d + 3*a*c*h + 5*a^2*j))/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^2*f + 20*a^2*c^3*f + b^3*(3*c^2*d + a^2*j) - 4*a*b*c*(6*
c^2*d + 3*a*c*h + 4*a^2*j) - (a*b^3*c^2*f - 52*a^2*b*c^3*f - 6*a*b^2*c*(5*c^2*d
- 3*a*c*h - 3*a^2*j) + b^4*(3*c^2*d - a^2*j) + 8*a^2*c^2*(21*c^2*d + 3*a*c*h + 5
*a^2*j))/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]]) - ((12*
c^5*e + 2*b^2*c^3*i - c^4*(6*b*g - 4*a*i) - b^5*k + 10*a*b^3*c*k - 30*a^2*b*c^2*
k)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^3*(b^2 - 4*a*c)^(5/2)) + (k*Lo
g[a + b*x^2 + c*x^4])/(4*c^3)

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Rubi [A]  time = 21.295, antiderivative size = 1179, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 50, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{x \left (\left (-\left (\frac{j a^2}{c^2}+d\right ) b^2+a f b+2 a \left (\frac{j a^2}{c}-h a+c d\right )\right ) c^2+\left (-a j b^3-c \left (-3 j a^2+c h a+c^2 d\right ) b+2 a c^3 f\right ) x^2\right )}{4 a c^2 \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )^2}+\frac{\left (\left (\frac{j a^2}{c}+3 c d\right ) b^3+a c f b^2-4 a \left (4 j a^2+3 c h a+6 c^2 d\right ) b+20 a^2 c^2 f+\frac{\left (3 c^2 d-a^2 j\right ) b^4+a c^2 f b^3-6 a c \left (-3 j a^2-3 c h a+5 c^2 d\right ) b^2-52 a^2 c^3 f b+8 a^2 c^2 \left (5 j 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{j a^2}{c}+3 c d\right ) b^3+a c f b^2-4 a \left (4 j a^2+3 c h a+6 c^2 d\right ) b+20 a^2 c^2 f-\frac{\left (3 c^2 d-a^2 j\right ) b^4+a c^2 f b^3-6 a c \left (-3 j a^2-3 c h a+5 c^2 d\right ) b^2-52 a^2 c^3 f b+8 a^2 c^2 \left (5 j 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 (-k b^5+10 a c k b^3+2 c^3 i b^2-30 a^2 c^2 k b+12 c^5 e-c^4 (6 b g-4 a i)\right ) \tanh ^{-1}\left (\frac{2 c x^2+b}{\sqrt{b^2-4 a c}}\right )}{2 c^3 \left (b^2-4 a c\right )^{5/2}}+\frac{k \log \left (c x^4+b x^2+a\right )}{4 c^3}+\frac{x \left (\left (\left (j a^2+3 c^2 d\right ) b^3+a c^2 f b^2-4 a c \left (4 j a^2+3 c h a+6 c^2 d\right ) b+20 a^2 c^3 f\right ) x^2+c \left (\left (3 d-\frac{2 a^2 j}{c^2}\right ) b^4+a f b^3-a \left (-\frac{11 j a^2}{c}+7 h a+25 c d\right ) b^2+8 a^2 c f b+4 a^2 \left (-9 j a^2+c h a+7 c^2 d\right )\right )\right )}{8 a^2 c \left (b^2-4 a c\right )^2 \left (c x^4+b x^2+a\right )}+\frac{-\frac{k b^6}{c}+11 a k b^4+c^2 i b^3-3 \left (g c^3+13 a^2 k c\right ) b^2+2 c^3 (3 c e+a i) b+2 \left (2 k b^5-15 a c k b^3+c^3 i b^2+25 a^2 c^2 k b+6 c^5 e-c^4 (3 b g-2 a i)\right ) x^2+32 a^3 c^2 k}{4 c^3 \left (b^2-4 a c\right )^2 \left (c x^4+b x^2+a\right )}-\frac{-a k b^4+4 a^2 c k b^2+c^3 (c e+a i) b+\left (-k b^5+5 a c k b^3+c^3 i b^2-5 a^2 c^2 k b+2 c^5 e-c^4 (b g+2 a i)\right ) x^2-2 a c^2 \left (k a^2+c^2 g\right )}{4 c^4 \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 + i*x^5 + j*x^8 + k*x^11)/(a + b*x^2 + c*x^4)^3,x]

