3.36 \(\int \frac{A+B x+C x^2}{x^3 \left (a+b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=534 \[ \frac{(2 A b-a C) \log \left (a+b x^2+c x^4\right )}{4 a^3}-\frac{\log (x) (2 A b-a C)}{a^3}-\frac{-6 a A c-a b C+2 A b^2}{2 a^2 x^2 \left (b^2-4 a c\right )}-\frac{B \left (3 b^2-10 a c\right )}{2 a^2 x \left (b^2-4 a c\right )}-\frac{B \sqrt{c} \left (\left (3 b^2-10 a c\right ) \sqrt{b^2-4 a c}-16 a b c+3 b^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{B \sqrt{c} \left (-\left (3 b^2-10 a c\right ) \sqrt{b^2-4 a c}-16 a b c+3 b^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{\left (2 A \left (6 a^2 c^2-6 a b^2 c+b^4\right )-a b C \left (b^2-6 a c\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^3 \left (b^2-4 a c\right )^{3/2}}+\frac{A \left (b^2-2 a c\right )+c x^2 (A b-2 a C)-a b C}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{B \left (-2 a c+b^2+b c x^2\right )}{2 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]

[Out]

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

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Rubi [A]  time = 4.17279, antiderivative size = 534, 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 \[ \frac{(2 A b-a C) \log \left (a+b x^2+c x^4\right )}{4 a^3}-\frac{\log (x) (2 A b-a C)}{a^3}-\frac{-6 a A c-a b C+2 A b^2}{2 a^2 x^2 \left (b^2-4 a c\right )}-\frac{B \left (3 b^2-10 a c\right )}{2 a^2 x \left (b^2-4 a c\right )}-\frac{B \sqrt{c} \left (\left (3 b^2-10 a c\right ) \sqrt{b^2-4 a c}-16 a b c+3 b^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{B \sqrt{c} \left (-\left (3 b^2-10 a c\right ) \sqrt{b^2-4 a c}-16 a b c+3 b^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{\left (2 A \left (6 a^2 c^2-6 a b^2 c+b^4\right )-a b C \left (b^2-6 a c\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^3 \left (b^2-4 a c\right )^{3/2}}+\frac{A \left (b^2-2 a c\right )+c x^2 (A b-2 a C)-a b C}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{B \left (-2 a c+b^2+b c x^2\right )}{2 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x + C*x^2)/(x^3*(a + b*x^2 + c*x^4)^2),x]

[Out]

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

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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**3/(c*x**4+b*x**2+a)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 5.20855, size = 655, normalized size = 1.23 \[ \frac{-\frac{2 a \left (2 a^2 c C+A \left (-3 a b c-2 a c^2 x^2+b^3+b^2 c x^2\right )-a \left (b^2 C+b c x (3 B+C x)+2 B c^2 x^3\right )+b^2 B x \left (b+c x^2\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (2 A \left (6 a^2 c^2-6 a b^2 c-4 a b c \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}+b^4\right )+a C \left (-b^2 \sqrt{b^2-4 a c}+4 a c \sqrt{b^2-4 a c}+6 a b c-b^3\right )\right ) \log \left (\sqrt{b^2-4 a c}-b-2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{\left (2 A \left (-6 a^2 c^2+6 a b^2 c-4 a b c \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}-b^4\right )+a C \left (-b^2 \sqrt{b^2-4 a c}+4 a c \sqrt{b^2-4 a c}-6 a b c+b^3\right )\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+4 \log (x) (a C-2 A b)-\frac{2 a A}{x^2}+\frac{\sqrt{2} a B \sqrt{c} \left (-3 b^2 \sqrt{b^2-4 a c}+10 a c \sqrt{b^2-4 a c}+16 a b c-3 b^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} a B \sqrt{c} \left (-3 b^2 \sqrt{b^2-4 a c}+10 a c \sqrt{b^2-4 a c}-16 a b c+3 b^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{4 a B}{x}}{4 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x + C*x^2)/(x^3*(a + b*x^2 + c*x^4)^2),x]

[Out]

((-2*a*A)/x^2 - (4*a*B)/x - (2*a*(2*a^2*c*C + b^2*B*x*(b + c*x^2) + A*(b^3 - 3*a
*b*c + b^2*c*x^2 - 2*a*c^2*x^2) - a*(b^2*C + 2*B*c^2*x^3 + b*c*x*(3*B + C*x))))/
((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*a*B*Sqrt[c]*(-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])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/S
qrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) +
(Sqrt[2]*a*B*Sqrt[c]*(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])*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]]) + 4*(-2*A*b + a*C)*Log[x] + ((2*A*(b^4
- 6*a*b^2*c + 6*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 4*a*b*c*Sqrt[b^2 - 4*a*c]) + a
*(-b^3 + 6*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + 4*a*c*Sqrt[b^2 - 4*a*c])*C)*Log[-b +
Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + ((2*A*(-b^4 + 6*a*b^2*c - 6*
a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 4*a*b*c*Sqrt[b^2 - 4*a*c]) + a*(b^3 - 6*a*b*c
- b^2*Sqrt[b^2 - 4*a*c] + 4*a*c*Sqrt[b^2 - 4*a*c])*C)*Log[b + Sqrt[b^2 - 4*a*c]
+ 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/(4*a^3)

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Maple [B]  time = 0.122, size = 6930, normalized size = 13. \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((C*x^2+B*x+A)/x^3/(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. \[ -\frac{{\left (3 \, B b^{2} c - 10 \, B a c^{2}\right )} x^{5} -{\left (6 \, A a c^{2} +{\left (C a b - 2 \, A b^{2}\right )} c\right )} x^{4} + A a b^{2} - 4 \, A a^{2} c +{\left (3 \, B b^{3} - 11 \, B a b c\right )} x^{3} -{\left (C a b^{2} - 2 \, A b^{3} -{\left (2 \, C a^{2} - 7 \, A a b\right )} c\right )} x^{2} + 2 \,{\left (B a b^{2} - 4 \, B a^{2} c\right )} x}{2 \,{\left ({\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{6} +{\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{4} +{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x^{2}\right )}} - \frac{\int \frac{3 \, B a b^{3} - 13 \, B a^{2} b c - 2 \,{\left (4 \,{\left (C a^{2} - 2 \, A a b\right )} c^{2} -{\left (C a b^{2} - 2 \, A b^{3}\right )} c\right )} x^{3} +{\left (3 \, B a b^{2} c - 10 \, B a^{2} c^{2}\right )} x^{2} + 2 \,{\left (C a b^{3} - 2 \, A b^{4} - 6 \, A a^{2} c^{2} - 5 \,{\left (C a^{2} b - 2 \, A a b^{2}\right )} c\right )} x}{c x^{4} + b x^{2} + a}\,{d x}}{2 \,{\left (a^{3} b^{2} - 4 \, a^{4} c\right )}} + \frac{{\left (C a - 2 \, A b\right )} \log \left (x\right )}{a^{3}} \]

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^3),x, algorithm="maxima")

[Out]

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

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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^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((C*x**2+B*x+A)/x**3/(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^3),x, algorithm="giac")

[Out]

Exception raised: TypeError