3.249 \(\int \frac{1}{\left (a-b x^2\right )^5} \, dx\)

Optimal. Leaf size=100 \[ \frac{35 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128 a^{9/2} \sqrt{b}}+\frac{35 x}{128 a^4 \left (a-b x^2\right )}+\frac{35 x}{192 a^3 \left (a-b x^2\right )^2}+\frac{7 x}{48 a^2 \left (a-b x^2\right )^3}+\frac{x}{8 a \left (a-b x^2\right )^4} \]

[Out]

x/(8*a*(a - b*x^2)^4) + (7*x)/(48*a^2*(a - b*x^2)^3) + (35*x)/(192*a^3*(a - b*x^
2)^2) + (35*x)/(128*a^4*(a - b*x^2)) + (35*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/(128*a^
(9/2)*Sqrt[b])

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Rubi [A]  time = 0.0899517, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{35 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128 a^{9/2} \sqrt{b}}+\frac{35 x}{128 a^4 \left (a-b x^2\right )}+\frac{35 x}{192 a^3 \left (a-b x^2\right )^2}+\frac{7 x}{48 a^2 \left (a-b x^2\right )^3}+\frac{x}{8 a \left (a-b x^2\right )^4} \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x^2)^(-5),x]

[Out]

x/(8*a*(a - b*x^2)^4) + (7*x)/(48*a^2*(a - b*x^2)^3) + (35*x)/(192*a^3*(a - b*x^
2)^2) + (35*x)/(128*a^4*(a - b*x^2)) + (35*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/(128*a^
(9/2)*Sqrt[b])

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Rubi in Sympy [A]  time = 11.4927, size = 88, normalized size = 0.88 \[ \frac{x}{8 a \left (a - b x^{2}\right )^{4}} + \frac{7 x}{48 a^{2} \left (a - b x^{2}\right )^{3}} + \frac{35 x}{192 a^{3} \left (a - b x^{2}\right )^{2}} + \frac{35 x}{128 a^{4} \left (a - b x^{2}\right )} + \frac{35 \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{128 a^{\frac{9}{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-b*x**2+a)**5,x)

[Out]

x/(8*a*(a - b*x**2)**4) + 7*x/(48*a**2*(a - b*x**2)**3) + 35*x/(192*a**3*(a - b*
x**2)**2) + 35*x/(128*a**4*(a - b*x**2)) + 35*atanh(sqrt(b)*x/sqrt(a))/(128*a**(
9/2)*sqrt(b))

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Mathematica [A]  time = 0.0850246, size = 79, normalized size = 0.79 \[ \frac{\frac{\sqrt{a} x \left (279 a^3-511 a^2 b x^2+385 a b^2 x^4-105 b^3 x^6\right )}{\left (a-b x^2\right )^4}+\frac{105 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}}{384 a^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - b*x^2)^(-5),x]

[Out]

((Sqrt[a]*x*(279*a^3 - 511*a^2*b*x^2 + 385*a*b^2*x^4 - 105*b^3*x^6))/(a - b*x^2)
^4 + (105*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b])/(384*a^(9/2))

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Maple [A]  time = 0.005, size = 107, normalized size = 1.1 \[{\frac{x}{8\,a \left ( b{x}^{2}-a \right ) ^{4}}}+{\frac{7}{8\,a} \left ( -{\frac{x}{6\,a \left ( b{x}^{2}-a \right ) ^{3}}}-{\frac{5}{6\,a} \left ( -{\frac{x}{4\,a \left ( b{x}^{2}-a \right ) ^{2}}}-{\frac{3}{4\,a} \left ( -{\frac{x}{2\,a \left ( b{x}^{2}-a \right ) }}+{\frac{1}{2\,a}{\it Artanh} \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \right ) } \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-b*x^2+a)^5,x)

[Out]