[Out]

-(x*(c^2*(a*b*f - b^2*(d + (a^2*j)/c^2) + 2*a*(c*d - a*h + (a^2*j)/c)) + (2*a*c^
3*f - a*b^3*j - b*c*(c^2*d + a*c*h - 3*a^2*j))*x^2))/(4*a*c^2*(b^2 - 4*a*c)*(a +
b*x^2 + c*x^4)^2) - (b*c^3*(c*e + a*i) - a*b^4*k + 4*a^2*b^2*c*k - 2*a*c^2*(c^2
*g + a^2*k) + (2*c^5*e + b^2*c^3*i - c^4*(b*g + 2*a*i) - b^5*k + 5*a*b^3*c*k - 5
*a^2*b*c^2*k)*x^2)/(4*c^4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (x*(c*(a*b^3*f
+ 8*a^2*b*c*f + 4*a^2*(7*c^2*d + a*c*h - 9*a^2*j) + b^4*(3*d - (2*a^2*j)/c^2) -
a*b^2*(25*c*d + 7*a*h - (11*a^2*j)/c)) + (a*b^2*c^2*f + 20*a^2*c^3*f + b^3*(3*c^
2*d + a^2*j) - 4*a*b*c*(6*c^2*d + 3*a*c*h + 4*a^2*j))*x^2))/(8*a^2*c*(b^2 - 4*a*
c)^2*(a + b*x^2 + c*x^4)) + (b^3*c^2*i + 2*b*c^3*(3*c*e + a*i) + 11*a*b^4*k - (b
^6*k)/c + 32*a^3*c^2*k - 3*b^2*(c^3*g + 13*a^2*c*k) + 2*(6*c^5*e + b^2*c^3*i - c
^4*(3*b*g - 2*a*i) + 2*b^5*k - 15*a*b^3*c*k + 25*a^2*b*c^2*k)*x^2)/(4*c^3*(b^2 -
4*a*c)^2*(a + b*x^2 + c*x^4)) + ((a*b^2*c*f + 20*a^2*c^2*f - 4*a*b*(6*c^2*d + 3
*a*c*h + 4*a^2*j) + b^3*(3*c*d + (a^2*j)/c) + (a*b^3*c^2*f - 52*a^2*b*c^3*f - 6*
a*b^2*c*(5*c^2*d - 3*a*c*h - 3*a^2*j) + b^4*(3*c^2*d - a^2*j) + 8*a^2*c^2*(21*c^
2*d + 3*a*c*h + 5*a^2*j))/(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 + 20*a^2*c^2*f - 4*a*b*(6*c^2*d + 3*a*c*h + 4*a^2*j)
+ b^3*(3*c*d + (a^2*j)/c) - (a*b^3*c^2*f - 52*a^2*b*c^3*f - 6*a*b^2*c*(5*c^2*d
- 3*a*c*h - 3*a^2*j) + b^4*(3*c^2*d - a^2*j) + 8*a^2*c^2*(21*c^2*d + 3*a*c*h + 5
*a^2*j))/(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]]) - (
(12*c^5*e + 2*b^2*c^3*i - c^4*(6*b*g - 4*a*i) - b^5*k + 10*a*b^3*c*k - 30*a^2*b*
c^2*k)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^3*(b^2 - 4*a*c)^(5/2)) + (
k*Log[a + b*x^2 + c*x^4])/(4*c^3)

