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

Optimal. Leaf size=412 \[ \frac{\left (-\frac{A c \left (4 a c+b^2\right )+b C \left (b^2-8 a c\right )}{\sqrt{b^2-4 a c}}+C \left (b^2-6 a c\right )+A b c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} c^{3/2} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (\frac{A c \left (4 a c+b^2\right )+b C \left (b^2-8 a c\right )}{\sqrt{b^2-4 a c}}+C \left (b^2-6 a c\right )+A b c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} c^{3/2} \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{x^3 \left (-2 a C+x^2 (2 A c-b C)+A b\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{x (2 A c-b C)}{2 c \left (b^2-4 a c\right )}+\frac{2 a B \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{B x^2 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]

[Out]

((2*A*c - b*C)*x)/(2*c*(b^2 - 4*a*c)) + (B*x^2*(2*a + b*x^2))/(2*(b^2 - 4*a*c)*(
a + b*x^2 + c*x^4)) - (x^3*(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)) + ((A*b*c + (b^2 - 6*a*c)*C - (A*c*(b^2 + 4*a*c) + b*(b^2 -
8*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]*c^(3/2)*(b^2 - 4*a*c)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((A*b*c +
(b^2 - 6*a*c)*C + (A*c*(b^2 + 4*a*c) + b*(b^2 - 8*a*c)*C)/Sqrt[b^2 - 4*a*c])*Arc
Tan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*c^(3/2)*(b^2 -
4*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (2*a*B*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 = 2.98195, antiderivative size = 412, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ \frac{\left (-\frac{A c \left (4 a c+b^2\right )+b C \left (b^2-8 a c\right )}{\sqrt{b^2-4 a c}}+C \left (b^2-6 a c\right )+A b c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} c^{3/2} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (\frac{A c \left (4 a c+b^2\right )+b C \left (b^2-8 a c\right )}{\sqrt{b^2-4 a c}}+C \left (b^2-6 a c\right )+A b c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} c^{3/2} \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{x^3 \left (-2 a C+x^2 (2 A c-b C)+A b\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{x (2 A c-b C)}{2 c \left (b^2-4 a c\right )}+\frac{2 a B \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{B x^2 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]

Antiderivative was successfully verified.

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

[Out]

((2*A*c - b*C)*x)/(2*c*(b^2 - 4*a*c)) + (B*x^2*(2*a + b*x^2))/(2*(b^2 - 4*a*c)*(
a + b*x^2 + c*x^4)) - (x^3*(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)) + ((A*b*c + (b^2 - 6*a*c)*C - (A*c*(b^2 + 4*a*c) + b*(b^2 -
8*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]*c^(3/2)*(b^2 - 4*a*c)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((A*b*c +
(b^2 - 6*a*c)*C + (A*c*(b^2 + 4*a*c) + b*(b^2 - 8*a*c)*C)/Sqrt[b^2 - 4*a*c])*Arc
Tan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*c^(3/2)*(b^2 -
4*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (2*a*B*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 = 179.049, size = 376, normalized size = 0.91 \[ \frac{2 B a \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} + \frac{x \left (2 B a c x + B b c x^{3} + a \left (2 A c - C b\right ) - x^{2} \left (- A b c - 2 C a c + C b^{2}\right )\right )}{2 c \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )} + \frac{\sqrt{2} \left (2 a c \left (2 A c - C b\right ) + b \left (A b c - 6 C a c + C b^{2}\right ) + \sqrt{- 4 a c + b^{2}} \left (A b c - 6 C a c + C b^{2}\right )\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{4 c^{\frac{3}{2}} \sqrt{b + \sqrt{- 4 a c + b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} - \frac{\sqrt{2} \left (2 a c \left (2 A c - C b\right ) + b \left (A b c - 6 C a c + C b^{2}\right ) - \sqrt{- 4 a c + b^{2}} \left (A b c - 6 C a c + C b^{2}\right )\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{4 c^{\frac{3}{2}} \sqrt{b - \sqrt{- 4 a c + b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*a*atanh((b + 2*c*x**2)/sqrt(-4*a*c + b**2))/(-4*a*c + b**2)**(3/2) + x*(2*B*
a*c*x + B*b*c*x**3 + a*(2*A*c - C*b) - x**2*(-A*b*c - 2*C*a*c + C*b**2))/(2*c*(-
4*a*c + b**2)*(a + b*x**2 + c*x**4)) + sqrt(2)*(2*a*c*(2*A*c - C*b) + b*(A*b*c -
6*C*a*c + C*b**2) + sqrt(-4*a*c + b**2)*(A*b*c - 6*C*a*c + C*b**2))*atan(sqrt(2
)*sqrt(c)*x/sqrt(b + sqrt(-4*a*c + b**2)))/(4*c**(3/2)*sqrt(b + sqrt(-4*a*c + b*
*2))*(-4*a*c + b**2)**(3/2)) - sqrt(2)*(2*a*c*(2*A*c - C*b) + b*(A*b*c - 6*C*a*c
+ C*b**2) - sqrt(-4*a*c + b**2)*(A*b*c - 6*C*a*c + C*b**2))*atan(sqrt(2)*sqrt(c
)*x/sqrt(b - sqrt(-4*a*c + b**2)))/(4*c**(3/2)*sqrt(b - sqrt(-4*a*c + b**2))*(-4
*a*c + b**2)**(3/2))

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Mathematica [A]  time = 2.80534, size = 444, normalized size = 1.08 \[ \frac{1}{4} \left (\frac{2 \left (a (b (B+C x)-2 c x (A+x (B+C x)))+b x^2 (b (B+C x)-A c x)\right )}{c \left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{2} \left (C \left (b^2 \sqrt{b^2-4 a c}-6 a c \sqrt{b^2-4 a c}+8 a b c-b^3\right )-A c \left (-b \sqrt{b^2-4 a c}+4 a c+b^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left (A c \left (b \sqrt{b^2-4 a c}+4 a c+b^2\right )+C \left (b^2 \sqrt{b^2-4 a c}-6 a c \sqrt{b^2-4 a c}-8 a b c+b^3\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{4 a B \log \left (\sqrt{b^2-4 a c}-b-2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{4 a B \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((2*(b*x^2*(-(A*c*x) + b*(B + C*x)) + a*(b*(B + C*x) - 2*c*x*(A + x*(B + C*x))))
)/(c*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(-(A*c*(b^2 + 4*a*c - b*Sqrt
[b^2 - 4*a*c])) + (-b^3 + 8*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 6*a*c*Sqrt[b^2 - 4*a
*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(c^(3/2)*(b^2 -
4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(A*c*(b^2 + 4*a*c + b*Sqrt
[b^2 - 4*a*c]) + (b^3 - 8*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 6*a*c*Sqrt[b^2 - 4*a*c
])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(c^(3/2)*(b^2 - 4
*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) - (4*a*B*Log[-b + Sqrt[b^2 - 4*a*c] - 2
*c*x^2])/(b^2 - 4*a*c)^(3/2) + (4*a*B*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2
- 4*a*c)^(3/2))/4

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Maple [B]  time = 0.13, size = 5283, normalized size = 12.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError