Optimal. Leaf size=50 \[ \frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x^{-n/2}}{\sqrt{b}}\right )}{a^{3/2} n}-\frac{2 x^{-n/2}}{a n} \]
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Rubi [A] time = 0.0249862, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {345, 193, 321, 205} \[ \frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x^{-n/2}}{\sqrt{b}}\right )}{a^{3/2} n}-\frac{2 x^{-n/2}}{a n} \]
Antiderivative was successfully verified.
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Rule 345
Rule 193
Rule 321
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{-1-\frac{n}{2}}}{a+b x^n} \, dx &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{a+\frac{b}{x^2}} \, dx,x,x^{-n/2}\right )}{n}\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{x^2}{b+a x^2} \, dx,x,x^{-n/2}\right )}{n}\\ &=-\frac{2 x^{-n/2}}{a n}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,x^{-n/2}\right )}{a n}\\ &=-\frac{2 x^{-n/2}}{a n}+\frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x^{-n/2}}{\sqrt{b}}\right )}{a^{3/2} n}\\ \end{align*}
Mathematica [C] time = 0.0065244, size = 32, normalized size = 0.64 \[ -\frac{2 x^{-n/2} \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\frac{b x^n}{a}\right )}{a n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 79, normalized size = 1.6 \begin{align*} -2\,{\frac{1}{an{x}^{n/2}}}+{\frac{1}{{a}^{2}n}\sqrt{-ab}\ln \left ({x}^{{\frac{n}{2}}}-{\frac{1}{b}\sqrt{-ab}} \right ) }-{\frac{1}{{a}^{2}n}\sqrt{-ab}\ln \left ({x}^{{\frac{n}{2}}}+{\frac{1}{b}\sqrt{-ab}} \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*} -b \int \frac{x^{\frac{1}{2} \, n}}{a b x x^{n} + a^{2} x}\,{d x} - \frac{2}{a n x^{\frac{1}{2} \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.05592, size = 282, normalized size = 5.64 \begin{align*} \left [-\frac{2 \, x x^{-\frac{1}{2} \, n - 1} - \sqrt{-\frac{b}{a}} \log \left (\frac{a x^{2} x^{-n - 2} + 2 \, a x x^{-\frac{1}{2} \, n - 1} \sqrt{-\frac{b}{a}} - b}{a x^{2} x^{-n - 2} + b}\right )}{a n}, -\frac{2 \,{\left (x x^{-\frac{1}{2} \, n - 1} + \sqrt{\frac{b}{a}} \arctan \left (\frac{\sqrt{\frac{b}{a}}}{x x^{-\frac{1}{2} \, n - 1}}\right )\right )}}{a n}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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}{2} \, n - 1}}{b x^{n} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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