Optimal. Leaf size=20 \[ \frac{f^{a+b x^n}}{b n \log (f)} \]
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Rubi [A] time = 0.0231736, antiderivative size = 20, 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 = {2209} \[ \frac{f^{a+b x^n}}{b n \log (f)} \]
Antiderivative was successfully verified.
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Rule 2209
Rubi steps
\begin{align*} \int f^{a+b x^n} x^{-1+n} \, dx &=\frac{f^{a+b x^n}}{b n \log (f)}\\ \end{align*}
Mathematica [A] time = 0.0042465, size = 20, normalized size = 1. \[ \frac{f^{a+b x^n}}{b n \log (f)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 25, normalized size = 1.3 \begin{align*}{\frac{{{\rm e}^{ \left ( b{{\rm e}^{n\ln \left ( x \right ) }}+a \right ) \ln \left ( f \right ) }}}{\ln \left ( f \right ) bn}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16261, size = 27, normalized size = 1.35 \begin{align*} \frac{f^{b x^{n} + a}}{b n \log \left (f\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56783, size = 57, normalized size = 2.85 \begin{align*} \frac{e^{\left (b x^{n} \log \left (f\right ) + a \log \left (f\right )\right )}}{b n \log \left (f\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 161.134, size = 39, normalized size = 1.95 \begin{align*} \begin{cases} \log{\left (x \right )} & \text{for}\: b = 0 \wedge f = 1 \wedge n = 0 \\f^{a + b} \log{\left (x \right )} & \text{for}\: n = 0 \\\frac{f^{a} x^{n}}{n} & \text{for}\: b = 0 \\\frac{x^{n}}{n} & \text{for}\: f = 1 \\\frac{f^{a} f^{b x^{n}}}{b n \log{\left (f \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25439, size = 27, normalized size = 1.35 \begin{align*} \frac{f^{b x^{n} + a}}{b n \log \left (f\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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