Optimal. Leaf size=43 \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2}}+\frac{b}{a^2 x}-\frac{1}{3 a x^3} \]
[Out]
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Rubi [A] time = 0.0517105, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2}}+\frac{b}{a^2 x}-\frac{1}{3 a x^3} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*(a*x + b*x^3)),x]
[Out]
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Rubi in Sympy [A] time = 10.7657, size = 37, normalized size = 0.86 \[ - \frac{1}{3 a x^{3}} + \frac{b}{a^{2} x} + \frac{b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(b*x**3+a*x),x)
[Out]
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Mathematica [A] time = 0.0337876, size = 43, normalized size = 1. \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2}}+\frac{b}{a^2 x}-\frac{1}{3 a x^3} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^3*(a*x + b*x^3)),x]
[Out]
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Maple [A] time = 0.007, size = 39, normalized size = 0.9 \[ -{\frac{1}{3\,a{x}^{3}}}+{\frac{b}{{a}^{2}x}}+{\frac{{b}^{2}}{{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(b*x^3+a*x),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)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.21424, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, b x^{3} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) + 6 \, b x^{2} - 2 \, a}{6 \, a^{2} x^{3}}, \frac{3 \, b x^{3} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right ) + 3 \, b x^{2} - a}{3 \, a^{2} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^3 + a*x)*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.52177, size = 87, normalized size = 2.02 \[ - \frac{\sqrt{- \frac{b^{3}}{a^{5}}} \log{\left (- \frac{a^{3} \sqrt{- \frac{b^{3}}{a^{5}}}}{b^{2}} + x \right )}}{2} + \frac{\sqrt{- \frac{b^{3}}{a^{5}}} \log{\left (\frac{a^{3} \sqrt{- \frac{b^{3}}{a^{5}}}}{b^{2}} + x \right )}}{2} + \frac{- a + 3 b x^{2}}{3 a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(b*x**3+a*x),x)
[Out]
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GIAC/XCAS [A] time = 0.217859, size = 54, normalized size = 1.26 \[ \frac{b^{2} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{2}} + \frac{3 \, b x^{2} - a}{3 \, a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^3 + a*x)*x^3),x, algorithm="giac")
[Out]