Optimal. Leaf size=68 \[ -\frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x^{-n/2}}{\sqrt{b}}\right )}{a^{5/2} n}+\frac{2 b x^{-n/2}}{a^2 n}-\frac{2 x^{-3 n/2}}{3 a n} \]
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Rubi [A] time = 0.0975609, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x^{-n/2}}{\sqrt{b}}\right )}{a^{5/2} n}+\frac{2 b x^{-n/2}}{a^2 n}-\frac{2 x^{-3 n/2}}{3 a n} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 - (3*n)/2)/(a + b*x^n),x]
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Rubi in Sympy [A] time = 15.5261, size = 56, normalized size = 0.82 \[ - \frac{2 x^{- \frac{3 n}{2}}}{3 a n} + \frac{2 b x^{- \frac{n}{2}}}{a^{2} n} - \frac{2 b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} x^{- \frac{n}{2}}}{\sqrt{b}} \right )}}{a^{\frac{5}{2}} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1-3/2*n)/(a+b*x**n),x)
[Out]
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Mathematica [A] time = 0.0911584, size = 62, normalized size = 0.91 \[ \frac{2 \left (\sqrt{a} x^{-3 n/2} \left (3 b x^n-a\right )-3 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x^{-n/2}}{\sqrt{b}}\right )\right )}{3 a^{5/2} n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 - (3*n)/2)/(a + b*x^n),x]
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Maple [A] time = 0.12, size = 97, normalized size = 1.4 \[ 2\,{\frac{b}{{a}^{2}n{x}^{n/2}}}-{\frac{2}{3\,an} \left ({x}^{{\frac{n}{2}}} \right ) ^{-3}}+{\frac{b}{{a}^{3}n}\sqrt{-ab}\ln \left ({x}^{{\frac{n}{2}}}+{\frac{1}{b}\sqrt{-ab}} \right ) }-{\frac{b}{{a}^{3}n}\sqrt{-ab}\ln \left ({x}^{{\frac{n}{2}}}-{\frac{1}{b}\sqrt{-ab}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1-3/2*n)/(a+b*x^n),x)
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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^(-3/2*n - 1)/(b*x^n + a),x, algorithm="maxima")
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Fricas [A] time = 0.261498, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, a x x^{-\frac{3}{2} \, n - 1} - 3 \, b \sqrt{-\frac{b}{a}} \log \left (-\frac{2 \, a x^{\frac{1}{3}} x^{-\frac{1}{2} \, n - \frac{1}{3}} \sqrt{-\frac{b}{a}} - a x^{\frac{2}{3}} x^{-n - \frac{2}{3}} + b}{a x^{\frac{2}{3}} x^{-n - \frac{2}{3}} + b}\right ) - 6 \, b x^{\frac{1}{3}} x^{-\frac{1}{2} \, n - \frac{1}{3}}}{3 \, a^{2} n}, -\frac{2 \,{\left (a x x^{-\frac{3}{2} \, n - 1} - 3 \, b \sqrt{\frac{b}{a}} \arctan \left (\frac{\sqrt{\frac{b}{a}}}{x^{\frac{1}{3}} x^{-\frac{1}{2} \, n - \frac{1}{3}}}\right ) - 3 \, b x^{\frac{1}{3}} x^{-\frac{1}{2} \, n - \frac{1}{3}}\right )}}{3 \, a^{2} n}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(-3/2*n - 1)/(b*x^n + a),x, algorithm="fricas")
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(-1-3/2*n)/(a+b*x**n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{-\frac{3}{2} \, n - 1}}{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(-3/2*n - 1)/(b*x^n + a),x, algorithm="giac")
[Out]