3.226 \(\int \frac{x^2}{a-b x^2} \, dx\)

Optimal. Leaf size=31 \[ \frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2}}-\frac{x}{b} \]

[Out]

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

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Rubi [A]  time = 0.012385, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {321, 208} \[ \frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2}}-\frac{x}{b} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a - b*x^2),x]

[Out]

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

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{x^2}{a-b x^2} \, dx &=-\frac{x}{b}+\frac{a \int \frac{1}{a-b x^2} \, dx}{b}\\ &=-\frac{x}{b}+\frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0089459, size = 31, normalized size = 1. \[ \frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2}}-\frac{x}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a - b*x^2),x]

[Out]

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

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Maple [A]  time = 0.003, size = 27, normalized size = 0.9 \begin{align*} -{\frac{x}{b}}+{\frac{a}{b}{\it Artanh} \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/b+1/b*a/(a*b)^(1/2)*arctanh(b*x/(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^2/(-b*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.33562, size = 165, normalized size = 5.32 \begin{align*} \left [\frac{\sqrt{\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{\frac{a}{b}} + a}{b x^{2} - a}\right ) - 2 \, x}{2 \, b}, -\frac{\sqrt{-\frac{a}{b}} \arctan \left (\frac{b x \sqrt{-\frac{a}{b}}}{a}\right ) + x}{b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.294116, size = 49, normalized size = 1.58 \begin{align*} - \frac{\sqrt{\frac{a}{b^{3}}} \log{\left (- b \sqrt{\frac{a}{b^{3}}} + x \right )}}{2} + \frac{\sqrt{\frac{a}{b^{3}}} \log{\left (b \sqrt{\frac{a}{b^{3}}} + x \right )}}{2} - \frac{x}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.7587, size = 39, normalized size = 1.26 \begin{align*} -\frac{a \arctan \left (\frac{b x}{\sqrt{-a b}}\right )}{\sqrt{-a b} b} - \frac{x}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(-b*x^2+a),x, algorithm="giac")

[Out]

-a*arctan(b*x/sqrt(-a*b))/(sqrt(-a*b)*b) - x/b