3.240 \(\int \frac{x^2}{\left (a-b x^2\right )^3} \, dx\)

Optimal. Leaf size=67 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{3/2} b^{3/2}}-\frac{x}{8 a b \left (a-b x^2\right )}+\frac{x}{4 b \left (a-b x^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.0619327, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{3/2} b^{3/2}}-\frac{x}{8 a b \left (a-b x^2\right )}+\frac{x}{4 b \left (a-b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a - b*x^2)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.84361, size = 51, normalized size = 0.76 \[ \frac{x}{4 b \left (a - b x^{2}\right )^{2}} - \frac{x}{8 a b \left (a - b x^{2}\right )} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 a^{\frac{3}{2}} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(4*b*(a - b*x**2)**2) - x/(8*a*b*(a - b*x**2)) - atanh(sqrt(b)*x/sqrt(a))/(8*a
**(3/2)*b**(3/2))

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Mathematica [A]  time = 0.0563439, size = 56, normalized size = 0.84 \[ \frac{x \left (a+b x^2\right )}{8 a b \left (a-b x^2\right )^2}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{3/2} b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a - b*x^2)^3,x]

[Out]

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

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Maple [A]  time = 0.011, size = 52, normalized size = 0.8 \[ -{\frac{1}{ \left ( b{x}^{2}-a \right ) ^{2}} \left ( -{\frac{{x}^{3}}{8\,a}}-{\frac{x}{8\,b}} \right ) }-{\frac{1}{8\,ab}{\it Artanh} \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(-b*x^2+a)^3,x)

[Out]

-(-1/8/a*x^3-1/8*x/b)/(b*x^2-a)^2-1/8/b/a/(a*b)^(1/2)*arctanh(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(-x^2/(b*x^2 - a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231819, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (b^{2} x^{4} - 2 \, a b x^{2} + a^{2}\right )} \log \left (-\frac{2 \, a b x -{\left (b x^{2} + a\right )} \sqrt{a b}}{b x^{2} - a}\right ) + 2 \,{\left (b x^{3} + a x\right )} \sqrt{a b}}{16 \,{\left (a b^{3} x^{4} - 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \sqrt{a b}}, -\frac{{\left (b^{2} x^{4} - 2 \, a b x^{2} + a^{2}\right )} \arctan \left (\frac{\sqrt{-a b} x}{a}\right ) -{\left (b x^{3} + a x\right )} \sqrt{-a b}}{8 \,{\left (a b^{3} x^{4} - 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \sqrt{-a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.85464, size = 104, normalized size = 1.55 \[ \frac{\sqrt{\frac{1}{a^{3} b^{3}}} \log{\left (- a^{2} b \sqrt{\frac{1}{a^{3} b^{3}}} + x \right )}}{16} - \frac{\sqrt{\frac{1}{a^{3} b^{3}}} \log{\left (a^{2} b \sqrt{\frac{1}{a^{3} b^{3}}} + x \right )}}{16} + \frac{a x + b x^{3}}{8 a^{3} b - 16 a^{2} b^{2} x^{2} + 8 a b^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(-b*x**2+a)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.206537, size = 72, normalized size = 1.07 \[ \frac{\arctan \left (\frac{b x}{\sqrt{-a b}}\right )}{8 \, \sqrt{-a b} a b} + \frac{b x^{3} + a x}{8 \,{\left (b x^{2} - a\right )}^{2} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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