3.279 \(\int \frac{1}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=1087 \[ \text{result too large to display} \]

[Out]

(e^4*x)/(2*d*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x^2)) - (x*(2*b^3*c*d*e - 6*a*b*c^
2*d*e - b^4*e^2 - b^2*c*(c*d^2 - 4*a*e^2) + 2*a*c^2*(c*d^2 - a*e^2) + c*(2*b^2*c
*d*e - 4*a*c^2*d*e - b^3*e^2 - b*c*(c*d^2 - 3*a*e^2))*x^2))/(2*a*(b^2 - 4*a*c)*(
c*d^2 - b*d*e + a*e^2)^2*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*e^2*(3*c^2*d^2
+ b*(b + Sqrt[b^2 - 4*a*c])*e^2 - c*e*(3*b*d + 2*Sqrt[b^2 - 4*a*c]*d + a*e))*Arc
Tan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b
- Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*e^2)^3) + (Sqrt[c]*(b^4*e^2 - b^3*e*(2*c
*d - Sqrt[b^2 - 4*a*c]*e) - 4*a*c^2*(3*c*d^2 - e*(Sqrt[b^2 - 4*a*c]*d + 3*a*e))
+ b^2*c*(c*d^2 - e*(2*Sqrt[b^2 - 4*a*c]*d + 9*a*e)) - b*c*(3*a*Sqrt[b^2 - 4*a*c]
*e^2 - c*d*(Sqrt[b^2 - 4*a*c]*d + 16*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b -
Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]
]*(c*d^2 - b*d*e + a*e^2)^2) - (Sqrt[2]*Sqrt[c]*e^2*(3*c^2*d^2 + b*(b - Sqrt[b^2
- 4*a*c])*e^2 - c*e*(3*b*d - 2*Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTan[(Sqrt[2]*Sqrt
[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*
c]]*(c*d^2 - b*d*e + a*e^2)^3) - (Sqrt[c]*(b^4*e^2 - b^3*e*(2*c*d + Sqrt[b^2 - 4
*a*c]*e) + b*c*(3*a*Sqrt[b^2 - 4*a*c]*e^2 - c*d*(Sqrt[b^2 - 4*a*c]*d - 16*a*e))
+ b^2*c*(c*d^2 + e*(2*Sqrt[b^2 - 4*a*c]*d - 9*a*e)) - 4*a*c^2*(3*c*d^2 + e*(Sqrt
[b^2 - 4*a*c]*d - 3*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]
]])/(2*Sqrt[2]*a*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e
+ a*e^2)^2) + (2*e^(7/2)*(2*c*d - b*e)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*(c*
d^2 - b*d*e + a*e^2)^3) + (e^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*(c*d^
2 - b*d*e + a*e^2)^2)

