Optimal. Leaf size=252 \[ \frac{b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt{a} x^{-n/2}+\sqrt{b}\right )}{\sqrt{2} a^{9/4} n}-\frac{b^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt{a} x^{-n/2}+\sqrt{b}\right )}{\sqrt{2} a^{9/4} n}+\frac{\sqrt{2} b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{9/4} n}-\frac{\sqrt{2} b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}+1\right )}{a^{9/4} n}+\frac{4 b x^{-n/4}}{a^2 n}-\frac{4 x^{-5 n/4}}{5 a n} \]
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Rubi [A] time = 0.446151, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526 \[ \frac{b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt{a} x^{-n/2}+\sqrt{b}\right )}{\sqrt{2} a^{9/4} n}-\frac{b^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt{a} x^{-n/2}+\sqrt{b}\right )}{\sqrt{2} a^{9/4} n}+\frac{\sqrt{2} b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{9/4} n}-\frac{\sqrt{2} b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}+1\right )}{a^{9/4} n}+\frac{4 b x^{-n/4}}{a^2 n}-\frac{4 x^{-5 n/4}}{5 a n} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 - (5*n)/4)/(a + b*x^n),x]
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Rubi in Sympy [A] time = 74.4801, size = 219, normalized size = 0.87 \[ - \frac{4 x^{- \frac{5 n}{4}}}{5 a n} + \frac{4 b x^{- \frac{n}{4}}}{a^{2} n} + \frac{\sqrt{2} b^{\frac{5}{4}} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{- \frac{n}{4}} + \sqrt{a} x^{- \frac{n}{2}} + \sqrt{b} \right )}}{2 a^{\frac{9}{4}} n} - \frac{\sqrt{2} b^{\frac{5}{4}} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{- \frac{n}{4}} + \sqrt{a} x^{- \frac{n}{2}} + \sqrt{b} \right )}}{2 a^{\frac{9}{4}} n} - \frac{\sqrt{2} b^{\frac{5}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{a} x^{- \frac{n}{4}}}{\sqrt [4]{b}} - 1 \right )}}{a^{\frac{9}{4}} n} - \frac{\sqrt{2} b^{\frac{5}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{a} x^{- \frac{n}{4}}}{\sqrt [4]{b}} + 1 \right )}}{a^{\frac{9}{4}} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1-5/4*n)/(a+b*x**n),x)
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Mathematica [C] time = 0.0803548, size = 70, normalized size = 0.28 \[ -\frac{5 b^2 \text{RootSum}\left [\text{$\#$1}^4 a+b\&,\frac{4 \log \left (x^{-n/4}-\text{$\#$1}\right )+n \log (x)}{\text{$\#$1}^3}\&\right ]+16 a x^{-5 n/4} \left (a-5 b x^n\right )}{20 a^3 n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 - (5*n)/4)/(a + b*x^n),x]
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Maple [C] time = 0.115, size = 73, normalized size = 0.3 \[ 4\,{\frac{b}{{a}^{2}n{x}^{n/4}}}-{\frac{4}{5\,an} \left ({x}^{{\frac{n}{4}}} \right ) ^{-5}}+\sum _{{\it \_R}={\it RootOf} \left ({a}^{9}{n}^{4}{{\it \_Z}}^{4}+{b}^{5} \right ) }{\it \_R}\,\ln \left ({x}^{{\frac{n}{4}}}+{\frac{{a}^{7}{n}^{3}{{\it \_R}}^{3}}{{b}^{4}}} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1-5/4*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^(-5/4*n - 1)/(b*x^n + a),x, algorithm="maxima")
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Fricas [A] time = 0.28204, size = 331, normalized size = 1.31 \[ \frac{20 \, a^{2} n \left (-\frac{b^{5}}{a^{9} n^{4}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{2} n x^{\frac{4}{5}} \left (-\frac{b^{5}}{a^{9} n^{4}}\right )^{\frac{1}{4}}}{b x x^{-\frac{1}{4} \, n - \frac{1}{5}} + x \sqrt{\frac{a^{4} n^{2} x^{\frac{3}{5}} \sqrt{-\frac{b^{5}}{a^{9} n^{4}}} + b^{2} x x^{-\frac{1}{2} \, n - \frac{2}{5}}}{x}}}\right ) - 5 \, a^{2} n \left (-\frac{b^{5}}{a^{9} n^{4}}\right )^{\frac{1}{4}} \log \left (\frac{a^{2} n x^{\frac{4}{5}} \left (-\frac{b^{5}}{a^{9} n^{4}}\right )^{\frac{1}{4}} + b x x^{-\frac{1}{4} \, n - \frac{1}{5}}}{x}\right ) + 5 \, a^{2} n \left (-\frac{b^{5}}{a^{9} n^{4}}\right )^{\frac{1}{4}} \log \left (-\frac{a^{2} n x^{\frac{4}{5}} \left (-\frac{b^{5}}{a^{9} n^{4}}\right )^{\frac{1}{4}} - b x x^{-\frac{1}{4} \, n - \frac{1}{5}}}{x}\right ) - 4 \, a x x^{-\frac{5}{4} \, n - 1} + 20 \, b x^{\frac{1}{5}} x^{-\frac{1}{4} \, n - \frac{1}{5}}}{5 \, a^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(-5/4*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**(-1-5/4*n)/(a+b*x**n),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{-\frac{5}{4} \, n - 1}}{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(-5/4*n - 1)/(b*x^n + a),x, algorithm="giac")
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