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

Optimal. Leaf size=78 \[ \frac{15 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{7/2}}-\frac{15}{8 a^3 x}+\frac{5}{8 a^2 x \left (a-b x^2\right )}+\frac{1}{4 a x \left (a-b x^2\right )^2} \]

[Out]

-15/(8*a^3*x) + 1/(4*a*x*(a - b*x^2)^2) + 5/(8*a^2*x*(a - b*x^2)) + (15*Sqrt[b]*
ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(7/2))

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Rubi [A]  time = 0.0838416, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{15 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{7/2}}-\frac{15}{8 a^3 x}+\frac{5}{8 a^2 x \left (a-b x^2\right )}+\frac{1}{4 a x \left (a-b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a - b*x^2)^3),x]

[Out]

-15/(8*a^3*x) + 1/(4*a*x*(a - b*x^2)^2) + 5/(8*a^2*x*(a - b*x^2)) + (15*Sqrt[b]*
ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(7/2))

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Rubi in Sympy [A]  time = 15.3825, size = 65, normalized size = 0.83 \[ \frac{1}{4 a x \left (a - b x^{2}\right )^{2}} + \frac{5}{8 a^{2} x \left (a - b x^{2}\right )} - \frac{15}{8 a^{3} x} + \frac{15 \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(4*a*x*(a - b*x**2)**2) + 5/(8*a**2*x*(a - b*x**2)) - 15/(8*a**3*x) + 15*sqrt(
b)*atanh(sqrt(b)*x/sqrt(a))/(8*a**(7/2))

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Mathematica [A]  time = 0.084513, size = 69, normalized size = 0.88 \[ \frac{15 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{7/2}}+\frac{-8 a^2+25 a b x^2-15 b^2 x^4}{8 a^3 x \left (a-b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(a - b*x^2)^3),x]

[Out]

(-8*a^2 + 25*a*b*x^2 - 15*b^2*x^4)/(8*a^3*x*(a - b*x^2)^2) + (15*Sqrt[b]*ArcTanh
[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(7/2))

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Maple [A]  time = 0.015, size = 56, normalized size = 0.7 \[ -{\frac{1}{{a}^{3}x}}-{\frac{b}{{a}^{3}} \left ({\frac{1}{ \left ( b{x}^{2}-a \right ) ^{2}} \left ({\frac{7\,b{x}^{3}}{8}}-{\frac{9\,ax}{8}} \right ) }-{\frac{15}{8}{\it Artanh} \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(-b*x^2+a)^3,x)

[Out]

-1/a^3/x-1/a^3*b*((7/8*b*x^3-9/8*a*x)/(b*x^2-a)^2-15/8/(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(-1/((b*x^2 - a)^3*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229946, size = 1, normalized size = 0.01 \[ \left [-\frac{30 \, b^{2} x^{4} - 50 \, a b x^{2} - 15 \,{\left (b^{2} x^{5} - 2 \, a b x^{3} + a^{2} x\right )} \sqrt{\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{\frac{b}{a}} + a}{b x^{2} - a}\right ) + 16 \, a^{2}}{16 \,{\left (a^{3} b^{2} x^{5} - 2 \, a^{4} b x^{3} + a^{5} x\right )}}, -\frac{15 \, b^{2} x^{4} - 25 \, a b x^{2} - 15 \,{\left (b^{2} x^{5} - 2 \, a b x^{3} + a^{2} x\right )} \sqrt{-\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{-\frac{b}{a}}}\right ) + 8 \, a^{2}}{8 \,{\left (a^{3} b^{2} x^{5} - 2 \, a^{4} b x^{3} + a^{5} x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/16*(30*b^2*x^4 - 50*a*b*x^2 - 15*(b^2*x^5 - 2*a*b*x^3 + a^2*x)*sqrt(b/a)*log
((b*x^2 + 2*a*x*sqrt(b/a) + a)/(b*x^2 - a)) + 16*a^2)/(a^3*b^2*x^5 - 2*a^4*b*x^3
+ a^5*x), -1/8*(15*b^2*x^4 - 25*a*b*x^2 - 15*(b^2*x^5 - 2*a*b*x^3 + a^2*x)*sqrt
(-b/a)*arctan(b*x/(a*sqrt(-b/a))) + 8*a^2)/(a^3*b^2*x^5 - 2*a^4*b*x^3 + a^5*x)]

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Sympy [A]  time = 2.52647, size = 107, normalized size = 1.37 \[ - \frac{15 \sqrt{\frac{b}{a^{7}}} \log{\left (- \frac{a^{4} \sqrt{\frac{b}{a^{7}}}}{b} + x \right )}}{16} + \frac{15 \sqrt{\frac{b}{a^{7}}} \log{\left (\frac{a^{4} \sqrt{\frac{b}{a^{7}}}}{b} + x \right )}}{16} - \frac{8 a^{2} - 25 a b x^{2} + 15 b^{2} x^{4}}{8 a^{5} x - 16 a^{4} b x^{3} + 8 a^{3} b^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15*sqrt(b/a**7)*log(-a**4*sqrt(b/a**7)/b + x)/16 + 15*sqrt(b/a**7)*log(a**4*sqr
t(b/a**7)/b + x)/16 - (8*a**2 - 25*a*b*x**2 + 15*b**2*x**4)/(8*a**5*x - 16*a**4*
b*x**3 + 8*a**3*b**2*x**5)

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GIAC/XCAS [A]  time = 0.21321, size = 82, normalized size = 1.05 \[ -\frac{15 \, b \arctan \left (\frac{b x}{\sqrt{-a b}}\right )}{8 \, \sqrt{-a b} a^{3}} - \frac{7 \, b^{2} x^{3} - 9 \, a b x}{8 \,{\left (b x^{2} - a\right )}^{2} a^{3}} - \frac{1}{a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15/8*b*arctan(b*x/sqrt(-a*b))/(sqrt(-a*b)*a^3) - 1/8*(7*b^2*x^3 - 9*a*b*x)/((b*
x^2 - a)^2*a^3) - 1/(a^3*x)