3.35 \(\int \frac{A+B x+C x^2}{x^2 \left (a+b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=514 \[ -\frac{-10 a A c-a b C+3 A b^2}{2 a^2 x \left (b^2-4 a c\right )}-\frac{\sqrt{c} \left (A \left (3 b^2 \sqrt{b^2-4 a c}-10 a c \sqrt{b^2-4 a c}-16 a b c+3 b^3\right )-a C \left (b \sqrt{b^2-4 a c}-12 a c+b^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left (-\frac{A \left (3 b^3-16 a b c\right )-a C \left (b^2-12 a c\right )}{\sqrt{b^2-4 a c}}-10 a A c-a b C+3 A b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{b B \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2}}-\frac{B \log \left (a+b x^2+c x^4\right )}{4 a^2}+\frac{B \log (x)}{a^2}+\frac{A \left (b^2-2 a c\right )+c x^2 (A b-2 a C)-a b C}{2 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{B \left (-2 a c+b^2+b c x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]

[Out]

-(3*A*b^2 - 10*a*A*c - a*b*C)/(2*a^2*(b^2 - 4*a*c)*x) + (B*(b^2 - 2*a*c + b*c*x^
2))/(2*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (A*(b^2 - 2*a*c) - a*b*C + c*(A*b
- 2*a*C)*x^2)/(2*a*(b^2 - 4*a*c)*x*(a + b*x^2 + c*x^4)) - (Sqrt[c]*(A*(3*b^3 - 1
6*a*b*c + 3*b^2*Sqrt[b^2 - 4*a*c] - 10*a*c*Sqrt[b^2 - 4*a*c]) - a*(b^2 - 12*a*c
+ b*Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]
])/(2*Sqrt[2]*a^2*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[c]*(3
*A*b^2 - 10*a*A*c - a*b*C - (A*(3*b^3 - 16*a*b*c) - a*(b^2 - 12*a*c)*C)/Sqrt[b^2
- 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a
^2*(b^2 - 4*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (b*B*(b^2 - 6*a*c)*ArcTanh[(b +
2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^2*(b^2 - 4*a*c)^(3/2)) + (B*Log[x])/a^2 - (B*L
og[a + b*x^2 + c*x^4])/(4*a^2)

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Rubi [A]  time = 3.79182, antiderivative size = 514, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 13, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.464 \[ -\frac{-10 a A c-a b C+3 A b^2}{2 a^2 x \left (b^2-4 a c\right )}-\frac{\sqrt{c} \left (A \left (3 b^2 \sqrt{b^2-4 a c}-10 a c \sqrt{b^2-4 a c}-16 a b c+3 b^3\right )-a C \left (b \sqrt{b^2-4 a c}-12 a c+b^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left (-\frac{A \left (3 b^3-16 a b c\right )-a C \left (b^2-12 a c\right )}{\sqrt{b^2-4 a c}}-10 a A c-a b C+3 A b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{b B \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2}}-\frac{B \log \left (a+b x^2+c x^4\right )}{4 a^2}+\frac{B \log (x)}{a^2}+\frac{A \left (b^2-2 a c\right )+c x^2 (A b-2 a C)-a b C}{2 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{B \left (-2 a c+b^2+b c x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x + C*x^2)/(x^2*(a + b*x^2 + c*x^4)^2),x]

[Out]

-(3*A*b^2 - 10*a*A*c - a*b*C)/(2*a^2*(b^2 - 4*a*c)*x) + (B*(b^2 - 2*a*c + b*c*x^
2))/(2*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (A*(b^2 - 2*a*c) - a*b*C + c*(A*b
- 2*a*C)*x^2)/(2*a*(b^2 - 4*a*c)*x*(a + b*x^2 + c*x^4)) - (Sqrt[c]*(A*(3*b^3 - 1
6*a*b*c + 3*b^2*Sqrt[b^2 - 4*a*c] - 10*a*c*Sqrt[b^2 - 4*a*c]) - a*(b^2 - 12*a*c
+ b*Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]
])/(2*Sqrt[2]*a^2*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[c]*(3
*A*b^2 - 10*a*A*c - a*b*C - (A*(3*b^3 - 16*a*b*c) - a*(b^2 - 12*a*c)*C)/Sqrt[b^2
- 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a
^2*(b^2 - 4*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (b*B*(b^2 - 6*a*c)*ArcTanh[(b +
2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^2*(b^2 - 4*a*c)^(3/2)) + (B*Log[x])/a^2 - (B*L
og[a + b*x^2 + c*x^4])/(4*a^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((C*x**2+B*x+A)/x**2/(c*x**4+b*x**2+a)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 4.33749, size = 559, normalized size = 1.09 \[ \frac{\frac{-4 a^2 c (B+C x)+2 a \left (b c x (3 A+x (B+C x))+2 A c^2 x^3+b^2 (B+C x)\right )-2 A b^2 x \left (b+c x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{2} \sqrt{c} \left (A \left (-3 b^2 \sqrt{b^2-4 a c}+10 a c \sqrt{b^2-4 a c}+16 a b c-3 b^3\right )+a C \left (b \sqrt{b^2-4 a c}-12 a c+b^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left (A \left (-3 b^2 \sqrt{b^2-4 a c}+10 a c \sqrt{b^2-4 a c}-16 a b c+3 b^3\right )+a C \left (b \sqrt{b^2-4 a c}+12 a c-b^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{B \left (b^2 \sqrt{b^2-4 a c}-4 a c \sqrt{b^2-4 a c}-6 a b c+b^3\right ) \log \left (\sqrt{b^2-4 a c}-b-2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{B \left (b^2 \sqrt{b^2-4 a c}-4 a c \sqrt{b^2-4 a c}+6 a b c-b^3\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{4 A}{x}+4 B \log (x)}{4 a^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x + C*x^2)/(x^2*(a + b*x^2 + c*x^4)^2),x]

