3.53 \(\int \frac{d+e x+f x^2}{\left (a+b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=621 \[ \frac{x \left (c x^2 \left (20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )+8 a^2 b c f+28 a^2 c^2 d+a b^3 f-25 a b^2 c d+3 b^4 d\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \left (-\frac{-52 a^2 b c f+168 a^2 c^2 d+a b^3 f-30 a b^2 c d+3 b^4 d}{\sqrt{b^2-4 a c}}+20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^2 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\sqrt{c} \left (4 a^2 c \left (5 f \sqrt{b^2-4 a c}+42 c d\right )-a b^2 \left (30 c d-f \sqrt{b^2-4 a c}\right )-4 a b c \left (6 d \sqrt{b^2-4 a c}+13 a f\right )+b^3 \left (3 d \sqrt{b^2-4 a c}+a f\right )+3 b^4 d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{6 c^2 e \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{3 c e \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{e \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

[Out]

-(e*(b + 2*c*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (x*(b^2*d - 2*a*c*d
- a*b*f + c*(b*d - 2*a*f)*x^2))/(4*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (3*
c*e*(b + 2*c*x^2))/(2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (x*(3*b^4*d - 25*a*
b^2*c*d + 28*a^2*c^2*d + a*b^3*f + 8*a^2*b*c*f + c*(3*b^3*d - 24*a*b*c*d + a*b^2
*f + 20*a^2*c*f)*x^2))/(8*a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (Sqrt[c]*(3
*b^4*d + b^3*(3*Sqrt[b^2 - 4*a*c]*d + a*f) - 4*a*b*c*(6*Sqrt[b^2 - 4*a*c]*d + 13
*a*f) - a*b^2*(30*c*d - Sqrt[b^2 - 4*a*c]*f) + 4*a^2*c*(42*c*d + 5*Sqrt[b^2 - 4*
a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2
*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*(3*b^3*d - 24*a*b*c
*d + a*b^2*f + 20*a^2*c*f - (3*b^4*d - 30*a*b^2*c*d + 168*a^2*c^2*d + a*b^3*f -
52*a^2*b*c*f)/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*(b^2 - 4*a*c)^2*Sqrt[b + Sqrt[b^2 - 4*a*c]]) - (6*c^2*e
*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 = 10.4248, antiderivative size = 621, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36 \[ \frac{x \left (c x^2 \left (20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )+8 a^2 b c f+28 a^2 c^2 d+a b^3 f-25 a b^2 c d+3 b^4 d\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \left (-\frac{-52 a^2 b c f+168 a^2 c^2 d+a b^3 f-30 a b^2 c d+3 b^4 d}{\sqrt{b^2-4 a c}}+20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^2 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\sqrt{c} \left (4 a^2 c \left (5 f \sqrt{b^2-4 a c}+42 c d\right )-a b^2 \left (30 c d-f \sqrt{b^2-4 a c}\right )-4 a b c \left (6 d \sqrt{b^2-4 a c}+13 a f\right )+b^3 \left (3 d \sqrt{b^2-4 a c}+a f\right )+3 b^4 d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{6 c^2 e \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{3 c e \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{e \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x + f*x^2)/(a + b*x^2 + c*x^4)^3,x]

[Out]

-(e*(b + 2*c*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (x*(b^2*d - 2*a*c*d
- a*b*f + c*(b*d - 2*a*f)*x^2))/(4*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (3*
c*e*(b + 2*c*x^2))/(2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (x*(3*b^4*d - 25*a*
b^2*c*d + 28*a^2*c^2*d + a*b^3*f + 8*a^2*b*c*f + c*(3*b^3*d - 24*a*b*c*d + a*b^2
*f + 20*a^2*c*f)*x^2))/(8*a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (Sqrt[c]*(3
*b^4*d + b^3*(3*Sqrt[b^2 - 4*a*c]*d + a*f) - 4*a*b*c*(6*Sqrt[b^2 - 4*a*c]*d + 13
*a*f) - a*b^2*(30*c*d - Sqrt[b^2 - 4*a*c]*f) + 4*a^2*c*(42*c*d + 5*Sqrt[b^2 - 4*
a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2
*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*(3*b^3*d - 24*a*b*c
*d + a*b^2*f + 20*a^2*c*f - (3*b^4*d - 30*a*b^2*c*d + 168*a^2*c^2*d + a*b^3*f -
52*a^2*b*c*f)/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*(b^2 - 4*a*c)^2*Sqrt[b + Sqrt[b^2 - 4*a*c]]) - (6*c^2*e
*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((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 = 6.6145, size = 683, normalized size = 1.1 \[ \frac{12 a^2 b c e+8 a^2 b c f x+28 a^2 c^2 d x+24 a^2 c^2 e x^2+20 a^2 c^2 f x^3+a b^3 f x-25 a b^2 c d x+a b^2 c f x^3-24 a b c^2 d x^3+3 b^4 d x+3 b^3 c d x^3}{8 a^2 \left (4 a c-b^2\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \left (20 a^2 c f \sqrt{b^2-4 a c}-52 a^2 b c f+168 a^2 c^2 d+a b^3 f-30 a b^2 c d-24 a b c d \sqrt{b^2-4 a c}+a b^2 f \sqrt{b^2-4 a c}+3 b^3 d \sqrt{b^2-4 a c}+3 b^4 d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left (20 a^2 c f \sqrt{b^2-4 a c}+52 a^2 b c f-168 a^2 c^2 d-a b^3 f+30 a b^2 c d-24 a b c d \sqrt{b^2-4 a c}+a b^2 f \sqrt{b^2-4 a c}+3 b^3 d \sqrt{b^2-4 a c}-3 b^4 d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{3 c^2 e \log \left (\sqrt{b^2-4 a c}-b-2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac{3 c^2 e \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{a b e+a b f x+2 a c d x+2 a c e x^2+2 a c f x^3-b^2 d x-b c d x^3}{4 a \left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x + f*x^2)/(a + b*x^2 + c*x^4)^3,x]

[Out]

(a*b*e - b^2*d*x + 2*a*c*d*x + a*b*f*x + 2*a*c*e*x^2 - b*c*d*x^3 + 2*a*c*f*x^3)/
(4*a*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)^2) + (12*a^2*b*c*e + 3*b^4*d*x - 25*a*b^
2*c*d*x + 28*a^2*c^2*d*x + a*b^3*f*x + 8*a^2*b*c*f*x + 24*a^2*c^2*e*x^2 + 3*b^3*
c*d*x^3 - 24*a*b*c^2*d*x^3 + a*b^2*c*f*x^3 + 20*a^2*c^2*f*x^3)/(8*a^2*(-b^2 + 4*
a*c)^2*(a + b*x^2 + c*x^4)) + (Sqrt[c]*(3*b^4*d - 30*a*b^2*c*d + 168*a^2*c^2*d +
3*b^3*Sqrt[b^2 - 4*a*c]*d - 24*a*b*c*Sqrt[b^2 - 4*a*c]*d + a*b^3*f - 52*a^2*b*c
*f + a*b^2*Sqrt[b^2 - 4*a*c]*f + 20*a^2*c*Sqrt[b^2 - 4*a*c]*f)*ArcTan[(Sqrt[2]*S
qrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*(b^2 - 4*a*c)^(5/2)*Sqrt[
b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*(-3*b^4*d + 30*a*b^2*c*d - 168*a^2*c^2*d + 3*
b^3*Sqrt[b^2 - 4*a*c]*d - 24*a*b*c*Sqrt[b^2 - 4*a*c]*d - a*b^3*f + 52*a^2*b*c*f
+ a*b^2*Sqrt[b^2 - 4*a*c]*f + 20*a^2*c*Sqrt[b^2 - 4*a*c]*f)*ArcTan[(Sqrt[2]*Sqrt
[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*(b^2 - 4*a*c)^(5/2)*Sqrt[b +
Sqrt[b^2 - 4*a*c]]) + (3*c^2*e*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*
a*c)^(5/2) - (3*c^2*e*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(5/2)

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Maple [B]  time = 0.398, size = 10809, normalized size = 17.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((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. \[ \frac{24 \, a^{2} c^{3} e x^{6} + 36 \, a^{2} b c^{2} e x^{4} +{\left (3 \,{\left (b^{3} c^{2} - 8 \, a b c^{3}\right )} d +{\left (a b^{2} c^{2} + 20 \, a^{2} c^{3}\right )} f\right )} x^{7} +{\left ({\left (6 \, b^{4} c - 49 \, a b^{2} c^{2} + 28 \, a^{2} c^{3}\right )} d + 2 \,{\left (a b^{3} c + 14 \, a^{2} b c^{2}\right )} f\right )} x^{5} + 8 \,{\left (a^{2} b^{2} c + 5 \, a^{3} c^{2}\right )} e x^{2} +{\left ({\left (3 \, b^{5} - 20 \, a b^{3} c - 4 \, a^{2} b c^{2}\right )} d +{\left (a b^{4} + 5 \, a^{2} b^{2} c + 36 \, a^{3} c^{2}\right )} f\right )} x^{3} - 2 \,{\left (a^{2} b^{3} - 10 \, a^{3} b c\right )} e +{\left ({\left (5 \, a b^{4} - 37 \, a^{2} b^{2} c + 44 \, a^{3} c^{2}\right )} d -{\left (a^{2} b^{3} - 16 \, a^{3} b c\right )} f\right )} x}{8 \,{\left ({\left (a^{2} b^{4} c^{2} - 8 \, a^{3} b^{2} c^{3} + 16 \, a^{4} c^{4}\right )} x^{8} + a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2} + 2 \,{\left (a^{2} b^{5} c - 8 \, a^{3} b^{3} c^{2} + 16 \, a^{4} b c^{3}\right )} x^{6} +{\left (a^{2} b^{6} - 6 \, a^{3} b^{4} c + 32 \, a^{5} c^{3}\right )} x^{4} + 2 \,{\left (a^{3} b^{5} - 8 \, a^{4} b^{3} c + 16 \, a^{5} b c^{2}\right )} x^{2}\right )}} + \frac{\int \frac{48 \, a^{2} c^{2} e x +{\left (3 \,{\left (b^{3} c - 8 \, a b c^{2}\right )} d +{\left (a b^{2} c + 20 \, a^{2} c^{2}\right )} f\right )} x^{2} + 3 \,{\left (b^{4} - 9 \, a b^{2} c + 28 \, a^{2} c^{2}\right )} d +{\left (a b^{3} - 16 \, a^{2} b c\right )} f}{c x^{4} + b x^{2} + a}\,{d x}}{8 \,{\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^2 + e*x + d)/(c*x^4 + b*x^2 + a)^3,x, algorithm="maxima")

[Out]

1/8*(24*a^2*c^3*e*x^6 + 36*a^2*b*c^2*e*x^4 + (3*(b^3*c^2 - 8*a*b*c^3)*d + (a*b^2
*c^2 + 20*a^2*c^3)*f)*x^7 + ((6*b^4*c - 49*a*b^2*c^2 + 28*a^2*c^3)*d + 2*(a*b^3*
c + 14*a^2*b*c^2)*f)*x^5 + 8*(a^2*b^2*c + 5*a^3*c^2)*e*x^2 + ((3*b^5 - 20*a*b^3*
c - 4*a^2*b*c^2)*d + (a*b^4 + 5*a^2*b^2*c + 36*a^3*c^2)*f)*x^3 - 2*(a^2*b^3 - 10
*a^3*b*c)*e + ((5*a*b^4 - 37*a^2*b^2*c + 44*a^3*c^2)*d - (a^2*b^3 - 16*a^3*b*c)*
f)*x)/((a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*x^8 + a^4*b^4 - 8*a^5*b^2*c +
16*a^6*c^2 + 2*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*x^6 + (a^2*b^6 - 6*a^3
*b^4*c + 32*a^5*c^3)*x^4 + 2*(a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*x^2) + 1/8*i
ntegrate((48*a^2*c^2*e*x + (3*(b^3*c - 8*a*b*c^2)*d + (a*b^2*c + 20*a^2*c^2)*f)*
x^2 + 3*(b^4 - 9*a*b^2*c + 28*a^2*c^2)*d + (a*b^3 - 16*a^2*b*c)*f)/(c*x^4 + b*x^
2 + a), x)/(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)

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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^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((f*x**2+e*x+d)/(c*x**4+b*x**2+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 50.9725, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done