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Rubi [A]  time = 24.4454, antiderivative size = 1088, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{x e^4}{2 d \left (c d^2-b e d+a e^2\right )^2 \left (e x^2+d\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) e^{7/2}}{2 d^{3/2} \left (c d^2-b e d+a e^2\right )^2}+\frac{2 (2 c d-b e) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) e^{7/2}}{\sqrt{d} \left (c d^2-b e d+a e^2\right )^3}+\frac{\sqrt{2} \sqrt{c} \left (3 c^2 d^2+b \left (b+\sqrt{b^2-4 a c}\right ) e^2-c e \left (3 b d+2 \sqrt{b^2-4 a c} d+a e\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) e^2}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}} \left (c d^2-b e d+a e^2\right )^3}-\frac{\sqrt{2} \sqrt{c} \left (3 c^2 d^2+b \left (b-\sqrt{b^2-4 a c}\right ) e^2-c e \left (3 b d-2 \sqrt{b^2-4 a c} d+a e\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right ) e^2}{\sqrt{b^2-4 a c} \sqrt{b+\sqrt{b^2-4 a c}} \left (c d^2-b e d+a e^2\right )^3}+\frac{\sqrt{c} \left (e^2 b^4-e \left (2 c d-\sqrt{b^2-4 a c} e\right ) b^3+c \left (c d^2-e \left (2 \sqrt{b^2-4 a c} d+9 a e\right )\right ) b^2-c \left (3 a \sqrt{b^2-4 a c} e^2-c d \left (\sqrt{b^2-4 a c} d+16 a e\right )\right ) b-4 a c^2 \left (3 c d^2-e \left (\sqrt{b^2-4 a c} d+3 a e\right )\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}} \left (c d^2-b e d+a e^2\right )^2}-\frac{\sqrt{c} \left (e^2 b^4-e \left (2 c d+\sqrt{b^2-4 a c} e\right ) b^3+c \left (c d^2+e \left (2 \sqrt{b^2-4 a c} d-9 a e\right )\right ) b^2+c \left (3 a \sqrt{b^2-4 a c} e^2-c d \left (\sqrt{b^2-4 a c} d-16 a e\right )\right ) b-4 a c^2 \left (3 c d^2+e \left (\sqrt{b^2-4 a c} d-3 a e\right )\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right )^{3/2} \sqrt{b+\sqrt{b^2-4 a c}} \left (c d^2-e (b d-a e)\right )^2}-\frac{x \left (-e^2 b^4+2 c d e b^3-c \left (c d^2-4 a e^2\right ) b^2-6 a c^2 d e b+c \left (-e^2 b^3+2 c d e b^2-c \left (c d^2-3 a e^2\right ) b-4 a c^2 d e\right ) x^2+2 a c^2 \left (c d^2-a e^2\right )\right )}{2 a \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )^2 \left (c x^4+b x^2+a\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x^2)^2*(a + b*x^2 + c*x^4)^2),x]

[Out]

