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

Optimal. Leaf size=24 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{c}} \]

[Out]

ArcTan[(Sqrt[c]*x)/Sqrt[b]]/(Sqrt[b]*Sqrt[c])

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Rubi [A]  time = 0.0117432, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1584, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{c}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(b*x^2 + c*x^4),x]

[Out]

ArcTan[(Sqrt[c]*x)/Sqrt[b]]/(Sqrt[b]*Sqrt[c])

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 205

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

Rubi steps

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

Mathematica [A]  time = 0.0042586, size = 24, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{c}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(b*x^2 + c*x^4),x]

[Out]

ArcTan[(Sqrt[c]*x)/Sqrt[b]]/(Sqrt[b]*Sqrt[c])

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Maple [A]  time = 0.044, size = 16, normalized size = 0.7 \begin{align*}{\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(b*c)^(1/2)*arctan(x*c/(b*c)^(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^2/(c*x^4+b*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.43387, size = 151, normalized size = 6.29 \begin{align*} \left [-\frac{\sqrt{-b c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-b c} x - b}{c x^{2} + b}\right )}{2 \, b c}, \frac{\sqrt{b c} \arctan \left (\frac{\sqrt{b c} x}{b}\right )}{b c}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.136575, size = 53, normalized size = 2.21 \begin{align*} - \frac{\sqrt{- \frac{1}{b c}} \log{\left (- b \sqrt{- \frac{1}{b c}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{b c}} \log{\left (b \sqrt{- \frac{1}{b c}} + x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.28577, size = 20, normalized size = 0.83 \begin{align*} \frac{\arctan \left (\frac{c x}{\sqrt{b c}}\right )}{\sqrt{b c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(c*x^4+b*x^2),x, algorithm="giac")

[Out]

arctan(c*x/sqrt(b*c))/sqrt(b*c)