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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((k*x**11+j*x**8+i*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 = 8.93561, size = 1649, normalized size = 1.4 \[ \frac{-a k x^2 b^5-a^2 k b^4-a c^2 j x^3 b^3+5 a^2 c k x^2 b^3+a c^3 i x^2 b^2+4 a^3 c k b^2-c^4 d x b^2-a^2 c^2 j x b^2-c^5 d x^3 b-a c^4 h x^3 b+3 a^2 c^3 j x^3 b-a c^4 g x^2 b-5 a^3 c^2 k x^2 b+a c^4 e b+a^2 c^3 i b+a c^4 f x b+2 a c^5 f x^3+2 a c^5 e x^2-2 a^2 c^4 i x^2-2 a^2 c^4 g-2 a^4 c^2 k+2 a c^5 d x-2 a^2 c^4 h x+2 a^3 c^3 j x}{4 a c^4 \left (4 a c-b^2\right ) \left (c x^4+b x^2+a\right )^2}+\frac{\left (40 c^2 j a^4+24 c^3 h a^3+18 b^2 c j a^3-16 b c \sqrt{b^2-4 a c} j 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-b^4 j a^2+b^3 \sqrt{b^2-4 a c} j 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 j a^4-24 c^3 h a^3-18 b^2 c j a^3-16 b c \sqrt{b^2-4 a c} j 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+b^4 j a^2+b^3 \sqrt{b^2-4 a c} j 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 (-k b^5+\sqrt{b^2-4 a c} k b^4+10 a c k b^3+2 c^3 i b^2-8 a c \sqrt{b^2-4 a c} k b^2-6 c^4 g b-30 a^2 c^2 k b+12 c^5 e+4 a c^4 i+16 a^2 c^2 \sqrt{b^2-4 a c} k\right ) \log \left (-2 c x^2-b+\sqrt{b^2-4 a c}\right )}{4 c^3 \left (b^2-4 a c\right )^{5/2}}+\frac{\left (k b^5+\sqrt{b^2-4 a c} k b^4-10 a c k b^3-2 c^3 i b^2-8 a c \sqrt{b^2-4 a c} k b^2+6 c^4 g b+30 a^2 c^2 k b-12 c^5 e-4 a c^4 i+16 a^2 c^2 \sqrt{b^2-4 a c} k\right ) \log \left (2 c x^2+b+\sqrt{b^2-4 a c}\right )}{4 c^3 \left (b^2-4 a c\right )^{5/2}}+\frac{-2 a^2 k b^6+8 a^2 c k x^2 b^5+22 a^3 c k b^4+3 c^4 d x b^4-2 a^2 c^2 j x b^4+3 c^5 d x^3 b^3+a^2 c^3 j x^3 b^3-60 a^3 c^2 k x^2 b^3+2 a^2 c^3 i b^3+a c^4 f x b^3+a c^5 f x^3 b^2+4 a^2 c^4 i x^2 b^2-6 a^2 c^4 g b^2-78 a^4 c^2 k b^2-25 a c^5 d x b^2-7 a^2 c^4 h x b^2+11 a^3 c^3 j x b^2-24 a c^6 d x^3 b-12 a^2 c^5 h x^3 b-16 a^3 c^4 j x^3 b-12 a^2 c^5 g x^2 b+100 a^4 c^3 k x^2 b+12 a^2 c^5 e b+4 a^3 c^4 i b+8 a^2 c^5 f x b+20 a^2 c^6 f x^3+24 a^2 c^6 e x^2+8 a^3 c^5 i x^2+64 a^5 c^3 k+28 a^2 c^6 d x+4 a^3 c^5 h x-36 a^4 c^4 j x}{8 a^2 c^4 \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 + i*x^5 + j*x^8 + k*x^11)/(a + b*x^2 + c*x^4)^3,x]

[Out]

