Optimal. Leaf size=68 \[ -\frac{2 c^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x^{-n/2}}{\sqrt{c}}\right )}{b^{5/2} n}+\frac{2 c x^{-n/2}}{b^2 n}-\frac{2 x^{-3 n/2}}{3 b n} \]
[Out]
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Rubi [A] time = 0.089816, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{2 c^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x^{-n/2}}{\sqrt{c}}\right )}{b^{5/2} n}+\frac{2 c x^{-n/2}}{b^2 n}-\frac{2 x^{-3 n/2}}{3 b n} \]
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 = 17.8674, size = 56, normalized size = 0.82 \[ - \frac{2 x^{- \frac{3 n}{2}}}{3 b n} + \frac{2 c x^{- \frac{n}{2}}}{b^{2} n} - \frac{2 c^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x^{- \frac{n}{2}}}{\sqrt{c}} \right )}}{b^{\frac{5}{2}} n} \]
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 = 0.0892205, size = 62, normalized size = 0.91 \[ \frac{2 \left (\sqrt{b} x^{-3 n/2} \left (3 c x^n-b\right )-3 c^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x^{-n/2}}{\sqrt{c}}\right )\right )}{3 b^{5/2} n} \]
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 = 0.122, size = 97, normalized size = 1.4 \[ 2\,{\frac{c}{n{b}^{2}{x}^{n/2}}}-{\frac{2}{3\,bn} \left ({x}^{{\frac{n}{2}}} \right ) ^{-3}}+{\frac{c}{{b}^{3}n}\sqrt{-bc}\ln \left ({x}^{{\frac{n}{2}}}+{\frac{1}{c}\sqrt{-bc}} \right ) }-{\frac{c}{{b}^{3}n}\sqrt{-bc}\ln \left ({x}^{{\frac{n}{2}}}-{\frac{1}{c}\sqrt{-bc}} \right ) } \]
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 = 0.300984, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, b x^{3} x^{-\frac{3}{2} \, n - 3} - 6 \, c x x^{-\frac{1}{2} \, n - 1} - 3 \, c \sqrt{-\frac{c}{b}} \log \left (\frac{b x^{2} x^{-n - 2} - 2 \, b x x^{-\frac{1}{2} \, n - 1} \sqrt{-\frac{c}{b}} - c}{b x^{2} x^{-n - 2} + c}\right )}{3 \, b^{2} n}, -\frac{2 \,{\left (b x^{3} x^{-\frac{3}{2} \, n - 3} - 3 \, c x x^{-\frac{1}{2} \, n - 1} - 3 \, c \sqrt{\frac{c}{b}} \arctan \left (\frac{\sqrt{\frac{c}{b}}}{x x^{-\frac{1}{2} \, n - 1}}\right )\right )}}{3 \, b^{2} n}\right ] \]
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]