Optimal. Leaf size=50 \[ \frac{2 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{b} x^{-n/2}}{\sqrt{c}}\right )}{b^{3/2} n}-\frac{2 x^{-n/2}}{b n} \]
[Out]
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Rubi [A] time = 0.0684434, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{b} x^{-n/2}}{\sqrt{c}}\right )}{b^{3/2} n}-\frac{2 x^{-n/2}}{b n} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 + n/2)/(b*x^n + c*x^(2*n)),x]
[Out]
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Rubi in Sympy [A] time = 13.4537, size = 39, normalized size = 0.78 \[ - \frac{2 x^{- \frac{n}{2}}}{b n} + \frac{2 \sqrt{c} \operatorname{atan}{\left (\frac{\sqrt{b} x^{- \frac{n}{2}}}{\sqrt{c}} \right )}}{b^{\frac{3}{2}} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1+1/2*n)/(b*x**n+c*x**(2*n)),x)
[Out]
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Mathematica [A] time = 0.0406458, size = 50, normalized size = 1. \[ \frac{2 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{b} x^{-n/2}}{\sqrt{c}}\right )}{b^{3/2} n}-\frac{2 x^{-n/2}}{b n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 + n/2)/(b*x^n + c*x^(2*n)),x]
[Out]
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Maple [A] time = 0.093, size = 79, normalized size = 1.6 \[ -2\,{\frac{1}{bn{x}^{n/2}}}+{\frac{1}{{b}^{2}n}\sqrt{-bc}\ln \left ({x}^{{\frac{n}{2}}}-{\frac{1}{c}\sqrt{-bc}} \right ) }-{\frac{1}{{b}^{2}n}\sqrt{-bc}\ln \left ({x}^{{\frac{n}{2}}}+{\frac{1}{c}\sqrt{-bc}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1+1/2*n)/(b*x^n+c*x^(2*n)),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(x^(1/2*n - 1)/(c*x^(2*n) + b*x^n),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.298401, size = 1, normalized size = 0.02 \[ \left [\frac{x x^{\frac{1}{2} \, n - 1} \sqrt{-\frac{c}{b}} \log \left (\frac{c x^{2} x^{n - 2} - 2 \, b x x^{\frac{1}{2} \, n - 1} \sqrt{-\frac{c}{b}} - b}{c x^{2} x^{n - 2} + b}\right ) - 2}{b n x x^{\frac{1}{2} \, n - 1}}, \frac{2 \,{\left (x x^{\frac{1}{2} \, n - 1} \sqrt{\frac{c}{b}} \arctan \left (\frac{b \sqrt{\frac{c}{b}}}{c x x^{\frac{1}{2} \, n - 1}}\right ) - 1\right )}}{b n x x^{\frac{1}{2} \, n - 1}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(1/2*n - 1)/(c*x^(2*n) + b*x^n),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(-1+1/2*n)/(b*x**n+c*x**(2*n)),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{1}{2} \, n - 1}}{c x^{2 \, n} + b x^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(1/2*n - 1)/(c*x^(2*n) + b*x^n),x, algorithm="giac")
[Out]