Optimal. Leaf size=236 \[ \frac{c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} x^{n/4}+\sqrt{b}+\sqrt{c} x^{n/2}\right )}{\sqrt{2} b^{7/4} n}-\frac{c^{3/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} x^{n/4}+\sqrt{b}+\sqrt{c} x^{n/2}\right )}{\sqrt{2} b^{7/4} n}+\frac{\sqrt{2} c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{b}}\right )}{b^{7/4} n}-\frac{\sqrt{2} c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{b}}+1\right )}{b^{7/4} n}-\frac{4 x^{-3 n/4}}{3 b n} \]
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Rubi [A] time = 0.386315, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36 \[ \frac{c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} x^{n/4}+\sqrt{b}+\sqrt{c} x^{n/2}\right )}{\sqrt{2} b^{7/4} n}-\frac{c^{3/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} x^{n/4}+\sqrt{b}+\sqrt{c} x^{n/2}\right )}{\sqrt{2} b^{7/4} n}+\frac{\sqrt{2} c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{b}}\right )}{b^{7/4} n}-\frac{\sqrt{2} c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{b}}+1\right )}{b^{7/4} n}-\frac{4 x^{-3 n/4}}{3 b n} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 + n/4)/(b*x^n + c*x^(2*n)),x]
[Out]
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Rubi in Sympy [A] time = 64.2371, size = 206, normalized size = 0.87 \[ - \frac{4 x^{- \frac{3 n}{4}}}{3 b n} + \frac{\sqrt{2} c^{\frac{3}{4}} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} x^{\frac{n}{4}} + \sqrt{b} + \sqrt{c} x^{\frac{n}{2}} \right )}}{2 b^{\frac{7}{4}} n} - \frac{\sqrt{2} c^{\frac{3}{4}} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} x^{\frac{n}{4}} + \sqrt{b} + \sqrt{c} x^{\frac{n}{2}} \right )}}{2 b^{\frac{7}{4}} n} + \frac{\sqrt{2} c^{\frac{3}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x^{\frac{n}{4}}}{\sqrt [4]{b}} \right )}}{b^{\frac{7}{4}} n} - \frac{\sqrt{2} c^{\frac{3}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x^{\frac{n}{4}}}{\sqrt [4]{b}} \right )}}{b^{\frac{7}{4}} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1+1/4*n)/(b*x**n+c*x**(2*n)),x)
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Mathematica [C] time = 0.0436591, size = 60, normalized size = 0.25 \[ \frac{3 c \text{RootSum}\left [\text{$\#$1}^4 b+c\&,\frac{4 \log \left (x^{-n/4}-\text{$\#$1}\right )+n \log (x)}{\text{$\#$1}}\&\right ]-16 b x^{-3 n/4}}{12 b^2 n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 + n/4)/(b*x^n + c*x^(2*n)),x]
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Maple [C] time = 0.322, size = 54, normalized size = 0.2 \[ -{\frac{4}{3\,bn} \left ({x}^{{\frac{n}{4}}} \right ) ^{-3}}+\sum _{{\it \_R}={\it RootOf} \left ({b}^{7}{n}^{4}{{\it \_Z}}^{4}+{c}^{3} \right ) }{\it \_R}\,\ln \left ({x}^{{\frac{n}{4}}}-{\frac{{b}^{2}n{\it \_R}}{c}} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1+1/4*n)/(b*x^n+c*x^(2*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^(1/4*n - 1)/(c*x^(2*n) + b*x^n),x, algorithm="maxima")
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Fricas [A] time = 0.306976, size = 335, normalized size = 1.42 \[ \frac{12 \, b n x^{3} x^{\frac{3}{4} \, n - 3} \left (-\frac{c^{3}}{b^{7} n^{4}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{2} n \left (-\frac{c^{3}}{b^{7} n^{4}}\right )^{\frac{1}{4}}}{c x x^{\frac{1}{4} \, n - 1} + x \sqrt{\frac{b^{4} n^{2} \sqrt{-\frac{c^{3}}{b^{7} n^{4}}} + c^{2} x^{2} x^{\frac{1}{2} \, n - 2}}{x^{2}}}}\right ) - 3 \, b n x^{3} x^{\frac{3}{4} \, n - 3} \left (-\frac{c^{3}}{b^{7} n^{4}}\right )^{\frac{1}{4}} \log \left (\frac{b^{2} n \left (-\frac{c^{3}}{b^{7} n^{4}}\right )^{\frac{1}{4}} + c x x^{\frac{1}{4} \, n - 1}}{x}\right ) + 3 \, b n x^{3} x^{\frac{3}{4} \, n - 3} \left (-\frac{c^{3}}{b^{7} n^{4}}\right )^{\frac{1}{4}} \log \left (-\frac{b^{2} n \left (-\frac{c^{3}}{b^{7} n^{4}}\right )^{\frac{1}{4}} - c x x^{\frac{1}{4} \, n - 1}}{x}\right ) - 4}{3 \, b n x^{3} x^{\frac{3}{4} \, n - 3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(1/4*n - 1)/(c*x^(2*n) + b*x^n),x, algorithm="fricas")
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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+1/4*n)/(b*x**n+c*x**(2*n)),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{1}{4} \, n - 1}}{c x^{2 \, n} + b x^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(1/4*n - 1)/(c*x^(2*n) + b*x^n),x, algorithm="giac")
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