3.67 \(\int \frac{d+e x^2+f x^4}{x^5 \left (a+b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=329 \[ -\frac{\log \left (a+b x^2+c x^4\right ) \left (-2 a b e-a (2 c d-a f)+3 b^2 d\right )}{4 a^4}+\frac{\log (x) \left (-2 a b e-a (2 c d-a f)+3 b^2 d\right )}{a^4}+\frac{2 b d-a e}{2 a^3 x^2}-\frac{d}{4 a^2 x^4}+\frac{c x^2 \left (2 a^2 c e-a b^2 e-a b (3 c d-a f)+b^3 d\right )+3 a^2 b c e+2 a^2 c (c d-a f)-a b^3 e-a b^2 (4 c d-a f)+b^4 d}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (-12 a^3 c^2 e+12 a^2 b^2 c e+6 a^2 b c (5 c d-a f)-2 a b^4 e-a b^3 (20 c d-a f)+3 b^5 d\right )}{2 a^4 \left (b^2-4 a c\right )^{3/2}} \]

[Out]

-d/(4*a^2*x^4) + (2*b*d - a*e)/(2*a^3*x^2) + (b^4*d - a*b^3*e + 3*a^2*b*c*e + 2*
a^2*c*(c*d - a*f) - a*b^2*(4*c*d - a*f) + c*(b^3*d - a*b^2*e + 2*a^2*c*e - a*b*(
3*c*d - a*f))*x^2)/(2*a^3*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + ((3*b^5*d - 2*a*b
^4*e + 12*a^2*b^2*c*e - 12*a^3*c^2*e + 6*a^2*b*c*(5*c*d - a*f) - a*b^3*(20*c*d -
a*f))*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^4*(b^2 - 4*a*c)^(3/2)) + (
(3*b^2*d - 2*a*b*e - a*(2*c*d - a*f))*Log[x])/a^4 - ((3*b^2*d - 2*a*b*e - a*(2*c
*d - a*f))*Log[a + b*x^2 + c*x^4])/(4*a^4)

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Rubi [A]  time = 2.36342, antiderivative size = 329, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.233 \[ -\frac{\log \left (a+b x^2+c x^4\right ) \left (-2 a b e-a (2 c d-a f)+3 b^2 d\right )}{4 a^4}+\frac{\log (x) \left (-2 a b e-a (2 c d-a f)+3 b^2 d\right )}{a^4}+\frac{2 b d-a e}{2 a^3 x^2}-\frac{d}{4 a^2 x^4}+\frac{c x^2 \left (2 a^2 c e-a b^2 e-a b (3 c d-a f)+b^3 d\right )+3 a^2 b c e+2 a^2 c (c d-a f)-a b^3 e-a b^2 (4 c d-a f)+b^4 d}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (-12 a^3 c^2 e+12 a^2 b^2 c e+6 a^2 b c (5 c d-a f)-2 a b^4 e-a b^3 (20 c d-a f)+3 b^5 d\right )}{2 a^4 \left (b^2-4 a c\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x^2 + f*x^4)/(x^5*(a + b*x^2 + c*x^4)^2),x]

[Out]

-d/(4*a^2*x^4) + (2*b*d - a*e)/(2*a^3*x^2) + (b^4*d - a*b^3*e + 3*a^2*b*c*e + 2*
a^2*c*(c*d - a*f) - a*b^2*(4*c*d - a*f) + c*(b^3*d - a*b^2*e + 2*a^2*c*e - a*b*(
3*c*d - a*f))*x^2)/(2*a^3*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + ((3*b^5*d - 2*a*b
^4*e + 12*a^2*b^2*c*e - 12*a^3*c^2*e + 6*a^2*b*c*(5*c*d - a*f) - a*b^3*(20*c*d -
a*f))*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^4*(b^2 - 4*a*c)^(3/2)) + (
(3*b^2*d - 2*a*b*e - a*(2*c*d - a*f))*Log[x])/a^4 - ((3*b^2*d - 2*a*b*e - a*(2*c
*d - a*f))*Log[a + b*x^2 + c*x^4])/(4*a^4)

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Rubi in Sympy [A]  time = 119.625, size = 384, normalized size = 1.17 \[ - \frac{d}{4 a^{2} x^{4}} - \frac{c \left (a^{2} b f + 2 a^{2} c e - a b^{2} e - 3 a b c d + b^{3} d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{a^{3} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} + \frac{- 2 a^{3} c f + a^{2} b^{2} f + 3 a^{2} b c e + 2 a^{2} c^{2} d - a b^{3} e - 4 a b^{2} c d + b^{4} d + c x^{2} \left (a^{2} b f + 2 a^{2} c e - a b^{2} e - 3 a b c d + b^{3} d\right )}{2 a^{3} \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )} - \frac{a e - 2 b d}{2 a^{3} x^{2}} - \frac{\left (\frac{a^{2} f}{2} - a b e - a c d + \frac{3 b^{2} d}{2}\right ) \log{\left (a + b x^{2} + c x^{4} \right )}}{2 a^{4}} + \frac{\left (a^{2} f - 2 a b e - 2 a c d + 3 b^{2} d\right ) \log{\left (x^{2} \right )}}{2 a^{4}} + \frac{\left (a^{2} b f + 2 a^{2} c e - 2 a b^{2} e - 6 a b c d + 3 b^{3} d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 a^{4} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**4+e*x**2+d)/x**5/(c*x**4+b*x**2+a)**2,x)

[Out]

-d/(4*a**2*x**4) - c*(a**2*b*f + 2*a**2*c*e - a*b**2*e - 3*a*b*c*d + b**3*d)*ata
nh((b + 2*c*x**2)/sqrt(-4*a*c + b**2))/(a**3*(-4*a*c + b**2)**(3/2)) + (-2*a**3*
c*f + a**2*b**2*f + 3*a**2*b*c*e + 2*a**2*c**2*d - a*b**3*e - 4*a*b**2*c*d + b**
4*d + c*x**2*(a**2*b*f + 2*a**2*c*e - a*b**2*e - 3*a*b*c*d + b**3*d))/(2*a**3*(-
4*a*c + b**2)*(a + b*x**2 + c*x**4)) - (a*e - 2*b*d)/(2*a**3*x**2) - (a**2*f/2 -
a*b*e - a*c*d + 3*b**2*d/2)*log(a + b*x**2 + c*x**4)/(2*a**4) + (a**2*f - 2*a*b
*e - 2*a*c*d + 3*b**2*d)*log(x**2)/(2*a**4) + (a**2*b*f + 2*a**2*c*e - 2*a*b**2*
e - 6*a*b*c*d + 3*b**3*d)*atanh((b + 2*c*x**2)/sqrt(-4*a*c + b**2))/(2*a**4*sqrt
(-4*a*c + b**2))

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Mathematica [A]  time = 3.25721, size = 592, normalized size = 1.8 \[ -\frac{\frac{2 a \left (2 a^2 c \left (a f-c \left (d+e x^2\right )\right )+b^3 \left (a e-c d x^2\right )+a b^2 \left (-a f+4 c d+c e x^2\right )-a b c \left (3 a e+a f x^2-3 c d x^2\right )+b^4 (-d)\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (2 a^2 b c \left (4 e \sqrt{b^2-4 a c}-3 a f+15 c d\right )-4 a^2 c \left (-2 c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}+3 a c e\right )+a b^2 \left (-14 c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}+12 a c e\right )+b^4 \left (3 d \sqrt{b^2-4 a c}-2 a e\right )+a b^3 \left (-2 e \sqrt{b^2-4 a c}+a f-20 c d\right )+3 b^5 d\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{\log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (2 a^2 b c \left (4 e \sqrt{b^2-4 a c}+3 a f-15 c d\right )+4 a^2 c \left (2 c d \sqrt{b^2-4 a c}-a f \sqrt{b^2-4 a c}+3 a c e\right )+a b^2 \left (a f \sqrt{b^2-4 a c}-2 c \left (7 d \sqrt{b^2-4 a c}+6 a e\right )\right )+b^4 \left (3 d \sqrt{b^2-4 a c}+2 a e\right )-a b^3 \left (2 e \sqrt{b^2-4 a c}+a f-20 c d\right )-3 b^5 d\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{a^2 d}{x^4}-4 \log (x) \left (-2 a b e+a (a f-2 c d)+3 b^2 d\right )+\frac{2 a (a e-2 b d)}{x^2}}{4 a^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x^2 + f*x^4)/(x^5*(a + b*x^2 + c*x^4)^2),x]

[Out]

-((a^2*d)/x^4 + (2*a*(-2*b*d + a*e))/x^2 + (2*a*(-(b^4*d) + b^3*(a*e - c*d*x^2)
+ a*b^2*(4*c*d - a*f + c*e*x^2) - a*b*c*(3*a*e - 3*c*d*x^2 + a*f*x^2) + 2*a^2*c*
(a*f - c*(d + e*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - 4*(3*b^2*d - 2*a*b
*e + a*(-2*c*d + a*f))*Log[x] + ((3*b^5*d + b^4*(3*Sqrt[b^2 - 4*a*c]*d - 2*a*e)
+ 2*a^2*b*c*(15*c*d + 4*Sqrt[b^2 - 4*a*c]*e - 3*a*f) + a*b^3*(-20*c*d - 2*Sqrt[b
^2 - 4*a*c]*e + a*f) - 4*a^2*c*(-2*c*Sqrt[b^2 - 4*a*c]*d + 3*a*c*e + a*Sqrt[b^2
- 4*a*c]*f) + a*b^2*(-14*c*Sqrt[b^2 - 4*a*c]*d + 12*a*c*e + a*Sqrt[b^2 - 4*a*c]*
f))*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + ((-3*b^5*d + b^4
*(3*Sqrt[b^2 - 4*a*c]*d + 2*a*e) - a*b^3*(-20*c*d + 2*Sqrt[b^2 - 4*a*c]*e + a*f)
+ 2*a^2*b*c*(-15*c*d + 4*Sqrt[b^2 - 4*a*c]*e + 3*a*f) + 4*a^2*c*(2*c*Sqrt[b^2 -
4*a*c]*d + 3*a*c*e - a*Sqrt[b^2 - 4*a*c]*f) + a*b^2*(-2*c*(7*Sqrt[b^2 - 4*a*c]*
d + 6*a*e) + a*Sqrt[b^2 - 4*a*c]*f))*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2
- 4*a*c)^(3/2))/(4*a^4)

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Maple [B]  time = 0.038, size = 1675, normalized size = 5.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^4+e*x^2+d)/x^5/(c*x^4+b*x^2+a)^2,x)

[Out]

-1/a/(c*x^4+b*x^2+a)/(4*a*c-b^2)*c^2*d-1/4*d/a^2/x^4+1/(c*x^4+b*x^2+a)/(4*a*c-b^
2)*c*f-2/a^3*ln(x)*b*e-2/a^3*ln(x)*c*d+3/a^4*ln(x)*b^2*d+1/a^3/x^2*b*d-1/2/a^3/(
c*x^4+b*x^2+a)/(4*a*c-b^2)*b^4*d-1/2/a/(c*x^4+b*x^2+a)/(4*a*c-b^2)*b^2*f-1/2/a/(
c*x^4+b*x^2+a)*c/(4*a*c-b^2)*x^2*b*f+1/2/a^2/(c*x^4+b*x^2+a)*c/(4*a*c-b^2)*x^2*b
^2*e+3/2/a^2/(c*x^4+b*x^2+a)*c^2/(4*a*c-b^2)*x^2*b*d-1/2/a^3/(c*x^4+b*x^2+a)*c/(
4*a*c-b^2)*x^2*b^3*d-1/a/(c*x^4+b*x^2+a)*c^2/(4*a*c-b^2)*x^2*e-3/2/a/(c*x^4+b*x^
2+a)/(4*a*c-b^2)*b*c*e+2/a^2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*b^2*c*d-3/a/(64*a^3*c^3
-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*(4*a*c-b^2)*c*x^2+(4*a*c-b^2)*b)
/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*b*c*f+6/a^2/(64*a^3*c^3-48*a^
2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*(4*a*c-b^2)*c*x^2+(4*a*c-b^2)*b)/(64*a
^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*b^2*c*e+15/a^2/(64*a^3*c^3-48*a^2*b
^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*(4*a*c-b^2)*c*x^2+(4*a*c-b^2)*b)/(64*a^3*
c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*b*c^2*d-10/a^3/(64*a^3*c^3-48*a^2*b^2*
c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*(4*a*c-b^2)*c*x^2+(4*a*c-b^2)*b)/(64*a^3*c^3
-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*b^3*c*d+2/a^2/(4*a*c-b^2)*c*ln((4*a*c-b^2
)*(c*x^4+b*x^2+a))*b*e-7/2/a^3/(4*a*c-b^2)*c*ln((4*a*c-b^2)*(c*x^4+b*x^2+a))*b^2
*d+1/2/a^2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*b^3*e-6/a/(64*a^3*c^3-48*a^2*b^2*c^2+12*a
*b^4*c-b^6)^(1/2)*arctan((2*(4*a*c-b^2)*c*x^2+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*
b^2*c^2+12*a*b^4*c-b^6)^(1/2))*c^2*e+1/2/a^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4
*c-b^6)^(1/2)*arctan((2*(4*a*c-b^2)*c*x^2+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*
c^2+12*a*b^4*c-b^6)^(1/2))*b^3*f-1/a^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6
)^(1/2)*arctan((2*(4*a*c-b^2)*c*x^2+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12
*a*b^4*c-b^6)^(1/2))*b^4*e-1/2/a^3/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^4+b*x^2+a))*b
^3*e+3/2/a^4/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*(4*a*c-b
^2)*c*x^2+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*b^5*d
-1/a/(4*a*c-b^2)*c*ln((4*a*c-b^2)*(c*x^4+b*x^2+a))*f+2/a^2/(4*a*c-b^2)*c^2*ln((4
*a*c-b^2)*(c*x^4+b*x^2+a))*d+3/4/a^4/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^4+b*x^2+a))
*b^4*d+1/4/a^2/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^4+b*x^2+a))*b^2*f-1/2/a^2/x^2*e+1
/a^2*ln(x)*f