(e^4*x)/(2*d*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x^2)) - (x*(2*b^3*c*d*e - 6*a*b*c^
2*d*e - b^4*e^2 - b^2*c*(c*d^2 - 4*a*e^2) + 2*a*c^2*(c*d^2 - a*e^2) + c*(2*b^2*c
*d*e - 4*a*c^2*d*e - b^3*e^2 - b*c*(c*d^2 - 3*a*e^2))*x^2))/(2*a*(b^2 - 4*a*c)*(
c*d^2 - b*d*e + a*e^2)^2*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*e^2*(3*c^2*d^2
+ b*(b + Sqrt[b^2 - 4*a*c])*e^2 - c*e*(3*b*d + 2*Sqrt[b^2 - 4*a*c]*d + a*e))*Arc
Tan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b
- Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*e^2)^3) + (Sqrt[c]*(b^4*e^2 - b^3*e*(2*c
*d - Sqrt[b^2 - 4*a*c]*e) - 4*a*c^2*(3*c*d^2 - e*(Sqrt[b^2 - 4*a*c]*d + 3*a*e))
+ b^2*c*(c*d^2 - e*(2*Sqrt[b^2 - 4*a*c]*d + 9*a*e)) - b*c*(3*a*Sqrt[b^2 - 4*a*c]
*e^2 - c*d*(Sqrt[b^2 - 4*a*c]*d + 16*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b -
Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]
]*(c*d^2 - b*d*e + a*e^2)^2) - (Sqrt[2]*Sqrt[c]*e^2*(3*c^2*d^2 + b*(b - Sqrt[b^2
- 4*a*c])*e^2 - c*e*(3*b*d - 2*Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTan[(Sqrt[2]*Sqrt
[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*
c]]*(c*d^2 - b*d*e + a*e^2)^3) - (Sqrt[c]*(b^4*e^2 - b^3*e*(2*c*d + Sqrt[b^2 - 4
*a*c]*e) + b*c*(3*a*Sqrt[b^2 - 4*a*c]*e^2 - c*d*(Sqrt[b^2 - 4*a*c]*d - 16*a*e))
+ b^2*c*(c*d^2 + e*(2*Sqrt[b^2 - 4*a*c]*d - 9*a*e)) - 4*a*c^2*(3*c*d^2 + e*(Sqrt
[b^2 - 4*a*c]*d - 3*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]
]])/(2*Sqrt[2]*a*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(c*d^2 - e*(b*d
- a*e))^2) + (2*e^(7/2)*(2*c*d - b*e)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*(c*
d^2 - b*d*e + a*e^2)^3) + (e^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*(c*d^
2 - b*d*e + a*e^2)^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(1/(e*x**2+d)**2/(c*x**4+b*x**2+a)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 8.0096, size = 1235, normalized size = 1.14 \[ \frac{x e^4}{2 d \left (c d^2-b e d+a e^2\right )^2 \left (e x^2+d\right )}+\frac{\left (9 c d^2-5 b e d+a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) e^{7/2}}{2 d^{3/2} \left (c d^2-b e d+a e^2\right )^3}+\frac{\sqrt{c} \left (d e^3 b^5-5 a e^4 b^4+\sqrt{b^2-4 a c} d e^3 b^4-3 c d^2 e^2 b^4-5 a \sqrt{b^2-4 a c} e^4 b^3+5 a c d e^3 b^3-3 c \sqrt{b^2-4 a c} d^2 e^2 b^3+3 c^2 d^3 e b^3-c^3 d^4 b^2+29 a^2 c e^4 b^2+7 a c \sqrt{b^2-4 a c} d e^3 b^2+12 a c^2 d^2 e^2 b^2+3 c^2 \sqrt{b^2-4 a c} d^3 e b^2-c^3 \sqrt{b^2-4 a c} d^4 b+19 a^2 c \sqrt{b^2-4 a c} e^4 b-52 a^2 c^2 d e^3 b+6 a c^2 \sqrt{b^2-4 a c} d^2 e^2 b-28 a c^3 d^3 e b+12 a c^4 d^4-28 a^3 c^2 e^4-36 a^2 c^2 \sqrt{b^2-4 a c} d e^3+48 a^2 c^3 d^2 e^2-4 a c^3 \sqrt{b^2-4 a c} d^3 e\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}} \left (-c d^2+b e d-a e^2\right )^3}+\frac{\sqrt{c} \left (-d e^3 b^5+5 a e^4 b^4+\sqrt{b^2-4 a c} d e^3 b^4+3 c d^2 e^2 b^4-5 a \sqrt{b^2-4 a c} e^4 b^3-5 a c d e^3 b^3-3 c \sqrt{b^2-4 a c} d^2 e^2 b^3-3 c^2 d^3 e b^3+c^3 d^4 b^2-29 a^2 c e^4 b^2+7 a c \sqrt{b^2-4 a c} d e^3 b^2-12 a c^2 d^2 e^2 b^2+3 c^2 \sqrt{b^2-4 a c} d^3 e b^2-c^3 \sqrt{b^2-4 a c} d^4 b+19 a^2 c \sqrt{b^2-4 a c} e^4 b+52 a^2 c^2 d e^3 b+6 a c^2 \sqrt{b^2-4 a c} d^2 e^2 b+28 a c^3 d^3 e b-12 a c^4 d^4+28 a^3 c^2 e^4-36 a^2 c^2 \sqrt{b^2-4 a c} d e^3-48 a^2 c^3 d^2 e^2-4 a c^3 \sqrt{b^2-4 a c} d^3 e\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right )^{3/2} \sqrt{b+\sqrt{b^2-4 a c}} \left (-c d^2+b e d-a e^2\right )^3}+\frac{-e^2 x b^4-c e^2 x^3 b^3+2 c d e x b^3+2 c^2 d e x^3 b^2-c^2 d^2 x b^2+4 a c e^2 x b^2-c^3 d^2 x^3 b+3 a c^2 e^2 x^3 b-6 a c^2 d e x b-4 a c^3 d e x^3+2 a c^3 d^2 x-2 a^2 c^2 e^2 x}{2 a \left (4 a c-b^2\right ) \left (c d^2-b e d+a e^2\right )^2 \left (c x^4+b x^2+a\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x^2)^2*(a + b*x^2 + c*x^4)^2),x]

[Out]