[Out]

((-4*A)/x + (-4*a^2*c*(B + C*x) - 2*A*b^2*x*(b + c*x^2) + 2*a*(2*A*c^2*x^3 + b^2
*(B + C*x) + b*c*x*(3*A + x*(B + C*x))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (
Sqrt[2]*Sqrt[c]*(A*(-3*b^3 + 16*a*b*c - 3*b^2*Sqrt[b^2 - 4*a*c] + 10*a*c*Sqrt[b^
2 - 4*a*c]) + a*(b^2 - 12*a*c + b*Sqrt[b^2 - 4*a*c])*C)*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]
]) + (Sqrt[2]*Sqrt[c]*(A*(3*b^3 - 16*a*b*c - 3*b^2*Sqrt[b^2 - 4*a*c] + 10*a*c*Sq
rt[b^2 - 4*a*c]) + a*(-b^2 + 12*a*c + b*Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sq
rt[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]]) + 4*B*Log[x] - (B*(b^3 - 6*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 4*a*c*Sqrt[b
^2 - 4*a*c])*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(3/2) - (B*(-b
^3 + 6*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 4*a*c*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2
- 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/(4*a^2)

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Maple [B]  time = 0.119, size = 6960, normalized size = 13.5 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((C*x^2+B*x+A)/x^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. \[ \frac{B a b c x^{3} +{\left (10 \, A a c^{2} +{\left (C a b - 3 \, A b^{2}\right )} c\right )} x^{4} - 2 \, A a b^{2} + 8 \, A a^{2} c +{\left (C a b^{2} - 3 \, A b^{3} -{\left (2 \, C a^{2} - 11 \, A a b\right )} c\right )} x^{2} +{\left (B a b^{2} - 2 \, B a^{2} c\right )} x}{2 \,{\left ({\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{5} +{\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{3} +{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x\right )}} + \frac{\int \frac{C a b^{2} - 3 \, A b^{3} - 2 \,{\left (B b^{2} c - 4 \, B a c^{2}\right )} x^{3} +{\left (10 \, A a c^{2} +{\left (C a b - 3 \, A b^{2}\right )} c\right )} x^{2} -{\left (6 \, C a^{2} - 13 \, A a b\right )} c - 2 \,{\left (B b^{3} - 5 \, B a b c\right )} x}{c x^{4} + b x^{2} + a}\,{d x}}{2 \,{\left (a^{2} b^{2} - 4 \, a^{3} c\right )}} + \frac{B \log \left (x\right )}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)/((c*x^4 + b*x^2 + a)^2*x^2),x, algorithm="maxima")

[Out]

1/2*(B*a*b*c*x^3 + (10*A*a*c^2 + (C*a*b - 3*A*b^2)*c)*x^4 - 2*A*a*b^2 + 8*A*a^2*
c + (C*a*b^2 - 3*A*b^3 - (2*C*a^2 - 11*A*a*b)*c)*x^2 + (B*a*b^2 - 2*B*a^2*c)*x)/
((a^2*b^2*c - 4*a^3*c^2)*x^5 + (a^2*b^3 - 4*a^3*b*c)*x^3 + (a^3*b^2 - 4*a^4*c)*x
) + 1/2*integrate((C*a*b^2 - 3*A*b^3 - 2*(B*b^2*c - 4*B*a*c^2)*x^3 + (10*A*a*c^2
+ (C*a*b - 3*A*b^2)*c)*x^2 - (6*C*a^2 - 13*A*a*b)*c - 2*(B*b^3 - 5*B*a*b*c)*x)/
(c*x^4 + b*x^2 + a), x)/(a^2*b^2 - 4*a^3*c) + B*log(x)/a^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((C*x^2 + B*x + A)/((c*x^4 + b*x^2 + a)^2*x^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((C*x**2+B*x+A)/x**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((C*x^2 + B*x + A)/((c*x^4 + b*x^2 + a)^2*x^2),x, algorithm="giac")

[Out]

Exception raised: TypeError