(a*b*c^4*e - 2*a^2*c^4*g + a^2*b*c^3*i - a^2*b^4*k + 4*a^3*b^2*c*k - 2*a^4*c^2*k
- b^2*c^4*d*x + 2*a*c^5*d*x + a*b*c^4*f*x - 2*a^2*c^4*h*x - a^2*b^2*c^2*j*x + 2
*a^3*c^3*j*x + 2*a*c^5*e*x^2 - a*b*c^4*g*x^2 + a*b^2*c^3*i*x^2 - 2*a^2*c^4*i*x^2
- a*b^5*k*x^2 + 5*a^2*b^3*c*k*x^2 - 5*a^3*b*c^2*k*x^2 - b*c^5*d*x^3 + 2*a*c^5*f
*x^3 - a*b*c^4*h*x^3 - a*b^3*c^2*j*x^3 + 3*a^2*b*c^3*j*x^3)/(4*a*c^4*(-b^2 + 4*a
*c)*(a + b*x^2 + c*x^4)^2) + (12*a^2*b*c^5*e - 6*a^2*b^2*c^4*g + 2*a^2*b^3*c^3*i
+ 4*a^3*b*c^4*i - 2*a^2*b^6*k + 22*a^3*b^4*c*k - 78*a^4*b^2*c^2*k + 64*a^5*c^3*
k + 3*b^4*c^4*d*x - 25*a*b^2*c^5*d*x + 28*a^2*c^6*d*x + a*b^3*c^4*f*x + 8*a^2*b*
c^5*f*x - 7*a^2*b^2*c^4*h*x + 4*a^3*c^5*h*x - 2*a^2*b^4*c^2*j*x + 11*a^3*b^2*c^3
*j*x - 36*a^4*c^4*j*x + 24*a^2*c^6*e*x^2 - 12*a^2*b*c^5*g*x^2 + 4*a^2*b^2*c^4*i*
x^2 + 8*a^3*c^5*i*x^2 + 8*a^2*b^5*c*k*x^2 - 60*a^3*b^3*c^2*k*x^2 + 100*a^4*b*c^3
*k*x^2 + 3*b^3*c^5*d*x^3 - 24*a*b*c^6*d*x^3 + a*b^2*c^5*f*x^3 + 20*a^2*c^6*f*x^3
- 12*a^2*b*c^5*h*x^3 + a^2*b^3*c^3*j*x^3 - 16*a^3*b*c^4*j*x^3)/(8*a^2*c^4*(-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
- 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 - a^2*b^
4*j + 18*a^3*b^2*c*j + 40*a^4*c^2*j + a^2*b^3*Sqrt[b^2 - 4*a*c]*j - 16*a^3*b*c*S
qrt[b^2 - 4*a*c]*j)*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 + a^2*b^4*j - 18*a^3*b^2*c*j - 40*a^4*c^2*j + a^2*b
^3*Sqrt[b^2 - 4*a*c]*j - 16*a^3*b*c*Sqrt[b^2 - 4*a*c]*j)*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)*Sqr
t[b + Sqrt[b^2 - 4*a*c]]) + ((12*c^5*e - 6*b*c^4*g + 2*b^2*c^3*i + 4*a*c^4*i - b
^5*k + 10*a*b^3*c*k - 30*a^2*b*c^2*k + b^4*Sqrt[b^2 - 4*a*c]*k - 8*a*b^2*c*Sqrt[
b^2 - 4*a*c]*k + 16*a^2*c^2*Sqrt[b^2 - 4*a*c]*k)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*
c*x^2])/(4*c^3*(b^2 - 4*a*c)^(5/2)) + ((-12*c^5*e + 6*b*c^4*g - 2*b^2*c^3*i - 4*
a*c^4*i + b^5*k - 10*a*b^3*c*k + 30*a^2*b*c^2*k + b^4*Sqrt[b^2 - 4*a*c]*k - 8*a*
b^2*c*Sqrt[b^2 - 4*a*c]*k + 16*a^2*c^2*Sqrt[b^2 - 4*a*c]*k)*Log[b + Sqrt[b^2 - 4
*a*c] + 2*c*x^2])/(4*c^3*(b^2 - 4*a*c)^(5/2))

