Optimal. Leaf size=236 \[ \frac{b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt{a}+\sqrt{b} x^{n/2}\right )}{\sqrt{2} a^{7/4} n}-\frac{b^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt{a}+\sqrt{b} x^{n/2}\right )}{\sqrt{2} a^{7/4} n}+\frac{\sqrt{2} b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}\right )}{a^{7/4} n}-\frac{\sqrt{2} b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}+1\right )}{a^{7/4} n}-\frac{4 x^{-3 n/4}}{3 a n} \]
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Rubi [A] time = 0.440814, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ \frac{b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt{a}+\sqrt{b} x^{n/2}\right )}{\sqrt{2} a^{7/4} n}-\frac{b^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt{a}+\sqrt{b} x^{n/2}\right )}{\sqrt{2} a^{7/4} n}+\frac{\sqrt{2} b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}\right )}{a^{7/4} n}-\frac{\sqrt{2} b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}+1\right )}{a^{7/4} n}-\frac{4 x^{-3 n/4}}{3 a n} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 - (3*n)/4)/(a + b*x^n),x]
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Rubi in Sympy [A] time = 65.3219, size = 206, normalized size = 0.87 \[ - \frac{4 x^{- \frac{3 n}{4}}}{3 a n} + \frac{\sqrt{2} b^{\frac{3}{4}} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{\frac{n}{4}} + \sqrt{a} + \sqrt{b} x^{\frac{n}{2}} \right )}}{2 a^{\frac{7}{4}} n} - \frac{\sqrt{2} b^{\frac{3}{4}} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{\frac{n}{4}} + \sqrt{a} + \sqrt{b} x^{\frac{n}{2}} \right )}}{2 a^{\frac{7}{4}} n} + \frac{\sqrt{2} b^{\frac{3}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x^{\frac{n}{4}}}{\sqrt [4]{a}} \right )}}{a^{\frac{7}{4}} n} - \frac{\sqrt{2} b^{\frac{3}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x^{\frac{n}{4}}}{\sqrt [4]{a}} \right )}}{a^{\frac{7}{4}} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1-3/4*n)/(a+b*x**n),x)
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Mathematica [C] time = 0.0438319, size = 60, normalized size = 0.25 \[ \frac{3 b \text{RootSum}\left [\text{$\#$1}^4 a+b\&,\frac{4 \log \left (x^{-n/4}-\text{$\#$1}\right )+n \log (x)}{\text{$\#$1}}\&\right ]-16 a x^{-3 n/4}}{12 a^2 n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 - (3*n)/4)/(a + b*x^n),x]
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Maple [C] time = 0.089, size = 54, normalized size = 0.2 \[ -{\frac{4}{3\,an} \left ({x}^{{\frac{n}{4}}} \right ) ^{-3}}+\sum _{{\it \_R}={\it RootOf} \left ({a}^{7}{n}^{4}{{\it \_Z}}^{4}+{b}^{3} \right ) }{\it \_R}\,\ln \left ({x}^{{\frac{n}{4}}}-{\frac{{a}^{2}n{\it \_R}}{b}} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1-3/4*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^(-3/4*n - 1)/(b*x^n + a),x, algorithm="maxima")
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Fricas [A] time = 0.342121, size = 324, normalized size = 1.37 \[ \frac{12 \, a n \left (-\frac{b^{3}}{a^{7} n^{4}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{5} n^{3} \left (-\frac{b^{3}}{a^{7} n^{4}}\right )^{\frac{3}{4}}}{b^{2} x^{\frac{1}{3}} x^{-\frac{1}{4} \, n - \frac{1}{3}} + x^{\frac{1}{3}} \sqrt{-\frac{a^{3} b^{3} n^{2} \sqrt{-\frac{b^{3}}{a^{7} n^{4}}} - b^{4} x^{\frac{2}{3}} x^{-\frac{1}{2} \, n - \frac{2}{3}}}{x^{\frac{2}{3}}}}}\right ) + 3 \, a n \left (-\frac{b^{3}}{a^{7} n^{4}}\right )^{\frac{1}{4}} \log \left (\frac{a^{5} n^{3} \left (-\frac{b^{3}}{a^{7} n^{4}}\right )^{\frac{3}{4}} + b^{2} x^{\frac{1}{3}} x^{-\frac{1}{4} \, n - \frac{1}{3}}}{x^{\frac{1}{3}}}\right ) - 3 \, a n \left (-\frac{b^{3}}{a^{7} n^{4}}\right )^{\frac{1}{4}} \log \left (-\frac{a^{5} n^{3} \left (-\frac{b^{3}}{a^{7} n^{4}}\right )^{\frac{3}{4}} - b^{2} x^{\frac{1}{3}} x^{-\frac{1}{4} \, n - \frac{1}{3}}}{x^{\frac{1}{3}}}\right ) - 4 \, x x^{-\frac{3}{4} \, n - 1}}{3 \, a n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(-3/4*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**(-1-3/4*n)/(a+b*x**n),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{-\frac{3}{4} \, n - 1}}{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(-3/4*n - 1)/(b*x^n + a),x, algorithm="giac")
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