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