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

Optimal. Leaf size=46 \[ \frac{x}{2 b \left (a-b x^2\right )}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0135534, antiderivative size = 46, 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 = {288, 208} \[ \frac{x}{2 b \left (a-b x^2\right )}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a - b*x^2)^2,x]

[Out]

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

Rule 288

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*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 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}{\left (a-b x^2\right )^2} \, dx &=\frac{x}{2 b \left (a-b x^2\right )}-\frac{\int \frac{1}{a-b x^2} \, dx}{2 b}\\ &=\frac{x}{2 b \left (a-b x^2\right )}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.029005, size = 47, normalized size = 1.02 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}}-\frac{x}{2 b \left (b x^2-a\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a - b*x^2)^2,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 38, normalized size = 0.8 \begin{align*} -{\frac{x}{2\,b \left ( b{x}^{2}-a \right ) }}-{\frac{1}{2\,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)^2,x)

[Out]

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

________________________________________________________________________________________

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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.30328, size = 263, normalized size = 5.72 \begin{align*} \left [-\frac{2 \, a b x -{\left (b x^{2} - a\right )} \sqrt{a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{a b} x + a}{b x^{2} - a}\right )}{4 \,{\left (a b^{3} x^{2} - a^{2} b^{2}\right )}}, -\frac{a b x -{\left (b x^{2} - a\right )} \sqrt{-a b} \arctan \left (\frac{\sqrt{-a b} x}{a}\right )}{2 \,{\left (a b^{3} x^{2} - a^{2} b^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(2*a*b*x - (b*x^2 - a)*sqrt(a*b)*log((b*x^2 - 2*sqrt(a*b)*x + a)/(b*x^2 - a)))/(a*b^3*x^2 - a^2*b^2), -1
/2*(a*b*x - (b*x^2 - a)*sqrt(-a*b)*arctan(sqrt(-a*b)*x/a))/(a*b^3*x^2 - a^2*b^2)]

________________________________________________________________________________________

Sympy [A]  time = 0.352352, size = 71, normalized size = 1.54 \begin{align*} - \frac{x}{- 2 a b + 2 b^{2} x^{2}} + \frac{\sqrt{\frac{1}{a b^{3}}} \log{\left (- a b \sqrt{\frac{1}{a b^{3}}} + x \right )}}{4} - \frac{\sqrt{\frac{1}{a b^{3}}} \log{\left (a b \sqrt{\frac{1}{a b^{3}}} + x \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 2.56562, size = 53, normalized size = 1.15 \begin{align*} \frac{\arctan \left (\frac{b x}{\sqrt{-a b}}\right )}{2 \, \sqrt{-a b} b} - \frac{x}{2 \,{\left (b x^{2} - a\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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