Optimal. Leaf size=33 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} (m+1)} \]
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Rubi [A] time = 0.0172386, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {345, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} (m+1)} \]
Antiderivative was successfully verified.
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Rule 345
Rule 205
Rubi steps
\begin{align*} \int \frac{x^m}{a+b x^{2+2 m}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,x^{1+m}\right )}{1+m}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{b} x^{1+m}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0118577, size = 33, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} (m+1)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.038, size = 61, normalized size = 1.9 \begin{align*} -{\frac{1}{2+2\,m}\ln \left ({x}^{m}-{\frac{a}{x}{\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}}+{\frac{1}{2+2\,m}\ln \left ({x}^{m}+{\frac{a}{x}{\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}} \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*} \int \frac{x^{m}}{b x^{2 \, m + 2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35306, size = 209, normalized size = 6.33 \begin{align*} \left [-\frac{\sqrt{-a b} \log \left (\frac{b x^{2} x^{2 \, m} - 2 \, \sqrt{-a b} x x^{m} - a}{b x^{2} x^{2 \, m} + a}\right )}{2 \,{\left (a b m + a b\right )}}, -\frac{\sqrt{a b} \arctan \left (\frac{\sqrt{a b}}{b x x^{m}}\right )}{a b m + a b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.62913, size = 197, normalized size = 5.97 \begin{align*} \frac{i \sqrt{\pi } a^{- \frac{m}{2 \left (m + 1\right )}} a^{- \frac{1}{2 \left (m + 1\right )}} \log{\left (1 - \frac{\sqrt{b} x x^{m} e^{\frac{i \pi }{2}}}{\sqrt{a}} \right )}}{4 \sqrt{b} m \Gamma \left (\frac{m}{2 \left (m + 1\right )} + 1 + \frac{1}{2 \left (m + 1\right )}\right ) + 4 \sqrt{b} \Gamma \left (\frac{m}{2 \left (m + 1\right )} + 1 + \frac{1}{2 \left (m + 1\right )}\right )} - \frac{i \sqrt{\pi } a^{- \frac{m}{2 \left (m + 1\right )}} a^{- \frac{1}{2 \left (m + 1\right )}} \log{\left (1 - \frac{\sqrt{b} x x^{m} e^{\frac{3 i \pi }{2}}}{\sqrt{a}} \right )}}{4 \sqrt{b} m \Gamma \left (\frac{m}{2 \left (m + 1\right )} + 1 + \frac{1}{2 \left (m + 1\right )}\right ) + 4 \sqrt{b} \Gamma \left (\frac{m}{2 \left (m + 1\right )} + 1 + \frac{1}{2 \left (m + 1\right )}\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{x^{m}}{b x^{2 \, m + 2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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