3.849 \(\int \frac{x^5}{a+b x^2+c x^4} \, dx\)

Optimal. Leaf size=81 \[ -\frac{\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^2 \sqrt{b^2-4 a c}}-\frac{b \log \left (a+b x^2+c x^4\right )}{4 c^2}+\frac{x^2}{2 c} \]

[Out]

x^2/(2*c) - ((b^2 - 2*a*c)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^2*Sqrt
[b^2 - 4*a*c]) - (b*Log[a + b*x^2 + c*x^4])/(4*c^2)

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Rubi [A]  time = 0.185071, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^2 \sqrt{b^2-4 a c}}-\frac{b \log \left (a+b x^2+c x^4\right )}{4 c^2}+\frac{x^2}{2 c} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + b*x^2 + c*x^4),x]

[Out]

x^2/(2*c) - ((b^2 - 2*a*c)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^2*Sqrt
[b^2 - 4*a*c]) - (b*Log[a + b*x^2 + c*x^4])/(4*c^2)

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Rubi in Sympy [A]  time = 26.5892, size = 73, normalized size = 0.9 \[ - \frac{b \log{\left (a + b x^{2} + c x^{4} \right )}}{4 c^{2}} + \frac{x^{2}}{2 c} - \frac{\left (- 2 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 c^{2} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(c*x**4+b*x**2+a),x)

[Out]

-b*log(a + b*x**2 + c*x**4)/(4*c**2) + x**2/(2*c) - (-2*a*c + b**2)*atanh((b + 2
*c*x**2)/sqrt(-4*a*c + b**2))/(2*c**2*sqrt(-4*a*c + b**2))

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Mathematica [A]  time = 0.075212, size = 78, normalized size = 0.96 \[ \frac{\frac{2 \left (b^2-2 a c\right ) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-b \log \left (a+b x^2+c x^4\right )+2 c x^2}{4 c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a + b*x^2 + c*x^4),x]

[Out]

(2*c*x^2 + (2*(b^2 - 2*a*c)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2
+ 4*a*c] - b*Log[a + b*x^2 + c*x^4])/(4*c^2)

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Maple [A]  time = 0.004, size = 111, normalized size = 1.4 \[{\frac{{x}^{2}}{2\,c}}-{\frac{b\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) }{4\,{c}^{2}}}-{\frac{a}{c}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{2}}{2\,{c}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(c*x^4+b*x^2+a),x)

[Out]

1/2*x^2/c-1/4*b*ln(c*x^4+b*x^2+a)/c^2-1/c/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(
4*a*c-b^2)^(1/2))*a+1/2/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/
2))*b^2

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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(x^5/(c*x^4 + b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.268004, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b^{2} - 2 \, a c\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) -{\left (2 \, c x^{2} - b \log \left (c x^{4} + b x^{2} + a\right )\right )} \sqrt{b^{2} - 4 \, a c}}{4 \, \sqrt{b^{2} - 4 \, a c} c^{2}}, \frac{2 \,{\left (b^{2} - 2 \, a c\right )} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) +{\left (2 \, c x^{2} - b \log \left (c x^{4} + b x^{2} + a\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{4 \, \sqrt{-b^{2} + 4 \, a c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(c*x^4 + b*x^2 + a),x, algorithm="fricas")

[Out]

[-1/4*((b^2 - 2*a*c)*log((b^3 - 4*a*b*c + 2*(b^2*c - 4*a*c^2)*x^2 + (2*c^2*x^4 +
2*b*c*x^2 + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^4 + b*x^2 + a)) - (2*c*x^2 - b
*log(c*x^4 + b*x^2 + a))*sqrt(b^2 - 4*a*c))/(sqrt(b^2 - 4*a*c)*c^2), 1/4*(2*(b^2
- 2*a*c)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + (2*c*x^2 - b
*log(c*x^4 + b*x^2 + a))*sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^2)]

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Sympy [A]  time = 5.3122, size = 316, normalized size = 3.9 \[ \left (- \frac{b}{4 c^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 c^{2} \left (4 a c - b^{2}\right )}\right ) \log{\left (x^{2} + \frac{- a b - 8 a c^{2} \left (- \frac{b}{4 c^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 c^{2} \left (4 a c - b^{2}\right )}\right ) + 2 b^{2} c \left (- \frac{b}{4 c^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 c^{2} \left (4 a c - b^{2}\right )}\right )}{2 a c - b^{2}} \right )} + \left (- \frac{b}{4 c^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 c^{2} \left (4 a c - b^{2}\right )}\right ) \log{\left (x^{2} + \frac{- a b - 8 a c^{2} \left (- \frac{b}{4 c^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 c^{2} \left (4 a c - b^{2}\right )}\right ) + 2 b^{2} c \left (- \frac{b}{4 c^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 c^{2} \left (4 a c - b^{2}\right )}\right )}{2 a c - b^{2}} \right )} + \frac{x^{2}}{2 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(c*x**4+b*x**2+a),x)

[Out]

(-b/(4*c**2) - sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(4*c**2*(4*a*c - b**2)))*log(x
**2 + (-a*b - 8*a*c**2*(-b/(4*c**2) - sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(4*c**2
*(4*a*c - b**2))) + 2*b**2*c*(-b/(4*c**2) - sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(
4*c**2*(4*a*c - b**2))))/(2*a*c - b**2)) + (-b/(4*c**2) + sqrt(-4*a*c + b**2)*(2
*a*c - b**2)/(4*c**2*(4*a*c - b**2)))*log(x**2 + (-a*b - 8*a*c**2*(-b/(4*c**2) +
sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(4*c**2*(4*a*c - b**2))) + 2*b**2*c*(-b/(4*c
**2) + sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(4*c**2*(4*a*c - b**2))))/(2*a*c - b**
2)) + x**2/(2*c)

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GIAC/XCAS [A]  time = 0.291988, size = 101, normalized size = 1.25 \[ \frac{x^{2}}{2 \, c} - \frac{b{\rm ln}\left (c x^{4} + b x^{2} + a\right )}{4 \, c^{2}} + \frac{{\left (b^{2} - 2 \, a c\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(c*x^4 + b*x^2 + a),x, algorithm="giac")

[Out]

1/2*x^2/c - 1/4*b*ln(c*x^4 + b*x^2 + a)/c^2 + 1/2*(b^2 - 2*a*c)*arctan((2*c*x^2
+ b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^2)