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^4 + e*x^2 + d)/((c*x^4 + b*x^2 + a)^2*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 6.93827, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^4 + e*x^2 + d)/((c*x^4 + b*x^2 + a)^2*x^5),x, algorithm="fricas")

[Out]

[1/4*((((3*b^5*c - 20*a*b^3*c^2 + 30*a^2*b*c^3)*d - 2*(a*b^4*c - 6*a^2*b^2*c^2 +
6*a^3*c^3)*e + (a^2*b^3*c - 6*a^3*b*c^2)*f)*x^8 + ((3*b^6 - 20*a*b^4*c + 30*a^2
*b^2*c^2)*d - 2*(a*b^5 - 6*a^2*b^3*c + 6*a^3*b*c^2)*e + (a^2*b^4 - 6*a^3*b^2*c)*
f)*x^6 + ((3*a*b^5 - 20*a^2*b^3*c + 30*a^3*b*c^2)*d - 2*(a^2*b^4 - 6*a^3*b^2*c +
6*a^4*c^2)*e + (a^3*b^3 - 6*a^4*b*c)*f)*x^4)*log((b^3 - 4*a*b*c + 2*(b^2*c - 4*
a*c^2)*x^2 + (2*c^2*x^4 + 2*b*c*x^2 + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^4 + b
*x^2 + a)) + (2*(a^3*b*c*f + (3*a*b^3*c - 11*a^2*b*c^2)*d - 2*(a^2*b^2*c - 3*a^3
*c^2)*e)*x^6 + ((6*a*b^4 - 25*a^2*b^2*c + 8*a^3*c^2)*d - 2*(2*a^2*b^3 - 7*a^3*b*
c)*e + 2*(a^3*b^2 - 2*a^4*c)*f)*x^4 + (3*(a^2*b^3 - 4*a^3*b*c)*d - 2*(a^3*b^2 -
4*a^4*c)*e)*x^2 - (a^3*b^2 - 4*a^4*c)*d - (((3*b^4*c - 14*a*b^2*c^2 + 8*a^2*c^3)
*d - 2*(a*b^3*c - 4*a^2*b*c^2)*e + (a^2*b^2*c - 4*a^3*c^2)*f)*x^8 + ((3*b^5 - 14
*a*b^3*c + 8*a^2*b*c^2)*d - 2*(a*b^4 - 4*a^2*b^2*c)*e + (a^2*b^3 - 4*a^3*b*c)*f)
*x^6 + ((3*a*b^4 - 14*a^2*b^2*c + 8*a^3*c^2)*d - 2*(a^2*b^3 - 4*a^3*b*c)*e + (a^
3*b^2 - 4*a^4*c)*f)*x^4)*log(c*x^4 + b*x^2 + a) + 4*(((3*b^4*c - 14*a*b^2*c^2 +
8*a^2*c^3)*d - 2*(a*b^3*c - 4*a^2*b*c^2)*e + (a^2*b^2*c - 4*a^3*c^2)*f)*x^8 + ((
3*b^5 - 14*a*b^3*c + 8*a^2*b*c^2)*d - 2*(a*b^4 - 4*a^2*b^2*c)*e + (a^2*b^3 - 4*a
^3*b*c)*f)*x^6 + ((3*a*b^4 - 14*a^2*b^2*c + 8*a^3*c^2)*d - 2*(a^2*b^3 - 4*a^3*b*
c)*e + (a^3*b^2 - 4*a^4*c)*f)*x^4)*log(x))*sqrt(b^2 - 4*a*c))/(((a^4*b^2*c - 4*a
^5*c^2)*x^8 + (a^4*b^3 - 4*a^5*b*c)*x^6 + (a^5*b^2 - 4*a^6*c)*x^4)*sqrt(b^2 - 4*
