3.242 \(\int \frac{1}{(a-b x^2)^3} \, dx\)

Optimal. Leaf size=64 \[ \frac{3 x}{8 a^2 \left (a-b x^2\right )}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b}}+\frac{x}{4 a \left (a-b x^2\right )^2} \]

[Out]

x/(4*a*(a - b*x^2)^2) + (3*x)/(8*a^2*(a - b*x^2)) + (3*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*Sqrt[b])

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Rubi [A]  time = 0.0179858, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {199, 208} \[ \frac{3 x}{8 a^2 \left (a-b x^2\right )}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b}}+\frac{x}{4 a \left (a-b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*x^2)^(-3),x]

[Out]

x/(4*a*(a - b*x^2)^2) + (3*x)/(8*a^2*(a - b*x^2)) + (3*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*Sqrt[b])

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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{1}{\left (a-b x^2\right )^3} \, dx &=\frac{x}{4 a \left (a-b x^2\right )^2}+\frac{3 \int \frac{1}{\left (a-b x^2\right )^2} \, dx}{4 a}\\ &=\frac{x}{4 a \left (a-b x^2\right )^2}+\frac{3 x}{8 a^2 \left (a-b x^2\right )}+\frac{3 \int \frac{1}{a-b x^2} \, dx}{8 a^2}\\ &=\frac{x}{4 a \left (a-b x^2\right )^2}+\frac{3 x}{8 a^2 \left (a-b x^2\right )}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.0383059, size = 56, normalized size = 0.88 \[ \frac{5 a x-3 b x^3}{8 a^2 \left (a-b x^2\right )^2}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - b*x^2)^(-3),x]

[Out]

(5*a*x - 3*b*x^3)/(8*a^2*(a - b*x^2)^2) + (3*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*Sqrt[b])

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Maple [A]  time = 0.004, size = 61, normalized size = 1. \begin{align*}{\frac{x}{4\,a \left ( b{x}^{2}-a \right ) ^{2}}}+{\frac{3}{4\,a} \left ( -{\frac{x}{2\,a \left ( b{x}^{2}-a \right ) }}+{\frac{1}{2\,a}{\it Artanh} \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-b*x^2+a)^3,x)

[Out]

1/4*x/a/(b*x^2-a)^2+3/4/a*(-1/2*x/a/(b*x^2-a)+1/2/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(1/(-b*x^2+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.28842, size = 404, normalized size = 6.31 \begin{align*} \left [-\frac{6 \, a b^{2} x^{3} - 10 \, a^{2} b x - 3 \,{\left (b^{2} x^{4} - 2 \, a b x^{2} + a^{2}\right )} \sqrt{a b} \log \left (\frac{b x^{2} + 2 \, \sqrt{a b} x + a}{b x^{2} - a}\right )}{16 \,{\left (a^{3} b^{3} x^{4} - 2 \, a^{4} b^{2} x^{2} + a^{5} b\right )}}, -\frac{3 \, a b^{2} x^{3} - 5 \, a^{2} b x + 3 \,{\left (b^{2} x^{4} - 2 \, a b x^{2} + a^{2}\right )} \sqrt{-a b} \arctan \left (\frac{\sqrt{-a b} x}{a}\right )}{8 \,{\left (a^{3} b^{3} x^{4} - 2 \, a^{4} b^{2} x^{2} + a^{5} b\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x^2+a)^3,x, algorithm="fricas")

[Out]

[-1/16*(6*a*b^2*x^3 - 10*a^2*b*x - 3*(b^2*x^4 - 2*a*b*x^2 + a^2)*sqrt(a*b)*log((b*x^2 + 2*sqrt(a*b)*x + a)/(b*
x^2 - a)))/(a^3*b^3*x^4 - 2*a^4*b^2*x^2 + a^5*b), -1/8*(3*a*b^2*x^3 - 5*a^2*b*x + 3*(b^2*x^4 - 2*a*b*x^2 + a^2
)*sqrt(-a*b)*arctan(sqrt(-a*b)*x/a))/(a^3*b^3*x^4 - 2*a^4*b^2*x^2 + a^5*b)]

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Sympy [A]  time = 0.489597, size = 99, normalized size = 1.55 \begin{align*} - \frac{3 \sqrt{\frac{1}{a^{5} b}} \log{\left (- a^{3} \sqrt{\frac{1}{a^{5} b}} + x \right )}}{16} + \frac{3 \sqrt{\frac{1}{a^{5} b}} \log{\left (a^{3} \sqrt{\frac{1}{a^{5} b}} + x \right )}}{16} - \frac{- 5 a x + 3 b x^{3}}{8 a^{4} - 16 a^{3} b x^{2} + 8 a^{2} b^{2} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x**2+a)**3,x)

[Out]

-3*sqrt(1/(a**5*b))*log(-a**3*sqrt(1/(a**5*b)) + x)/16 + 3*sqrt(1/(a**5*b))*log(a**3*sqrt(1/(a**5*b)) + x)/16
- (-5*a*x + 3*b*x**3)/(8*a**4 - 16*a**3*b*x**2 + 8*a**2*b**2*x**4)

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Giac [A]  time = 1.61313, size = 66, normalized size = 1.03 \begin{align*} -\frac{3 \, \arctan \left (\frac{b x}{\sqrt{-a b}}\right )}{8 \, \sqrt{-a b} a^{2}} - \frac{3 \, b x^{3} - 5 \, a x}{8 \,{\left (b x^{2} - a\right )}^{2} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x^2+a)^3,x, algorithm="giac")

[Out]

-3/8*arctan(b*x/sqrt(-a*b))/(sqrt(-a*b)*a^2) - 1/8*(3*b*x^3 - 5*a*x)/((b*x^2 - a)^2*a^2)