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Maple [B]  time = 0.361, size = 35336, normalized size = 30. \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((k*x^11+j*x^8+i*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((k*x^11 + j*x^8 + i*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^4*b*c^3*i - (12*a^2*b*c^5*h - 3*(b^3*c^5 - 8*a*b*c^6)*d - (a*b^2*c^5 +
20*a^2*c^6)*f - (a^2*b^3*c^3 - 16*a^3*b*c^4)*j)*x^7 + 4*(6*a^2*c^6*e - 3*a^2*b*
c^5*g + (a^2*b^2*c^4 + 2*a^3*c^5)*i + (2*a^2*b^5*c - 15*a^3*b^3*c^2 + 25*a^4*b*c
^3)*k)*x^6 + ((6*b^4*c^4 - 49*a*b^2*c^5 + 28*a^2*c^6)*d + 2*(a*b^3*c^4 + 14*a^2*
b*c^5)*f - (19*a^2*b^2*c^4 - 4*a^3*c^5)*h - (a^2*b^4*c^2 + 5*a^3*b^2*c^3 + 36*a^
4*c^4)*j)*x^5 + 2*(18*a^2*b*c^5*e - 9*a^2*b^2*c^4*g + 3*(a^2*b^3*c^3 + 2*a^3*b*c
^4)*i + (3*a^2*b^6 - 19*a^3*b^4*c + 11*a^4*b^2*c^2 + 32*a^5*c^3)*k)*x^4 + ((3*b^
5*c^3 - 20*a*b^3*c^4 - 4*a^2*b*c^5)*d + (a*b^4*c^3 + 5*a^2*b^2*c^4 + 36*a^3*c^5)
*f - (5*a^2*b^3*c^3 + 16*a^3*b*c^4)*h - 2*(a^3*b^3*c^2 + 14*a^4*b*c^3)*j)*x^3 +
4*(2*(a^2*b^2*c^4 + 5*a^3*c^5)*e - (a^2*b^3*c^3 + 5*a^3*b*c^4)*g + (5*a^3*b^2*c^
3 - 2*a^4*c^4)*i + (3*a^3*b^5 - 22*a^4*b^3*c + 31*a^5*b*c^2)*k)*x^2 - 2*(a^2*b^3
*c^3 - 10*a^3*b*c^4)*e - 2*(a^3*b^2*c^3 + 8*a^4*c^4)*g + 6*(a^4*b^4 - 7*a^5*b^2*
c + 8*a^6*c^2)*k + ((5*a*b^4*c^3 - 37*a^2*b^2*c^4 + 44*a^3*c^5)*d - (a^2*b^3*c^3
- 16*a^3*b*c^4)*f - 3*(a^3*b^2*c^3 + 4*a^4*c^4)*h - (a^4*b^2*c^2 + 20*a^5*c^3)*
j)*x)/(a^4*b^4*c^3 - 8*a^5*b^2*c^4 + 16*a^6*c^5 + (a^2*b^4*c^5 - 8*a^3*b^2*c^6 +
16*a^4*c^7)*x^8 + 2*(a^2*b^5*c^4 - 8*a^3*b^3*c^5 + 16*a^4*b*c^6)*x^6 + (a^2*b^6
*c^3 - 6*a^3*b^4*c^4 + 32*a^5*c^6)*x^4 + 2*(a^3*b^5*c^3 - 8*a^4*b^3*c^4 + 16*a^5
*b*c^5)*x^2) + 1/8*integrate((8*(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*k*x^3 - (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 - (a^2*b^3
*c - 16*a^3*b*c^2)*j)*x^2 + 3*(b^4*c^2 - 9*a*b^2*c^3 + 28*a^2*c^4)*d + (a*b^3*c^
2 - 16*a^2*b*c^3)*f + 3*(a^2*b^2*c^2 + 4*a^3*c^3)*h + (a^3*b^2*c + 20*a^4*c^2)*j
+ 8*(6*a^2*c^4*e - 3*a^2*b*c^3*g + (a^2*b^2*c^2 + 2*a^3*c^3)*i + (a^3*b^3 - 7*a
^4*b*c)*k)*x)/(c*x^4 + b*x^2 + a), x)/(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)

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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((k*x^11 + j*x^8 + i*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((k*x**11+j*x**8+i*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(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((k*x^11 + j*x^8 + i*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]

Timed out