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

Optimal. Leaf size=31 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.0255772, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1107, 618, 204} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b + 2*a*x^2 + a*x^4),x]

[Out]

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

Rule 1107

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{x}{a+b+2 a x^2+a x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{a+b+2 a x+a x^2} \, dx,x,x^2\right )\\ &=-\operatorname{Subst}\left (\int \frac{1}{-4 a b-x^2} \, dx,x,2 a \left (1+x^2\right )\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{a} \left (1+x^2\right )}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.0072065, size = 31, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.042, size = 26, normalized size = 0.8 \begin{align*}{\frac{1}{2}\arctan \left ({\frac{2\,a{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a*x^4+2*a*x^2+a+b),x)

[Out]

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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.48632, size = 217, normalized size = 7. \begin{align*} \left [-\frac{\sqrt{-a b} \log \left (\frac{a x^{4} + 2 \, a x^{2} - 2 \, \sqrt{-a b}{\left (x^{2} + 1\right )} + a - b}{a x^{4} + 2 \, a x^{2} + a + b}\right )}{4 \, a b}, -\frac{\sqrt{a b} \arctan \left (\frac{\sqrt{a b}}{a x^{2} + a}\right )}{2 \, a b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.278275, size = 60, normalized size = 1.94 \begin{align*} - \frac{\sqrt{- \frac{1}{a b}} \log{\left (- b \sqrt{- \frac{1}{a b}} + x^{2} + 1 \right )}}{4} + \frac{\sqrt{- \frac{1}{a b}} \log{\left (b \sqrt{- \frac{1}{a b}} + x^{2} + 1 \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a*x**4+2*a*x**2+a+b),x)

[Out]

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

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Giac [A]  time = 3.37943, size = 28, normalized size = 0.9 \begin{align*} \frac{\arctan \left (\frac{a x^{2} + a}{\sqrt{a b}}\right )}{2 \, \sqrt{a b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(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)