Optimal. Leaf size=315 \[ -\frac{\sqrt [4]{b} x^{n/4} (c x)^{-n/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt{a} x^{-n/2}+\sqrt{b}\right )}{\sqrt{2} a^{5/4} c n}+\frac{\sqrt [4]{b} x^{n/4} (c x)^{-n/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt{a} x^{-n/2}+\sqrt{b}\right )}{\sqrt{2} a^{5/4} c n}-\frac{\sqrt{2} \sqrt [4]{b} x^{n/4} (c x)^{-n/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{5/4} c n}+\frac{\sqrt{2} \sqrt [4]{b} x^{n/4} (c x)^{-n/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}+1\right )}{a^{5/4} c n}-\frac{4 (c x)^{-n/4}}{a c n} \]
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Rubi [A] time = 0.228068, antiderivative size = 315, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {347, 345, 193, 321, 211, 1165, 628, 1162, 617, 204} \[ -\frac{\sqrt [4]{b} x^{n/4} (c x)^{-n/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt{a} x^{-n/2}+\sqrt{b}\right )}{\sqrt{2} a^{5/4} c n}+\frac{\sqrt [4]{b} x^{n/4} (c x)^{-n/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt{a} x^{-n/2}+\sqrt{b}\right )}{\sqrt{2} a^{5/4} c n}-\frac{\sqrt{2} \sqrt [4]{b} x^{n/4} (c x)^{-n/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{5/4} c n}+\frac{\sqrt{2} \sqrt [4]{b} x^{n/4} (c x)^{-n/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}+1\right )}{a^{5/4} c n}-\frac{4 (c x)^{-n/4}}{a c n} \]
Antiderivative was successfully verified.
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Rule 347
Rule 345
Rule 193
Rule 321
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{(c x)^{-1-\frac{n}{4}}}{a+b x^n} \, dx &=\frac{\left (x^{n/4} (c x)^{-n/4}\right ) \int \frac{x^{-1-\frac{n}{4}}}{a+b x^n} \, dx}{c}\\ &=-\frac{\left (4 x^{n/4} (c x)^{-n/4}\right ) \operatorname{Subst}\left (\int \frac{1}{a+\frac{b}{x^4}} \, dx,x,x^{-n/4}\right )}{c n}\\ &=-\frac{\left (4 x^{n/4} (c x)^{-n/4}\right ) \operatorname{Subst}\left (\int \frac{x^4}{b+a x^4} \, dx,x,x^{-n/4}\right )}{c n}\\ &=-\frac{4 (c x)^{-n/4}}{a c n}+\frac{\left (4 b x^{n/4} (c x)^{-n/4}\right ) \operatorname{Subst}\left (\int \frac{1}{b+a x^4} \, dx,x,x^{-n/4}\right )}{a c n}\\ &=-\frac{4 (c x)^{-n/4}}{a c n}+\frac{\left (2 \sqrt{b} x^{n/4} (c x)^{-n/4}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b}-\sqrt{a} x^2}{b+a x^4} \, dx,x,x^{-n/4}\right )}{a c n}+\frac{\left (2 \sqrt{b} x^{n/4} (c x)^{-n/4}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b}+\sqrt{a} x^2}{b+a x^4} \, dx,x,x^{-n/4}\right )}{a c n}\\ &=-\frac{4 (c x)^{-n/4}}{a c n}-\frac{\left (\sqrt [4]{b} x^{n/4} (c x)^{-n/4}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{b}}{\sqrt [4]{a}}+2 x}{-\frac{\sqrt{b}}{\sqrt{a}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}-x^2} \, dx,x,x^{-n/4}\right )}{\sqrt{2} a^{5/4} c n}-\frac{\left (\sqrt [4]{b} x^{n/4} (c x)^{-n/4}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{b}}{\sqrt [4]{a}}-2 x}{-\frac{\sqrt{b}}{\sqrt{a}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}-x^2} \, dx,x,x^{-n/4}\right )}{\sqrt{2} a^{5/4} c n}+\frac{\left (\sqrt{b} x^{n/4} (c x)^{-n/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{b}}{\sqrt{a}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+x^2} \, dx,x,x^{-n/4}\right )}{a^{3/2} c n}+\frac{\left (\sqrt{b} x^{n/4} (c x)^{-n/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{b}}{\sqrt{a}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+x^2} \, dx,x,x^{-n/4}\right )}{a^{3/2} c n}\\ &=-\frac{4 (c x)^{-n/4}}{a c n}-\frac{\sqrt [4]{b} x^{n/4} (c x)^{-n/4} \log \left (\sqrt{b}+\sqrt{a} x^{-n/2}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}\right )}{\sqrt{2} a^{5/4} c n}+\frac{\sqrt [4]{b} x^{n/4} (c x)^{-n/4} \log \left (\sqrt{b}+\sqrt{a} x^{-n/2}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}\right )}{\sqrt{2} a^{5/4} c n}+\frac{\left (\sqrt{2} \sqrt [4]{b} x^{n/4} (c x)^{-n/4}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{5/4} c n}-\frac{\left (\sqrt{2} \sqrt [4]{b} x^{n/4} (c x)^{-n/4}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{5/4} c n}\\ &=-\frac{4 (c x)^{-n/4}}{a c n}-\frac{\sqrt{2} \sqrt [4]{b} x^{n/4} (c x)^{-n/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{5/4} c n}+\frac{\sqrt{2} \sqrt [4]{b} x^{n/4} (c x)^{-n/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{5/4} c n}-\frac{\sqrt [4]{b} x^{n/4} (c x)^{-n/4} \log \left (\sqrt{b}+\sqrt{a} x^{-n/2}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}\right )}{\sqrt{2} a^{5/4} c n}+\frac{\sqrt [4]{b} x^{n/4} (c x)^{-n/4} \log \left (\sqrt{b}+\sqrt{a} x^{-n/2}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}\right )}{\sqrt{2} a^{5/4} c n}\\ \end{align*}