1/8*x/a/(b*x^2-a)^4+7/8/a*(-1/6*x/a/(b*x^2-a)^3-5/6/a*(-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(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)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22514, size = 1, normalized size = 0.01 \[ \left [\frac{105 \,{\left (b^{4} x^{8} - 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} - 4 \, a^{3} b x^{2} + a^{4}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} + a\right )} \sqrt{a b}}{b x^{2} - a}\right ) - 2 \,{\left (105 \, b^{3} x^{7} - 385 \, a b^{2} x^{5} + 511 \, a^{2} b x^{3} - 279 \, a^{3} x\right )} \sqrt{a b}}{768 \,{\left (a^{4} b^{4} x^{8} - 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} - 4 \, a^{7} b x^{2} + a^{8}\right )} \sqrt{a b}}, \frac{105 \,{\left (b^{4} x^{8} - 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} - 4 \, a^{3} b x^{2} + a^{4}\right )} \arctan \left (\frac{\sqrt{-a b} x}{a}\right ) -{\left (105 \, b^{3} x^{7} - 385 \, a b^{2} x^{5} + 511 \, a^{2} b x^{3} - 279 \, a^{3} x\right )} \sqrt{-a b}}{384 \,{\left (a^{4} b^{4} x^{8} - 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} - 4 \, a^{7} b x^{2} + a^{8}\right )} \sqrt{-a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(b*x^2 - a)^5,x, algorithm="fricas")

[Out]

[1/768*(105*(b^4*x^8 - 4*a*b^3*x^6 + 6*a^2*b^2*x^4 - 4*a^3*b*x^2 + a^4)*log((2*a
*b*x + (b*x^2 + a)*sqrt(a*b))/(b*x^2 - a)) - 2*(105*b^3*x^7 - 385*a*b^2*x^5 + 51
1*a^2*b*x^3 - 279*a^3*x)*sqrt(a*b))/((a^4*b^4*x^8 - 4*a^5*b^3*x^6 + 6*a^6*b^2*x^
4 - 4*a^7*b*x^2 + a^8)*sqrt(a*b)), 1/384*(105*(b^4*x^8 - 4*a*b^3*x^6 + 6*a^2*b^2
*x^4 - 4*a^3*b*x^2 + a^4)*arctan(sqrt(-a*b)*x/a) - (105*b^3*x^7 - 385*a*b^2*x^5
+ 511*a^2*b*x^3 - 279*a^3*x)*sqrt(-a*b))/((a^4*b^4*x^8 - 4*a^5*b^3*x^6 + 6*a^6*b
^2*x^4 - 4*a^7*b*x^2 + a^8)*sqrt(-a*b))]

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Sympy [A]  time = 3.13283, size = 146, normalized size = 1.46 \[ - \frac{35 \sqrt{\frac{1}{a^{9} b}} \log{\left (- a^{5} \sqrt{\frac{1}{a^{9} b}} + x \right )}}{256} + \frac{35 \sqrt{\frac{1}{a^{9} b}} \log{\left (a^{5} \sqrt{\frac{1}{a^{9} b}} + x \right )}}{256} - \frac{- 279 a^{3} x + 511 a^{2} b x^{3} - 385 a b^{2} x^{5} + 105 b^{3} x^{7}}{384 a^{8} - 1536 a^{7} b x^{2} + 2304 a^{6} b^{2} x^{4} - 1536 a^{5} b^{3} x^{6} + 384 a^{4} b^{4} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-b*x**2+a)**5,x)

[Out]

-35*sqrt(1/(a**9*b))*log(-a**5*sqrt(1/(a**9*b)) + x)/256 + 35*sqrt(1/(a**9*b))*l
og(a**5*sqrt(1/(a**9*b)) + x)/256 - (-279*a**3*x + 511*a**2*b*x**3 - 385*a*b**2*
x**5 + 105*b**3*x**7)/(384*a**8 - 1536*a**7*b*x**2 + 2304*a**6*b**2*x**4 - 1536*
a**5*b**3*x**6 + 384*a**4*b**4*x**8)

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GIAC/XCAS [A]  time = 0.21048, size = 96, normalized size = 0.96 \[ -\frac{35 \, \arctan \left (\frac{b x}{\sqrt{-a b}}\right )}{128 \, \sqrt{-a b} a^{4}} - \frac{105 \, b^{3} x^{7} - 385 \, a b^{2} x^{5} + 511 \, a^{2} b x^{3} - 279 \, a^{3} x}{384 \,{\left (b x^{2} - a\right )}^{4} a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(b*x^2 - a)^5,x, algorithm="giac")

[Out]

-35/128*arctan(b*x/sqrt(-a*b))/(sqrt(-a*b)*a^4) - 1/384*(105*b^3*x^7 - 385*a*b^2
*x^5 + 511*a^2*b*x^3 - 279*a^3*x)/((b*x^2 - a)^4*a^4)