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 = 0.206301, 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, Rules used = {1584, 362, 345, 211, 1165, 628, 1162, 617, 204} \[ \frac{c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} x^{n/4}+\sqrt{b}+\sqrt{c} x^{n/2}\right )}{\sqrt{2} b^{7/4} n}-\frac{c^{3/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} x^{n/4}+\sqrt{b}+\sqrt{c} x^{n/2}\right )}{\sqrt{2} b^{7/4} n}+\frac{\sqrt{2} c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{b}}\right )}{b^{7/4} n}-\frac{\sqrt{2} c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{b}}+1\right )}{b^{7/4} n}-\frac{4 x^{-3 n/4}}{3 b n} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 362
Rule 345
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{x^{-1+\frac{n}{4}}}{b x^n+c x^{2 n}} \, dx &=\int \frac{x^{-1-\frac{3 n}{4}}}{b+c x^n} \, dx\\ &=-\frac{4 x^{-3 n/4}}{3 b n}-\frac{c \int \frac{x^{\frac{1}{4} (-4+n)}}{b+c x^n} \, dx}{b}\\ &=-\frac{4 x^{-3 n/4}}{3 b n}-\frac{(4 c) \operatorname{Subst}\left (\int \frac{1}{b+c x^4} \, dx,x,x^{1+\frac{1}{4} (-4+n)}\right )}{b n}\\ &=-\frac{4 x^{-3 n/4}}{3 b n}-\frac{(2 c) \operatorname{Subst}\left (\int \frac{\sqrt{b}-\sqrt{c} x^2}{b+c x^4} \, dx,x,x^{1+\frac{1}{4} (-4+n)}\right )}{b^{3/2} n}-\frac{(2 c) \operatorname{Subst}\left (\int \frac{\sqrt{b}+\sqrt{c} x^2}{b+c x^4} \, dx,x,x^{1+\frac{1}{4} (-4+n)}\right )}{b^{3/2} n}\\ &=-\frac{4 x^{-3 n/4}}{3 b n}-\frac{\sqrt{c} \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{b}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,x^{1+\frac{1}{4} (-4+n)}\right )}{b^{3/2} n}-\frac{\sqrt{c} \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{b}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,x^{1+\frac{1}{4} (-4+n)}\right )}{b^{3/2} n}+\frac{c^{3/4} \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{b}}{\sqrt [4]{c}}+2 x}{-\frac{\sqrt{b}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,x^{1+\frac{1}{4} (-4+n)}\right )}{\sqrt{2} b^{7/4} n}+\frac{c^{3/4} \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{b}}{\sqrt [4]{c}}-2 x}{-\frac{\sqrt{b}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,x^{1+\frac{1}{4} (-4+n)}\right )}{\sqrt{2} b^{7/4} n}\\ &=-\frac{4 x^{-3 n/4}}{3 b n}+\frac{c^{3/4} \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} x^{n/4}+\sqrt{c} x^{n/2}\right )}{\sqrt{2} b^{7/4} n}-\frac{c^{3/4} \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} x^{n/4}+\sqrt{c} x^{n/2}\right )}{\sqrt{2} b^{7/4} n}-\frac{\left (\sqrt{2} c^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} x^{1+\frac{1}{4} (-4+n)}}{\sqrt [4]{b}}\right )}{b^{7/4} n}+\frac{\left (\sqrt{2} c^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} x^{1+\frac{1}{4} (-4+n)}}{\sqrt [4]{b}}\right )}{b^{7/4} n}\\ &=-\frac{4 x^{-3 n/4}}{3 b 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 (1+\frac{\sqrt{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{b}}\right )}{b^{7/4} n}+\frac{c^{3/4} \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} x^{n/4}+\sqrt{c} x^{n/2}\right )}{\sqrt{2} b^{7/4} n}-\frac{c^{3/4} \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} x^{n/4}+\sqrt{c} x^{n/2}\right )}{\sqrt{2} b^{7/4} n}\\ \end{align*}
Mathematica [C] time = 0.0093805, size = 34, normalized size = 0.14 \[ -\frac{4 x^{-3 n/4} \, _2F_1\left (-\frac{3}{4},1;\frac{1}{4};-\frac{c x^n}{b}\right )}{3 b n} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.087, size = 54, normalized size = 0.2 \begin{align*} -{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -c \int \frac{x^{\frac{1}{4} \, n}}{b c x x^{n} + b^{2} x}\,{d x} - \frac{4}{3 \, b n x^{\frac{3}{4} \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70372, size = 625, normalized size = 2.65 \begin{align*} -\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^{5} c n^{3} x x^{\frac{1}{4} \, n - 1} \left (-\frac{c^{3}}{b^{7} n^{4}}\right )^{\frac{3}{4}} - b^{5} n^{3} 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}}} \left (-\frac{c^{3}}{b^{7} n^{4}}\right )^{\frac{3}{4}}}{c^{3}}\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{- n} x^{\frac{n}{4} - 1}}{b + c x^{n}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{1}{4} \, n - 1}}{c x^{2 \, n} + b x^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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