Optimal. Leaf size=19 \[ \frac{\log \left (a+b x^{m+1}\right )}{b (m+1)} \]
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Rubi [A] time = 0.0051135, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {260} \[ \frac{\log \left (a+b x^{m+1}\right )}{b (m+1)} \]
Antiderivative was successfully verified.
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Rule 260
Rubi steps
\begin{align*} \int \frac{x^m}{a+b x^{1+m}} \, dx &=\frac{\log \left (a+b x^{1+m}\right )}{b (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0054525, size = 19, normalized size = 1. \[ \frac{\log \left (a+b x^{m+1}\right )}{b (m+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 21, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( a+bx{{\rm e}^{m\ln \left ( x \right ) }} \right ) }{b \left ( 1+m \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.966832, size = 26, normalized size = 1.37 \begin{align*} \frac{\log \left (b x^{m + 1} + a\right )}{b{\left (m + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35528, size = 43, normalized size = 2.26 \begin{align*} \frac{\log \left (b x^{m + 1} + a\right )}{b m + b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.773872, size = 37, normalized size = 1.95 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \wedge m = -1 \\\frac{x x^{m}}{a \left (m + 1\right )} & \text{for}\: b = 0 \\\frac{\log{\left (x \right )}}{a + b} & \text{for}\: m = -1 \\\frac{\log{\left (\frac{a}{b} + x x^{m} \right )}}{b m + b} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0839, size = 26, normalized size = 1.37 \begin{align*} \frac{\log \left ({\left | b x^{m + 1} + a \right |}\right )}{b m + b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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