Mathematica [C] time = 0.0104571, size = 37, normalized size = 0.12 \[ -\frac{4 x (c x)^{-\frac{n}{4}-1} \, _2F_1\left (-\frac{1}{4},1;\frac{3}{4};-\frac{b x^n}{a}\right )}{a n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.065, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{a+b{x}^{n}} \left ( cx \right ) ^{-1-{\frac{n}{4}}}}\, dx \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*} -b \int \frac{x^{\frac{3}{4} \, n}}{a b c^{\frac{1}{4} \, n + 1} x x^{n} + a^{2} c^{\frac{1}{4} \, n + 1} x}\,{d x} - \frac{4 \, c^{-\frac{1}{4} \, n - 1}}{a n x^{\frac{1}{4} \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45725, size = 838, normalized size = 2.66 \begin{align*} \frac{4 \, a n \left (-\frac{b c^{-n - 4}}{a^{5} n^{4}}\right )^{\frac{1}{4}} \arctan \left (-\frac{a^{4} n^{3} x \left (-\frac{b c^{-n - 4}}{a^{5} n^{4}}\right )^{\frac{3}{4}} e^{\left (-\frac{1}{4} \,{\left (n + 4\right )} \log \left (c\right ) - \frac{1}{4} \,{\left (n + 4\right )} \log \left (x\right )\right )} - a^{4} n^{3} x \sqrt{\frac{a^{2} n^{2} \sqrt{-\frac{b c^{-n - 4}}{a^{5} n^{4}}} + x^{2} e^{\left (-\frac{1}{2} \,{\left (n + 4\right )} \log \left (c\right ) - \frac{1}{2} \,{\left (n + 4\right )} \log \left (x\right )\right )}}{x^{2}}} \left (-\frac{b c^{-n - 4}}{a^{5} n^{4}}\right )^{\frac{3}{4}}}{b c^{-n - 4}}\right ) + a n \left (-\frac{b c^{-n - 4}}{a^{5} n^{4}}\right )^{\frac{1}{4}} \log \left (\frac{a n \left (-\frac{b c^{-n - 4}}{a^{5} n^{4}}\right )^{\frac{1}{4}} + x e^{\left (-\frac{1}{4} \,{\left (n + 4\right )} \log \left (c\right ) - \frac{1}{4} \,{\left (n + 4\right )} \log \left (x\right )\right )}}{x}\right ) - a n \left (-\frac{b c^{-n - 4}}{a^{5} n^{4}}\right )^{\frac{1}{4}} \log \left (-\frac{a n \left (-\frac{b c^{-n - 4}}{a^{5} n^{4}}\right )^{\frac{1}{4}} - x e^{\left (-\frac{1}{4} \,{\left (n + 4\right )} \log \left (c\right ) - \frac{1}{4} \,{\left (n + 4\right )} \log \left (x\right )\right )}}{x}\right ) - 4 \, x e^{\left (-\frac{1}{4} \,{\left (n + 4\right )} \log \left (c\right ) - \frac{1}{4} \,{\left (n + 4\right )} \log \left (x\right )\right )}}{a n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 5.14964, size = 299, normalized size = 0.95 \begin{align*} \frac{c^{- \frac{n}{4}} x^{- \frac{n}{4}} \Gamma \left (- \frac{1}{4}\right )}{a c n \Gamma \left (\frac{3}{4}\right )} - \frac{\sqrt [4]{b} c^{- \frac{n}{4}} e^{- \frac{3 i \pi }{4}} \log{\left (1 - \frac{\sqrt [4]{b} x^{\frac{n}{4}} e^{\frac{i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac{1}{4}\right )}{4 a^{\frac{5}{4}} c n \Gamma \left (\frac{3}{4}\right )} - \frac{i \sqrt [4]{b} c^{- \frac{n}{4}} e^{- \frac{3 i \pi }{4}} \log{\left (1 - \frac{\sqrt [4]{b} x^{\frac{n}{4}} e^{\frac{3 i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac{1}{4}\right )}{4 a^{\frac{5}{4}} c n \Gamma \left (\frac{3}{4}\right )} + \frac{\sqrt [4]{b} c^{- \frac{n}{4}} e^{- \frac{3 i \pi }{4}} \log{\left (1 - \frac{\sqrt [4]{b} x^{\frac{n}{4}} e^{\frac{5 i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac{1}{4}\right )}{4 a^{\frac{5}{4}} c n \Gamma \left (\frac{3}{4}\right )} + \frac{i \sqrt [4]{b} c^{- \frac{n}{4}} e^{- \frac{3 i \pi }{4}} \log{\left (1 - \frac{\sqrt [4]{b} x^{\frac{n}{4}} e^{\frac{7 i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac{1}{4}\right )}{4 a^{\frac{5}{4}} c n \Gamma \left (\frac{3}{4}\right )} \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{\left (c x\right )^{-\frac{1}{4} \, n - 1}}{b x^{n} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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