Optimal. Leaf size=41 \[ \frac{x^m f^{a c} (-b c x \log (f))^{-m} \text{Gamma}(m+1,-b c x \log (f))}{b c \log (f)} \]
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Rubi [A] time = 0.0217245, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2181} \[ \frac{x^m f^{a c} (-b c x \log (f))^{-m} \text{Gamma}(m+1,-b c x \log (f))}{b c \log (f)} \]
Antiderivative was successfully verified.
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Rule 2181
Rubi steps
\begin{align*} \int f^{c (a+b x)} x^m \, dx &=\frac{f^{a c} x^m \Gamma (1+m,-b c x \log (f)) (-b c x \log (f))^{-m}}{b c \log (f)}\\ \end{align*}
Mathematica [A] time = 0.0078306, size = 36, normalized size = 0.88 \[ x^{m+1} \left (-f^{a c}\right ) (-b c x \log (f))^{-m-1} \text{Gamma}(m+1,-b c x \log (f)) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.036, size = 117, normalized size = 2.9 \begin{align*} -{\frac{{f}^{ac} \left ( -bc \right ) ^{-m} \left ( \ln \left ( f \right ) \right ) ^{-m-1} \left ({x}^{m} \left ( -bc \right ) ^{m} \left ( \ln \left ( f \right ) \right ) ^{m}m\Gamma \left ( m \right ) \left ( -bcx\ln \left ( f \right ) \right ) ^{-m}-{x}^{m} \left ( -bc \right ) ^{m} \left ( \ln \left ( f \right ) \right ) ^{m}{{\rm e}^{bcx\ln \left ( f \right ) }}-{x}^{m} \left ( -bc \right ) ^{m} \left ( \ln \left ( f \right ) \right ) ^{m}m \left ( -bcx\ln \left ( f \right ) \right ) ^{-m}\Gamma \left ( m,-bcx\ln \left ( f \right ) \right ) \right ) }{bc}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.26006, size = 49, normalized size = 1.2 \begin{align*} -\left (-b c x \log \left (f\right )\right )^{-m - 1} f^{a c} x^{m + 1} \Gamma \left (m + 1, -b c x \log \left (f\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5318, size = 105, normalized size = 2.56 \begin{align*} \frac{e^{\left (a c \log \left (f\right ) - m \log \left (-b c \log \left (f\right )\right )\right )} \Gamma \left (m + 1, -b c x \log \left (f\right )\right )}{b c \log \left (f\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{c \left (a + b x\right )} x^{m}\, dx \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 f^{{\left (b x + a\right )} c} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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