a*c)), -1/4*(2*(((3*b^5*c - 20*a*b^3*c^2 + 30*a^2*b*c^3)*d - 2*(a*b^4*c - 6*a^2*
b^2*c^2 + 6*a^3*c^3)*e + (a^2*b^3*c - 6*a^3*b*c^2)*f)*x^8 + ((3*b^6 - 20*a*b^4*c
+ 30*a^2*b^2*c^2)*d - 2*(a*b^5 - 6*a^2*b^3*c + 6*a^3*b*c^2)*e + (a^2*b^4 - 6*a^
3*b^2*c)*f)*x^6 + ((3*a*b^5 - 20*a^2*b^3*c + 30*a^3*b*c^2)*d - 2*(a^2*b^4 - 6*a^
3*b^2*c + 6*a^4*c^2)*e + (a^3*b^3 - 6*a^4*b*c)*f)*x^4)*arctan(-(2*c*x^2 + b)*sqr
t(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - (2*(a^3*b*c*f + (3*a*b^3*c - 11*a^2*b*c^2)*d -
2*(a^2*b^2*c - 3*a^3*c^2)*e)*x^6 + ((6*a*b^4 - 25*a^2*b^2*c + 8*a^3*c^2)*d - 2*(
2*a^2*b^3 - 7*a^3*b*c)*e + 2*(a^3*b^2 - 2*a^4*c)*f)*x^4 + (3*(a^2*b^3 - 4*a^3*b*
c)*d - 2*(a^3*b^2 - 4*a^4*c)*e)*x^2 - (a^3*b^2 - 4*a^4*c)*d - (((3*b^4*c - 14*a*
b^2*c^2 + 8*a^2*c^3)*d - 2*(a*b^3*c - 4*a^2*b*c^2)*e + (a^2*b^2*c - 4*a^3*c^2)*f
)*x^8 + ((3*b^5 - 14*a*b^3*c + 8*a^2*b*c^2)*d - 2*(a*b^4 - 4*a^2*b^2*c)*e + (a^2
*b^3 - 4*a^3*b*c)*f)*x^6 + ((3*a*b^4 - 14*a^2*b^2*c + 8*a^3*c^2)*d - 2*(a^2*b^3
- 4*a^3*b*c)*e + (a^3*b^2 - 4*a^4*c)*f)*x^4)*log(c*x^4 + b*x^2 + a) + 4*(((3*b^4
*c - 14*a*b^2*c^2 + 8*a^2*c^3)*d - 2*(a*b^3*c - 4*a^2*b*c^2)*e + (a^2*b^2*c - 4*
a^3*c^2)*f)*x^8 + ((3*b^5 - 14*a*b^3*c + 8*a^2*b*c^2)*d - 2*(a*b^4 - 4*a^2*b^2*c
)*e + (a^2*b^3 - 4*a^3*b*c)*f)*x^6 + ((3*a*b^4 - 14*a^2*b^2*c + 8*a^3*c^2)*d - 2
*(a^2*b^3 - 4*a^3*b*c)*e + (a^3*b^2 - 4*a^4*c)*f)*x^4)*log(x))*sqrt(-b^2 + 4*a*c
))/(((a^4*b^2*c - 4*a^5*c^2)*x^8 + (a^4*b^3 - 4*a^5*b*c)*x^6 + (a^5*b^2 - 4*a^6*
c)*x^4)*sqrt(-b^2 + 4*a*c))]

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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((f*x**4+e*x**2+d)/x**5/(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((f*x^4 + e*x^2 + d)/((c*x^4 + b*x^2 + a)^2*x^5),x, algorithm="giac")

[Out]

Exception raised: TypeError