Optimal. Leaf size=76 \[ -\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{7/2}}-\frac{15}{8 a^3 x}+\frac{5}{8 a^2 x \left (a+b x^2\right )}+\frac{1}{4 a x \left (a+b x^2\right )^2} \]
[Out]
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Rubi [A] time = 0.0768378, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{7/2}}-\frac{15}{8 a^3 x}+\frac{5}{8 a^2 x \left (a+b x^2\right )}+\frac{1}{4 a x \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*(a + b*x^2)^3),x]
[Out]
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Rubi in Sympy [A] time = 14.17, size = 65, normalized size = 0.86 \[ \frac{1}{4 a x \left (a + b x^{2}\right )^{2}} + \frac{5}{8 a^{2} x \left (a + b x^{2}\right )} - \frac{15}{8 a^{3} x} - \frac{15 \sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 a^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(b*x**2+a)**3,x)
[Out]
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Mathematica [A] time = 0.0768561, size = 68, normalized size = 0.89 \[ -\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{7/2}}-\frac{8 a^2+25 a b x^2+15 b^2 x^4}{8 a^3 x \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*(a + b*x^2)^3),x]
[Out]
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Maple [A] time = 0.016, size = 66, normalized size = 0.9 \[ -{\frac{1}{{a}^{3}x}}-{\frac{7\,{b}^{2}{x}^{3}}{8\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{9\,bx}{8\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{15\,b}{8\,{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(b*x^2+a)^3,x)
[Out]
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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]
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Fricas [A] time = 0.226484, 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]
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Sympy [A] time = 2.41412, size = 114, normalized size = 1.5 \[ \frac{15 \sqrt{- \frac{b}{a^{7}}} \log{\left (- \frac{a^{4} \sqrt{- \frac{b}{a^{7}}}}{b} + x \right )}}{16} - \frac{15 \sqrt{- \frac{b}{a^{7}}} \log{\left (\frac{a^{4} \sqrt{- \frac{b}{a^{7}}}}{b} + x \right )}}{16} - \frac{8 a^{2} + 25 a b x^{2} + 15 b^{2} x^{4}}{8 a^{5} x + 16 a^{4} b x^{3} + 8 a^{3} b^{2} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(b*x**2+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.206678, size = 77, normalized size = 1.01 \[ -\frac{15 \, b \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{3}} - \frac{7 \, b^{2} x^{3} + 9 \, a b x}{8 \,{\left (b x^{2} + a\right )}^{2} a^{3}} - \frac{1}{a^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^3*x^2),x, algorithm="giac")
[Out]