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

Optimal. Leaf size=69 \[ -\frac{(a+b) \tanh ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 a^{3/2} \sqrt{b}}-\frac{\log \left (a x^4+2 a x^2+a-b\right )}{2 a}+\frac{x^2}{2 a} \]

[Out]

x^2/(2*a) - ((a + b)*ArcTanh[(Sqrt[a]*(1 + x^2))/Sqrt[b]])/(2*a^(3/2)*Sqrt[b]) -
Log[a - b + 2*a*x^2 + a*x^4]/(2*a)

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Rubi [A]  time = 0.18313, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{(a+b) \tanh ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 a^{3/2} \sqrt{b}}-\frac{\log \left (a x^4+2 a x^2+a-b\right )}{2 a}+\frac{x^2}{2 a} \]

Antiderivative was successfully verified.

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

[Out]

x^2/(2*a) - ((a + b)*ArcTanh[(Sqrt[a]*(1 + x^2))/Sqrt[b]])/(2*a^(3/2)*Sqrt[b]) -
Log[a - b + 2*a*x^2 + a*x^4]/(2*a)

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Rubi in Sympy [A]  time = 30.7995, size = 58, normalized size = 0.84 \[ \frac{x^{2}}{2 a} - \frac{\log{\left (a x^{4} + 2 a x^{2} + a - b \right )}}{2 a} - \frac{\left (a + b\right ) \operatorname{atanh}{\left (\frac{\sqrt{a} \left (x^{2} + 1\right )}{\sqrt{b}} \right )}}{2 a^{\frac{3}{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2/(2*a) - log(a*x**4 + 2*a*x**2 + a - b)/(2*a) - (a + b)*atanh(sqrt(a)*(x**2
+ 1)/sqrt(b))/(2*a**(3/2)*sqrt(b))

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Mathematica [A]  time = 0.0622546, size = 62, normalized size = 0.9 \[ \frac{x^2-\log \left (a \left (x^2+1\right )^2-b\right )}{2 a}-\frac{(a+b) \tanh ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 a^{3/2} \sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

-((a + b)*ArcTanh[(Sqrt[a]*(1 + x^2))/Sqrt[b]])/(2*a^(3/2)*Sqrt[b]) + (x^2 - Log
[-b + a*(1 + x^2)^2])/(2*a)

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Maple [A]  time = 0.005, size = 86, normalized size = 1.3 \[{\frac{{x}^{2}}{2\,a}}-{\frac{\ln \left ( a{x}^{4}+2\,a{x}^{2}+a-b \right ) }{2\,a}}-{\frac{1}{2}{\it Artanh} \left ({\frac{2\,a{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{b}{2\,a}{\it Artanh} \left ({\frac{2\,a{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(a*x^4+2*a*x^2+a-b),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.278857, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (a + b\right )} \log \left (-\frac{2 \, a b x^{2} + 2 \, a b -{\left (a x^{4} + 2 \, a x^{2} + a + b\right )} \sqrt{a b}}{a x^{4} + 2 \, a x^{2} + a - b}\right ) + 2 \, \sqrt{a b}{\left (x^{2} - \log \left (a x^{4} + 2 \, a x^{2} + a - b\right )\right )}}{4 \, \sqrt{a b} a}, \frac{{\left (a + b\right )} \arctan \left (\frac{b}{\sqrt{-a b}{\left (x^{2} + 1\right )}}\right ) + \sqrt{-a b}{\left (x^{2} - \log \left (a x^{4} + 2 \, a x^{2} + a - b\right )\right )}}{2 \, \sqrt{-a b} a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.68541, size = 138, normalized size = 2. \[ \left (- \frac{1}{2 a} - \frac{\sqrt{a^{3} b} \left (a + b\right )}{4 a^{3} b}\right ) \log{\left (x^{2} + \frac{- 4 a b \left (- \frac{1}{2 a} - \frac{\sqrt{a^{3} b} \left (a + b\right )}{4 a^{3} b}\right ) + a - b}{a + b} \right )} + \left (- \frac{1}{2 a} + \frac{\sqrt{a^{3} b} \left (a + b\right )}{4 a^{3} b}\right ) \log{\left (x^{2} + \frac{- 4 a b \left (- \frac{1}{2 a} + \frac{\sqrt{a^{3} b} \left (a + b\right )}{4 a^{3} b}\right ) + a - b}{a + b} \right )} + \frac{x^{2}}{2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-1/(2*a) - sqrt(a**3*b)*(a + b)/(4*a**3*b))*log(x**2 + (-4*a*b*(-1/(2*a) - sqrt
(a**3*b)*(a + b)/(4*a**3*b)) + a - b)/(a + b)) + (-1/(2*a) + sqrt(a**3*b)*(a + b
)/(4*a**3*b))*log(x**2 + (-4*a*b*(-1/(2*a) + sqrt(a**3*b)*(a + b)/(4*a**3*b)) +
a - b)/(a + b)) + x**2/(2*a)

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GIAC/XCAS [A]  time = 0.54788, size = 81, normalized size = 1.17 \[ \frac{x^{2}}{2 \, a} + \frac{{\left (a + b\right )} \arctan \left (\frac{a x^{2} + a}{\sqrt{-a b}}\right )}{2 \, \sqrt{-a b} a} - \frac{{\rm ln}\left (a x^{4} + 2 \, a x^{2} + a - b\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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