Optimal. Leaf size=68 \[ \frac{5 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2}}+\frac{5 b}{2 a^3 x}-\frac{5}{6 a^2 x^3}+\frac{1}{2 a x^3 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.0703671, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{5 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2}}+\frac{5 b}{2 a^3 x}-\frac{5}{6 a^2 x^3}+\frac{1}{2 a x^3 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*(a*x + b*x^3)^2),x]
[Out]
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Rubi in Sympy [A] time = 14.5361, size = 61, normalized size = 0.9 \[ \frac{1}{2 a x^{3} \left (a + b x^{2}\right )} - \frac{5}{6 a^{2} x^{3}} + \frac{5 b}{2 a^{3} x} + \frac{5 b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 a^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(b*x**3+a*x)**2,x)
[Out]
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Mathematica [A] time = 0.0701812, size = 67, normalized size = 0.99 \[ \frac{5 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2}}+\frac{b^2 x}{2 a^3 \left (a+b x^2\right )}+\frac{2 b}{a^3 x}-\frac{1}{3 a^2 x^3} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*(a*x + b*x^3)^2),x]
[Out]
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Maple [A] time = 0.016, size = 59, normalized size = 0.9 \[ -{\frac{1}{3\,{a}^{2}{x}^{3}}}+2\,{\frac{b}{{a}^{3}x}}+{\frac{{b}^{2}x}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{5\,{b}^{2}}{2\,{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(b*x^3+a*x)^2,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^3 + a*x)^2*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.215621, size = 1, normalized size = 0.01 \[ \left [\frac{30 \, b^{2} x^{4} + 20 \, a b x^{2} + 15 \,{\left (b^{2} x^{5} + a b x^{3}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) - 4 \, a^{2}}{12 \,{\left (a^{3} b x^{5} + a^{4} x^{3}\right )}}, \frac{15 \, b^{2} x^{4} + 10 \, a b x^{2} + 15 \,{\left (b^{2} x^{5} + a b x^{3}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right ) - 2 \, a^{2}}{6 \,{\left (a^{3} b x^{5} + a^{4} x^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^3 + a*x)^2*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.12167, size = 114, normalized size = 1.68 \[ - \frac{5 \sqrt{- \frac{b^{3}}{a^{7}}} \log{\left (- \frac{a^{4} \sqrt{- \frac{b^{3}}{a^{7}}}}{b^{2}} + x \right )}}{4} + \frac{5 \sqrt{- \frac{b^{3}}{a^{7}}} \log{\left (\frac{a^{4} \sqrt{- \frac{b^{3}}{a^{7}}}}{b^{2}} + x \right )}}{4} + \frac{- 2 a^{2} + 10 a b x^{2} + 15 b^{2} x^{4}}{6 a^{4} x^{3} + 6 a^{3} b x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(b*x**3+a*x)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.219085, size = 80, normalized size = 1.18 \[ \frac{5 \, b^{2} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{3}} + \frac{b^{2} x}{2 \,{\left (b x^{2} + a\right )} a^{3}} + \frac{6 \, b x^{2} - a}{3 \, a^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^3 + a*x)^2*x^2),x, algorithm="giac")
[Out]