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

Optimal. Leaf size=62 \[ \frac{3 x}{8 a^2 \left (a+b x^2\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b}}+\frac{x}{4 a \left (a+b x^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.0163593, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {199, 205} \[ \frac{3 x}{8 a^2 \left (a+b x^2\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b}}+\frac{x}{4 a \left (a+b x^2\right )^2} \]

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*ArcTan[(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 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{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 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.0345738, size = 55, normalized size = 0.89 \[ \frac{5 a x+3 b x^3}{8 a^2 \left (a+b x^2\right )^2}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b}} \]

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*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*Sqrt[b])

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Maple [A]  time = 0.003, size = 51, normalized size = 0.8 \begin{align*}{\frac{x}{4\,a \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{3\,x}{8\,{a}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{3}{8\,{a}^{2}}\arctan \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(1/(b*x^2+a)^3,x)

[Out]

1/4*x/a/(b*x^2+a)^2+3/8*x/a^2/(b*x^2+a)+3/8/a^2/(a*b)^(1/2)*arctan(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 = 1.24934, size = 401, normalized size = 6.47 \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 = 0.508068, size = 105, normalized size = 1.69 \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 = 2.86886, size = 61, normalized size = 0.98 \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)