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

Optimal. Leaf size=660 \[ -\frac{\sqrt{c} e^2 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} \sqrt{b-\sqrt{b^2-4 a c}} \left (a e^2-b d e+c d^2\right )^2}-\frac{\sqrt{c} e^2 \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} \sqrt{\sqrt{b^2-4 a c}+b} \left (a e^2-b d e+c d^2\right )^2}+\frac{x \left (c x^2 \left (2 a c e+b^2 (-e)+b c d\right )+3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right ) \left (a e^2-b d e+c d^2\right )}+\frac{\sqrt{c} \left (\frac{8 a b c e-12 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+2 a c e+b^2 (-e)+b c d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}} \left (a e^2-b d e+c d^2\right )}+\frac{\sqrt{c} \left (-\frac{8 a b c e-12 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+2 a c e+b^2 (-e)+b c d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b} \left (a e^2-b d e+c d^2\right )}+\frac{e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \left (a e^2-b d e+c d^2\right )^2} \]

[Out]

(x*(b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e + c*(b*c*d - b^2*e + 2*a*c*e)*x^2))/
(2*a*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x^2 + c*x^4)) - (Sqrt[c]*e^2*(
e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^
2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*e^2)^2) +
(Sqrt[c]*(b*c*d - b^2*e + 2*a*c*e + (b^2*c*d - 12*a*c^2*d - b^3*e + 8*a*b*c*e)/S
qrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sq
rt[2]*a*(b^2 - 4*a*c)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*e^2)) - (Sq
rt[c]*e^2*(e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[
b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a
*e^2)^2) + (Sqrt[c]*(b*c*d - b^2*e + 2*a*c*e - (b^2*c*d - 12*a*c^2*d - b^3*e + 8
*a*b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*
c]]])/(2*Sqrt[2]*a*(b^2 - 4*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*
e^2)) + (e^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*(c*d^2 - b*d*e + a*e^2)^2
)

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Rubi [A]  time = 6.51583, antiderivative size = 660, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{\sqrt{c} e^2 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} \sqrt{b-\sqrt{b^2-4 a c}} \left (a e^2-b d e+c d^2\right )^2}-\frac{\sqrt{c} e^2 \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} \sqrt{\sqrt{b^2-4 a c}+b} \left (a e^2-b d e+c d^2\right )^2}+\frac{x \left (c x^2 \left (2 a c e+b^2 (-e)+b c d\right )+3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right ) \left (a e^2-b d e+c d^2\right )}+\frac{\sqrt{c} \left (\frac{8 a b c e-12 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+2 a c e+b^2 (-e)+b c d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}} \left (a e^2-b d e+c d^2\right )}+\frac{\sqrt{c} \left (-\frac{8 a b c e-12 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+2 a c e+b^2 (-e)+b c d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b} \left (a e^2-b d e+c d^2\right )}+\frac{e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \left (a e^2-b d e+c d^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(x*(b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e + c*(b*c*d - b^2*e + 2*a*c*e)*x^2))/
(2*a*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x^2 + c*x^4)) - (Sqrt[c]*e^2*(
e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^
2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*e^2)^2) +
(Sqrt[c]*(b*c*d - b^2*e + 2*a*c*e + (b^2*c*d - 12*a*c^2*d - b^3*e + 8*a*b*c*e)/S
qrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sq
rt[2]*a*(b^2 - 4*a*c)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*e^2)) - (Sq
rt[c]*e^2*(e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[
b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a
*e^2)^2) + (Sqrt[c]*(b*c*d - b^2*e + 2*a*c*e - (b^2*c*d - 12*a*c^2*d - b^3*e + 8
*a*b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*
c]]])/(2*Sqrt[2]*a*(b^2 - 4*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*
e^2)) + (e^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*(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)/(c*x**4+b*x**2+a)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 6.28336, size = 708, normalized size = 1.07 \[ \frac{\frac{2 x \left (e (a e-b d)+c d^2\right ) \left (-b c \left (3 a e+c d x^2\right )+2 a c^2 \left (d-e x^2\right )+b^3 e+b^2 c \left (e x^2-d\right )\right )}{a \left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{2} \sqrt{c} \left (b^2 \left (-c d e \left (2 d \sqrt{b^2-4 a c}+3 a e\right )-3 a e^3 \sqrt{b^2-4 a c}+c^2 d^3\right )+2 a c \left (c d e \left (d \sqrt{b^2-4 a c}-14 a e\right )+5 a e^3 \sqrt{b^2-4 a c}-6 c^2 d^3\right )+b c \left (c d^2 \left (d \sqrt{b^2-4 a c}+20 a e\right )+a e^2 \left (16 a e-d \sqrt{b^2-4 a c}\right )\right )+b^3 e \left (e \left (d \sqrt{b^2-4 a c}-3 a e\right )-2 c d^2\right )+b^4 d e^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{a \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left (-b^2 \left (c d e \left (2 d \sqrt{b^2-4 a c}-3 a e\right )+3 a e^3 \sqrt{b^2-4 a c}+c^2 d^3\right )+2 a c \left (c d e \left (d \sqrt{b^2-4 a c}+14 a e\right )+5 a e^3 \sqrt{b^2-4 a c}+6 c^2 d^3\right )+b c \left (c d^2 \left (d \sqrt{b^2-4 a c}-20 a e\right )-a e^2 \left (d \sqrt{b^2-4 a c}+16 a e\right )\right )+b^3 e \left (e \left (d \sqrt{b^2-4 a c}+3 a e\right )+2 c d^2\right )+b^4 (-d) e^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{a \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{4 e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d}}}{4 \left (e (a e-b d)+c d^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

