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

Optimal. Leaf size=575 \[ \frac{2 b d-a e}{a^3 x}-\frac{d}{3 a^2 x^3}+\frac{x \left (a^2 \left (\frac{b^4 d}{a^2}-\frac{b^2 (b e+4 c d)}{a}-2 a c f+b^2 f+3 b c e+2 c^2 d\right )+c x^2 \left (2 a^2 c e-a b^2 e-a b (3 c d-a f)+b^3 d\right )\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (2 a^2 c \left (5 e \sqrt{b^2-4 a c}-6 a f+14 c d\right )-a b^2 \left (3 e \sqrt{b^2-4 a c}-a f+29 c d\right )-a b \left (19 c d \sqrt{b^2-4 a c}-a f \sqrt{b^2-4 a c}-16 a c e\right )+b^3 \left (5 d \sqrt{b^2-4 a c}-3 a e\right )+5 b^4 d\right )}{2 \sqrt{2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (2 a^2 c \left (-5 e \sqrt{b^2-4 a c}-6 a f+14 c d\right )-a b^2 \left (-3 e \sqrt{b^2-4 a c}-a f+29 c d\right )+a b \left (19 c d \sqrt{b^2-4 a c}-a f \sqrt{b^2-4 a c}+16 a c e\right )-b^3 \left (5 d \sqrt{b^2-4 a c}+3 a e\right )+5 b^4 d\right )}{2 \sqrt{2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}} \]

[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 - 2*a*c*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)) + (Sqrt[c]*(5
*b^4*d + b^3*(5*Sqrt[b^2 - 4*a*c]*d - 3*a*e) + 2*a^2*c*(14*c*d + 5*Sqrt[b^2 - 4*
a*c]*e - 6*a*f) - 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))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/S
qrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a^3*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^
2 - 4*a*c]]) - (Sqrt[c]*(5*b^4*d - b^3*(5*Sqrt[b^2 - 4*a*c]*d + 3*a*e) + 2*a^2*c
*(14*c*d - 5*Sqrt[b^2 - 4*a*c]*e - 6*a*f) - 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))*Ar
cTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a^3*(b^2 - 4*a
*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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Rubi [A]  time = 18.2041, antiderivative size = 575, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 b d-a e}{a^3 x}-\frac{d}{3 a^2 x^3}+\frac{x \left (a^2 \left (\frac{b^4 d}{a^2}-\frac{b^2 (b e+4 c d)}{a}-2 a c f+b^2 f+3 b c e+2 c^2 d\right )+c x^2 \left (2 a^2 c e-a b^2 e-a b (3 c d-a f)+b^3 d\right )\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (2 a^2 c \left (5 e \sqrt{b^2-4 a c}-6 a f+14 c d\right )-a b^2 \left (3 e \sqrt{b^2-4 a c}-a f+29 c d\right )-a b \left (19 c d \sqrt{b^2-4 a c}-a f \sqrt{b^2-4 a c}-16 a c e\right )+b^3 \left (5 d \sqrt{b^2-4 a c}-3 a e\right )+5 b^4 d\right )}{2 \sqrt{2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (2 a^2 c \left (-5 e \sqrt{b^2-4 a c}-6 a f+14 c d\right )-a b^2 \left (-3 e \sqrt{b^2-4 a c}-a f+29 c d\right )+a b \left (19 c d \sqrt{b^2-4 a c}-a f \sqrt{b^2-4 a c}+16 a c e\right )-b^3 \left (5 d \sqrt{b^2-4 a c}+3 a e\right )+5 b^4 d\right )}{2 \sqrt{2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x^2 + f*x^4)/(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 - 2*a*c*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)) + (Sqrt[c]*(5
*b^4*d + b^3*(5*Sqrt[b^2 - 4*a*c]*d - 3*a*e) + 2*a^2*c*(14*c*d + 5*Sqrt[b^2 - 4*
a*c]*e - 6*a*f) - 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))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/S
qrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a^3*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^
2 - 4*a*c]]) - (Sqrt[c]*(5*b^4*d - b^3*(5*Sqrt[b^2 - 4*a*c]*d + 3*a*e) + 2*a^2*c
*(14*c*d - 5*Sqrt[b^2 - 4*a*c]*e - 6*a*f) - 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))*Ar
cTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a^3*(b^2 - 4*a
*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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Rubi in Sympy [A]  time = 76.1512, size = 216, normalized size = 0.38 \[ - \frac{f}{3 a c x^{3}} + \frac{b f}{a^{2} c x} - \frac{\sqrt{2} f \left (- 2 a c + b^{2} - b \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{2 a^{2} \sqrt{c} \sqrt{b + \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} + \frac{\sqrt{2} f \left (- 2 a c + b^{2} + b \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{2 a^{2} \sqrt{c} \sqrt{b - \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-f/(3*a*c*x**3) + b*f/(a**2*c*x) - sqrt(2)*f*(-2*a*c + b**2 - b*sqrt(-4*a*c + b*
*2))*atan(sqrt(2)*sqrt(c)*x/sqrt(b + sqrt(-4*a*c + b**2)))/(2*a**2*sqrt(c)*sqrt(
b + sqrt(-4*a*c + b**2))*sqrt(-4*a*c + b**2)) + sqrt(2)*f*(-2*a*c + b**2 + b*sqr
t(-4*a*c + b**2))*atan(sqrt(2)*sqrt(c)*x/sqrt(b - sqrt(-4*a*c + b**2)))/(2*a**2*
sqrt(c)*sqrt(b - sqrt(-4*a*c + b**2))*sqrt(-4*a*c + b**2))

