3.907 \(\int \frac{x^3}{a+b+2 a x^2+a x^4} \, dx\)

Optimal. Leaf size=54 \[ \frac{\log \left (a x^4+2 a x^2+a+b\right )}{4 a}-\frac{\tan ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.103545, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\log \left (a x^4+2 a x^2+a+b\right )}{4 a}-\frac{\tan ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.2146, size = 48, normalized size = 0.89 \[ \frac{\log{\left (a x^{4} + 2 a x^{2} + a + b \right )}}{4 a} - \frac{\operatorname{atan}{\left (\frac{\sqrt{a} \left (x^{2} + 1\right )}{\sqrt{b}} \right )}}{2 \sqrt{a} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(a*x**4+2*a*x**2+a+b),x)

[Out]

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

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Mathematica [A]  time = 0.0297325, size = 49, normalized size = 0.91 \[ \frac{\log \left (a \left (x^2+1\right )^2+b\right )-\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{\sqrt{b}}}{4 a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 47, normalized size = 0.9 \[{\frac{\ln \left ( a{x}^{4}+2\,a{x}^{2}+a+b \right ) }{4\,a}}-{\frac{1}{2}\arctan \left ({\frac{2\,a{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(a*x^4+2*a*x^2+a+b),x)

[Out]

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

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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^3/(a*x^4 + 2*a*x^2 + a + b),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.279579, size = 1, normalized size = 0.02 \[ \left [\frac{a \log \left (-\frac{2 \, a b x^{2} + 2 \, a b -{\left (a x^{4} + 2 \, a x^{2} + a - b\right )} \sqrt{-a b}}{a x^{4} + 2 \, a x^{2} + a + b}\right ) + \sqrt{-a b} \log \left (a x^{4} + 2 \, a x^{2} + a + b\right )}{4 \, \sqrt{-a b} a}, \frac{2 \, a \arctan \left (\frac{b}{\sqrt{a b}{\left (x^{2} + 1\right )}}\right ) + \sqrt{a b} \log \left (a x^{4} + 2 \, a x^{2} + a + b\right )}{4 \, \sqrt{a b} a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(a*x^4 + 2*a*x^2 + a + b),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.40943, size = 117, normalized size = 2.17 \[ \left (\frac{1}{4 a} - \frac{\sqrt{- a^{3} b}}{4 a^{2} b}\right ) \log{\left (x^{2} + \frac{- 4 a b \left (\frac{1}{4 a} - \frac{\sqrt{- a^{3} b}}{4 a^{2} b}\right ) + a + b}{a} \right )} + \left (\frac{1}{4 a} + \frac{\sqrt{- a^{3} b}}{4 a^{2} b}\right ) \log{\left (x^{2} + \frac{- 4 a b \left (\frac{1}{4 a} + \frac{\sqrt{- a^{3} b}}{4 a^{2} b}\right ) + a + b}{a} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(a*x**4+2*a*x**2+a+b),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.549735, size = 57, normalized size = 1.06 \[ -\frac{\arctan \left (\frac{a x^{2} + a}{\sqrt{a b}}\right )}{2 \, \sqrt{a b}} + \frac{{\rm ln}\left (a x^{4} + 2 \, a x^{2} + a + b\right )}{4 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(a*x^4 + 2*a*x^2 + a + b),x, algorithm="giac")

[Out]

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