(e^4*x)/(2*d*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x^2)) + (-(b^2*c^2*d^2*x) + 2*a*c^
3*d^2*x + 2*b^3*c*d*e*x - 6*a*b*c^2*d*e*x - b^4*e^2*x + 4*a*b^2*c*e^2*x - 2*a^2*
c^2*e^2*x - b*c^3*d^2*x^3 + 2*b^2*c^2*d*e*x^3 - 4*a*c^3*d*e*x^3 - b^3*c*e^2*x^3
+ 3*a*b*c^2*e^2*x^3)/(2*a*(-b^2 + 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x^2 +
c*x^4)) + (Sqrt[c]*(-(b^2*c^3*d^4) + 12*a*c^4*d^4 - b*c^3*Sqrt[b^2 - 4*a*c]*d^4
+ 3*b^3*c^2*d^3*e - 28*a*b*c^3*d^3*e + 3*b^2*c^2*Sqrt[b^2 - 4*a*c]*d^3*e - 4*a*c
^3*Sqrt[b^2 - 4*a*c]*d^3*e - 3*b^4*c*d^2*e^2 + 12*a*b^2*c^2*d^2*e^2 + 48*a^2*c^3
*d^2*e^2 - 3*b^3*c*Sqrt[b^2 - 4*a*c]*d^2*e^2 + 6*a*b*c^2*Sqrt[b^2 - 4*a*c]*d^2*e
^2 + b^5*d*e^3 + 5*a*b^3*c*d*e^3 - 52*a^2*b*c^2*d*e^3 + b^4*Sqrt[b^2 - 4*a*c]*d*
e^3 + 7*a*b^2*c*Sqrt[b^2 - 4*a*c]*d*e^3 - 36*a^2*c^2*Sqrt[b^2 - 4*a*c]*d*e^3 - 5
*a*b^4*e^4 + 29*a^2*b^2*c*e^4 - 28*a^3*c^2*e^4 - 5*a*b^3*Sqrt[b^2 - 4*a*c]*e^4 +
19*a^2*b*c*Sqrt[b^2 - 4*a*c]*e^4)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2
- 4*a*c]]])/(2*Sqrt[2]*a*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(-(c*d^
2) + b*d*e - a*e^2)^3) + (Sqrt[c]*(b^2*c^3*d^4 - 12*a*c^4*d^4 - b*c^3*Sqrt[b^2 -
4*a*c]*d^4 - 3*b^3*c^2*d^3*e + 28*a*b*c^3*d^3*e + 3*b^2*c^2*Sqrt[b^2 - 4*a*c]*d
^3*e - 4*a*c^3*Sqrt[b^2 - 4*a*c]*d^3*e + 3*b^4*c*d^2*e^2 - 12*a*b^2*c^2*d^2*e^2
- 48*a^2*c^3*d^2*e^2 - 3*b^3*c*Sqrt[b^2 - 4*a*c]*d^2*e^2 + 6*a*b*c^2*Sqrt[b^2 -
4*a*c]*d^2*e^2 - b^5*d*e^3 - 5*a*b^3*c*d*e^3 + 52*a^2*b*c^2*d*e^3 + b^4*Sqrt[b^2
- 4*a*c]*d*e^3 + 7*a*b^2*c*Sqrt[b^2 - 4*a*c]*d*e^3 - 36*a^2*c^2*Sqrt[b^2 - 4*a*
c]*d*e^3 + 5*a*b^4*e^4 - 29*a^2*b^2*c*e^4 + 28*a^3*c^2*e^4 - 5*a*b^3*Sqrt[b^2 -
4*a*c]*e^4 + 19*a^2*b*c*Sqrt[b^2 - 4*a*c]*e^4)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b
+ Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a
*c]]*(-(c*d^2) + b*d*e - a*e^2)^3) + (e^(7/2)*(9*c*d^2 - 5*b*d*e + a*e^2)*ArcTan
[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*(c*d^2 - b*d*e + a*e^2)^3)

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Maple [B]  time = 0.193, size = 14860, normalized size = 13.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x^2+d)^2/(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. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2 + a)^2*(e*x^2 + d)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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(1/((c*x^4 + b*x^2 + a)^2*(e*x^2 + d)^2),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(1/(e*x**2+d)**2/(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(1/((c*x^4 + b*x^2 + a)^2*(e*x^2 + d)^2),x, algorithm="giac")

[Out]

Exception raised: TypeError