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Mathematica [A]  time = 3.60531, size = 548, normalized size = 0.95 \[ \frac{\frac{6 x \left (2 a^2 c \left (c \left (d+e x^2\right )-a f\right )+b^3 \left (c d x^2-a e\right )+a b^2 \left (a f-c \left (4 d+e x^2\right )\right )+a b c \left (3 a e+a f x^2-3 c d x^2\right )+b^4 d\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{3 \sqrt{2} \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (2 a^2 c \left (5 e \sqrt{b^2-4 a c}-6 a f+14 c d\right )+a b^2 \left (-3 e \sqrt{b^2-4 a c}+a f-29 c d\right )+a b \left (-19 c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}+16 a c e\right )+b^3 \left (5 d \sqrt{b^2-4 a c}-3 a e\right )+5 b^4 d\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{2} \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (2 a^2 c \left (5 e \sqrt{b^2-4 a c}+6 a f-14 c d\right )-a b^2 \left (3 e \sqrt{b^2-4 a c}+a f-29 c d\right )+a b \left (-19 c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}-16 a c e\right )+b^3 \left (5 d \sqrt{b^2-4 a c}+3 a e\right )-5 b^4 d\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{24 b d-12 a e}{x}-\frac{4 a d}{x^3}}{12 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x^2 + f*x^4)/(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
*c*(3*a*e - 3*c*d*x^2 + a*f*x^2) + 2*a^2*c*(-(a*f) + c*(d + e*x^2)) + a*b^2*(a*f
- c*(4*d + e*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) + 2*a^2*c*(14*c*d + 5*Sqrt[b^2 - 4
*a*c]*e - 6*a*f) + a*b^2*(-29*c*d - 3*Sqrt[b^2 - 4*a*c]*e + a*f) + a*b*(-19*c*Sq
rt[b^2 - 4*a*c]*d + 16*a*c*e + a*Sqrt[b^2 - 4*a*c]*f))*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) + 2*a^2*c*(-14*c*d + 5*Sqrt[b^2 - 4*a*c]*
e + 6*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))
*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*S
qrt[b + Sqrt[b^2 - 4*a*c]]))/(12*a^3)

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Maple [B]  time = 0.095, size = 6122, normalized size = 10.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^4+e*x^2+d)/x^4/(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. \[ \frac{3 \,{\left (a^{2} b c f +{\left (5 \, b^{3} c - 19 \, a b c^{2}\right )} d -{\left (3 \, a b^{2} c - 10 \, a^{2} c^{2}\right )} e\right )} x^{6} +{\left ({\left (15 \, b^{4} - 62 \, a b^{2} c + 14 \, a^{2} c^{2}\right )} d - 3 \,{\left (3 \, a b^{3} - 11 \, a^{2} b c\right )} e + 3 \,{\left (a^{2} b^{2} - 2 \, a^{3} c\right )} f\right )} x^{4} + 2 \,{\left (5 \,{\left (a b^{3} - 4 \, a^{2} b c\right )} d - 3 \,{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} e\right )} x^{2} - 2 \,{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d}{6 \,{\left ({\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} x^{7} +{\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x^{5} +{\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{3}\right )}} + \frac{\int \frac{{\left (a^{2} b c f +{\left (5 \, b^{3} c - 19 \, a b c^{2}\right )} d -{\left (3 \, a b^{2} c - 10 \, a^{2} c^{2}\right )} e\right )} x^{2} +{\left (5 \, b^{4} - 24 \, a b^{2} c + 14 \, a^{2} c^{2}\right )} d -{\left (3 \, a b^{3} - 13 \, a^{2} b c\right )} e +{\left (a^{2} b^{2} - 6 \, a^{3} c\right )} f}{c x^{4} + b x^{2} + a}\,{d x}}{2 \,{\left (a^{3} b^{2} - 4 \, a^{4} c\right )}} \]

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^4),x, algorithm="maxima")

[Out]

1/6*(3*(a^2*b*c*f + (5*b^3*c - 19*a*b*c^2)*d - (3*a*b^2*c - 10*a^2*c^2)*e)*x^6 +
((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^2*b*c*f + (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. \[ \text{Timed out} \]

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^4),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((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/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^4),x, algorithm="giac")

[Out]

Exception raised: TypeError