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

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

[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 = 0.0458219, 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 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b}}+\frac{3 x}{8 a^2 \left (a+b x^2\right )}+\frac{x}{4 a \left (a+b x^2\right )^2} \]

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])

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Rubi in Sympy [A]  time = 5.70246, size = 54, normalized size = 0.87 \[ \frac{x}{4 a \left (a + b x^{2}\right )^{2}} + \frac{3 x}{8 a^{2} \left (a + b x^{2}\right )} + \frac{3 \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**2+a)**3,x)

[Out]

x/(4*a*(a + b*x**2)**2) + 3*x/(8*a**2*(a + b*x**2)) + 3*atan(sqrt(b)*x/sqrt(a))/
(8*a**(5/2)*sqrt(b))

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Mathematica [A]  time = 0.077456, size = 55, normalized size = 0.89 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b}}+\frac{5 a x+3 b x^3}{8 a^2 \left (a+b x^2\right )^2} \]

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 = 0.006, size = 51, normalized size = 0.8 \[{\frac{x}{4\,a \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{3\,x}{8\,{a}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{3}{8\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(-3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.215088, size = 1, normalized size = 0.02 \[ \left [\frac{3 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (3 \, b x^{3} + 5 \, a x\right )} \sqrt{-a b}}{16 \,{\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )} \sqrt{-a b}}, \frac{3 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (3 \, b x^{3} + 5 \, a x\right )} \sqrt{a b}}{8 \,{\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(-3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.84603, size = 105, normalized size = 1.69 \[ - \frac{3 \sqrt{- \frac{1}{a^{5} b}} \log{\left (- a^{3} \sqrt{- \frac{1}{a^{5} b}} + x \right )}}{16} + \frac{3 \sqrt{- \frac{1}{a^{5} b}} \log{\left (a^{3} \sqrt{- \frac{1}{a^{5} b}} + x \right )}}{16} + \frac{5 a x + 3 b x^{3}}{8 a^{4} + 16 a^{3} b x^{2} + 8 a^{2} b^{2} x^{4}} \]

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))*l
og(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/XCAS [A]  time = 0.21021, size = 61, normalized size = 0.98 \[ \frac{3 \, \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{2}} + \frac{3 \, b x^{3} + 5 \, a x}{8 \,{\left (b x^{2} + a\right )}^{2} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((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)