3.130 \(\int \frac{d+e x^2+f x^4+g x^6}{x^4 \left (a+b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=542 \[ \frac{2 b d-a e}{a^3 x}-\frac{d}{3 a^2 x^3}+\frac{x \left (a^2 \left (\frac{b^4 d}{a^2}-\frac{b^2 (b e+4 c d)}{a}-a (b g+2 c f)+b^2 f+3 b c e+2 c^2 d\right )+c x^2 \left (2 a^2 (c e-a g)-a b^2 e-a b (3 c d-a f)+b^3 d\right )\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (\frac{4 a^2 b (a g+4 c e)+4 a^2 c (7 c d-3 a f)-3 a b^3 e-a b^2 (29 c d-a f)+5 b^4 d}{\sqrt{b^2-4 a c}}+2 a^2 (5 c e-a g)-3 a b^2 e-a b (19 c d-a f)+5 b^3 d\right )}{2 \sqrt{2} a^3 \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (-\frac{4 a^2 b (a g+4 c e)+4 a^2 c (7 c d-3 a f)-3 a b^3 e-a b^2 (29 c d-a f)+5 b^4 d}{\sqrt{b^2-4 a c}}+2 a^2 (5 c e-a g)-3 a b^2 e-a b (19 c d-a f)+5 b^3 d\right )}{2 \sqrt{2} a^3 \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}} \]

[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 - a*(2*c*f + b*g)) + c*(b^3*d - a*b^2*e - a*b*
(3*c*d - a*f) + 2*a^2*(c*e - a*g))*x^2))/(2*a^3*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4
)) + (Sqrt[c]*(5*b^3*d - 3*a*b^2*e - a*b*(19*c*d - a*f) + 2*a^2*(5*c*e - a*g) +
(5*b^4*d - 3*a*b^3*e + 4*a^2*c*(7*c*d - 3*a*f) - a*b^2*(29*c*d - a*f) + 4*a^2*b*
(4*c*e + a*g))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 -
4*a*c]]])/(2*Sqrt[2]*a^3*(b^2 - 4*a*c)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*
(5*b^3*d - 3*a*b^2*e - a*b*(19*c*d - a*f) + 2*a^2*(5*c*e - a*g) - (5*b^4*d - 3*a
*b^3*e + 4*a^2*c*(7*c*d - 3*a*f) - a*b^2*(29*c*d - a*f) + 4*a^2*b*(4*c*e + a*g))
/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*
Sqrt[2]*a^3*(b^2 - 4*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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Rubi [A]  time = 19.3327, antiderivative size = 542, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114 \[ \frac{2 b d-a e}{a^3 x}-\frac{d}{3 a^2 x^3}+\frac{x \left (a^2 \left (\frac{b^4 d}{a^2}-\frac{b^2 (b e+4 c d)}{a}-a (b g+2 c f)+b^2 f+3 b c e+2 c^2 d\right )+c x^2 \left (2 a^2 (c e-a g)-a b^2 e-a b (3 c d-a f)+b^3 d\right )\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (\frac{4 a^2 b (a g+4 c e)+4 a^2 c (7 c d-3 a f)-3 a b^3 e-a b^2 (29 c d-a f)+5 b^4 d}{\sqrt{b^2-4 a c}}+2 a^2 (5 c e-a g)-3 a b^2 e-a b (19 c d-a f)+5 b^3 d\right )}{2 \sqrt{2} a^3 \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (-\frac{4 a^2 b (a g+4 c e)+4 a^2 c (7 c d-3 a f)-3 a b^3 e-a b^2 (29 c d-a f)+5 b^4 d}{\sqrt{b^2-4 a c}}+2 a^2 (5 c e-a g)-3 a b^2 e-a b (19 c d-a f)+5 b^3 d\right )}{2 \sqrt{2} a^3 \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x^2 + f*x^4 + g*x^6)/(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 - a*(2*c*f + b*g)) + c*(b^3*d - a*b^2*e - a*b*
(3*c*d - a*f) + 2*a^2*(c*e - a*g))*x^2))/(2*a^3*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4
)) + (Sqrt[c]*(5*b^3*d - 3*a*b^2*e - a*b*(19*c*d - a*f) + 2*a^2*(5*c*e - a*g) +
(5*b^4*d - 3*a*b^3*e + 4*a^2*c*(7*c*d - 3*a*f) - a*b^2*(29*c*d - a*f) + 4*a^2*b*
(4*c*e + a*g))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 -
4*a*c]]])/(2*Sqrt[2]*a^3*(b^2 - 4*a*c)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*
(5*b^3*d - 3*a*b^2*e - a*b*(19*c*d - a*f) + 2*a^2*(5*c*e - a*g) - (5*b^4*d - 3*a
*b^3*e + 4*a^2*c*(7*c*d - 3*a*f) - a*b^2*(29*c*d - a*f) + 4*a^2*b*(4*c*e + a*g))
/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*
Sqrt[2]*a^3*(b^2 - 4*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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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((g*x**6+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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Mathematica [A]  time = 4.88511, size = 612, normalized size = 1.13 \[ \frac{\frac{6 x \left (a b \left (a^2 (-g)+a c \left (3 e+f x^2\right )-3 c^2 d x^2\right )+2 a^2 c \left (c \left (d+e x^2\right )-a \left (f+g x^2\right )\right )+b^3 \left (c d x^2-a e\right )+a b^2 \left (a f-c \left (4 d+e x^2\right )\right )+b^4 d\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{3 \sqrt{2} \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (-2 a^2 \left (-5 c e \sqrt{b^2-4 a c}+a g \sqrt{b^2-4 a c}+6 a c f-14 c^2 d\right )+a b \left (4 a^2 g-19 c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}+16 a c e\right )+a b^2 \left (-3 e \sqrt{b^2-4 a c}+a f-29 c d\right )+b^3 \left (5 d \sqrt{b^2-4 a c}-3 a e\right )+5 b^4 d\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{2} \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (2 a^2 \left (-5 c e \sqrt{b^2-4 a c}+a g \sqrt{b^2-4 a c}-6 a c f+14 c^2 d\right )+a b \left (4 a^2 g+19 c d \sqrt{b^2-4 a c}-a f \sqrt{b^2-4 a c}+16 a c e\right )+a b^2 \left (3 e \sqrt{b^2-4 a c}+a f-29 c d\right )-b^3 \left (5 d \sqrt{b^2-4 a c}+3 a e\right )+5 b^4 d\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{24 b d-12 a e}{x}-\frac{4 a d}{x^3}}{12 a^3} \]

Antiderivative was successfully verified.

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

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Maple [B]  time = 0.095, size = 7512, normalized size = 13.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x^6 + 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 - 2*a^3*c*g + (5*b^3*c - 19*a*b*c^2)*d - (3*a*b^2*c - 10*a^2*c
^2)*e)*x^6 - (3*a^3*b*g - (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^3*b*g
+ (a^2*b*c*f - 2*a^3*c*g + (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((g*x^6 + 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((g*x**6+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((g*x^6 + f*x^4 + e*x^2 + d)/((c*x^4 + b*x^2 + a)^2*x^4),x, algorithm="giac")

[Out]

Exception raised: TypeError