((2*(c*d^2 + e*(-(b*d) + a*e))*x*(b^3*e - b*c*(3*a*e + c*d*x^2) + 2*a*c^2*(d - e
*x^2) + b^2*c*(-d + e*x^2)))/(a*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*S
qrt[c]*(b^4*d*e^2 + 2*a*c*(-6*c^2*d^3 + 5*a*Sqrt[b^2 - 4*a*c]*e^3 + c*d*e*(Sqrt[
b^2 - 4*a*c]*d - 14*a*e)) + b^3*e*(-2*c*d^2 + e*(Sqrt[b^2 - 4*a*c]*d - 3*a*e)) +
b^2*(c^2*d^3 - 3*a*Sqrt[b^2 - 4*a*c]*e^3 - c*d*e*(2*Sqrt[b^2 - 4*a*c]*d + 3*a*e
)) + b*c*(a*e^2*(-(Sqrt[b^2 - 4*a*c]*d) + 16*a*e) + c*d^2*(Sqrt[b^2 - 4*a*c]*d +
20*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*
a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(-(b^4*d*e^2) - b^2*(
c^2*d^3 + 3*a*Sqrt[b^2 - 4*a*c]*e^3 + c*d*e*(2*Sqrt[b^2 - 4*a*c]*d - 3*a*e)) + b
^3*e*(2*c*d^2 + e*(Sqrt[b^2 - 4*a*c]*d + 3*a*e)) + 2*a*c*(6*c^2*d^3 + 5*a*Sqrt[b
^2 - 4*a*c]*e^3 + c*d*e*(Sqrt[b^2 - 4*a*c]*d + 14*a*e)) + b*c*(c*d^2*(Sqrt[b^2 -
4*a*c]*d - 20*a*e) - a*e^2*(Sqrt[b^2 - 4*a*c]*d + 16*a*e)))*ArcTan[(Sqrt[2]*Sqr
t[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 -
4*a*c]]) + (4*e^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d])/(4*(c*d^2 + e*(-(b*
d) + a*e))^2)

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Maple [B]  time = 0.153, size = 10749, normalized size = 16.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x^2+d)/(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)),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)),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)/(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)),x, algorithm="giac")

[Out]

Exception raised: TypeError