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

Optimal. Leaf size=317 \[ \frac{(2 A c-b C) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{-2 a C+x^2 (2 A c-b C)+A b}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{B x \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{B \sqrt{c} \left (2 b-\sqrt{b^2-4 a c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{B \sqrt{c} \left (\sqrt{b^2-4 a c}+2 b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}} \]

[Out]

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

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Rubi [A]  time = 0.99192, antiderivative size = 317, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346 \[ \frac{(2 A c-b C) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{-2 a C+x^2 (2 A c-b C)+A b}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{B x \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{B \sqrt{c} \left (2 b-\sqrt{b^2-4 a c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{B \sqrt{c} \left (\sqrt{b^2-4 a c}+2 b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 98.3809, size = 291, normalized size = 0.92 \[ - \frac{\sqrt{2} B \sqrt{c} \left (2 b + \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{2 \sqrt{b + \sqrt{- 4 a c + b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} + \frac{\sqrt{2} B \sqrt{c} \left (2 b - \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{2 \sqrt{b - \sqrt{- 4 a c + b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} + \frac{\left (2 A c - C b\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} + \frac{x \left (- B a b - 2 B a c x^{2} + c x^{3} \left (A b - 2 C a\right ) + x \left (- 2 A a c + A b^{2} - C a b\right )\right )}{2 a \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*B*sqrt(c)*(2*b + sqrt(-4*a*c + b**2))*atan(sqrt(2)*sqrt(c)*x/sqrt(b + s
qrt(-4*a*c + b**2)))/(2*sqrt(b + sqrt(-4*a*c + b**2))*(-4*a*c + b**2)**(3/2)) +
sqrt(2)*B*sqrt(c)*(2*b - sqrt(-4*a*c + b**2))*atan(sqrt(2)*sqrt(c)*x/sqrt(b - sq
rt(-4*a*c + b**2)))/(2*sqrt(b - sqrt(-4*a*c + b**2))*(-4*a*c + b**2)**(3/2)) + (
2*A*c - C*b)*atanh((b + 2*c*x**2)/sqrt(-4*a*c + b**2))/(-4*a*c + b**2)**(3/2) +
x*(-B*a*b - 2*B*a*c*x**2 + c*x**3*(A*b - 2*C*a) + x*(-2*A*a*c + A*b**2 - C*a*b))
/(2*a*(-4*a*c + b**2)*(a + b*x**2 + c*x**4))

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Mathematica [A]  time = 3.315, size = 335, normalized size = 1.06 \[ \frac{1}{2} \left (\frac{2 a C-A \left (b+2 c x^2\right )+x \left (-b B+b C x-2 B c x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{(b C-2 A c) \log \left (\sqrt{b^2-4 a c}-b-2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{(2 A c-b C) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{\sqrt{2} B \sqrt{c} \left (\sqrt{b^2-4 a c}-2 b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} B \sqrt{c} \left (\sqrt{b^2-4 a c}+2 b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.162, size = 1344, normalized size = 4.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c/(4*a*c-b^2)^2*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)/(
(-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*(-4*a*c+b^2)^(1/2)*b*B-1/2*c/(4*a*c-b^2)^2*2^(
1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))
*c)^(1/2))*b^2*B+1/4/c/(4*a*c-b^2)^2/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))*C*b^
3-1/2/(4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)*A*b^2-1/2/(4*a*c-b^2)
^2/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))*A*b^2-c/(4*a*c-b^2)^2*2^(1/2)/((b+(-4*
a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*(-
4*a*c+b^2)^(1/2)*b*B-c/(4*a*c-b^2)^2*A*(-4*a*c+b^2)^(1/2)*ln(-2*c*x^2+(-4*a*c+b^
2)^(1/2)-b)-1/2/(4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)*B*x*b^2-1/(
4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)*C*(-4*a*c+b^2)^(1/2)*a-1/(4*
a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)*C*a*b-1/2/(4*a*c-b^2)^2*C*(-4*
a*c+b^2)^(1/2)*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)*b-1/2/(4*a*c-b^2)^2/(x^2+1/2*b/c
-1/2/c*(-4*a*c+b^2)^(1/2))*B*x*b^2+1/(4*a*c-b^2)^2/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^
2)^(1/2))*C*(-4*a*c+b^2)^(1/2)*a-1/(4*a*c-b^2)^2/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)
^(1/2))*C*a*b+1/2/(4*a*c-b^2)^2*C*(-4*a*c+b^2)^(1/2)*ln(-2*c*x^2+(-4*a*c+b^2)^(1
/2)-b)*b+2*c/(4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)*a*A+1/4/c/(4*a
*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)*C*b^3+c/(4*a*c-b^2)^2*A*(-4*a*c
+b^2)^(1/2)*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)+2*c/(4*a*c-b^2)^2/(x^2+1/2*b/c-1/2/
c*(-4*a*c+b^2)^(1/2))*a*A+2*c/(4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/
c)*B*a*x+1/4/c/(4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)*C*(-4*a*c+b^
2)^(1/2)*b^2+2*c/(4*a*c-b^2)^2/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))*B*a*x-1/4/
c/(4*a*c-b^2)^2/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))*C*(-4*a*c+b^2)^(1/2)*b^2-
2*c^2/(4*a*c-b^2)^2*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2
)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*B+1/2*c/(4*a*c-b^2)^2*2^(1/2)/((-b+(-4*a*
c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^
2*B+2*c^2/(4*a*c-b^2)^2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1
/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*B

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{2 \, B c x^{3} + B b x -{\left (C b - 2 \, A c\right )} x^{2} - 2 \, C a + A b}{2 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}\right )}} - \frac{\int \frac{2 \, B c x^{2} - B b - 2 \,{\left (C b - 2 \, A c\right )} x}{c x^{4} + b x^{2} + a}\,{d x}}{2 \,{\left (b^{2} - 4 \, a c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(2*B*c*x^3 + B*b*x - (C*b - 2*A*c)*x^2 - 2*C*a + A*b)/((b^2*c - 4*a*c^2)*x^
4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*x^2) - 1/2*integrate((2*B*c*x^2 - B*b - 2*
(C*b - 2*A*c)*x)/(c*x^4 + b*x^2 + a), x)/(b^2 - 4*a*c)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError

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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(x*(C*x**2+B*x+A)/(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)*x/(c*x^4 + b*x^2 + a)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError