Optimal. Leaf size=41 \[ \frac {f^{a c} x^m \Gamma (1+m,-b c x \log (f)) (-b c x \log (f))^{-m}}{b c \log (f)} \]
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Rubi [A]
time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, 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 = {2212}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
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*}
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Mathematica [A]
time = 0.02, size = 36, normalized size = 0.88 \begin {gather*} -f^{a c} x^{1+m} \Gamma (1+m,-b c x \log (f)) (-b c x \log (f))^{-1-m} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(116\) vs.
\(2(41)=82\).
time = 0.03, size = 117, normalized size = 2.85
method | result | size |
meijerg | \(-\frac {f^{c a} \left (-c b \right )^{-m} \ln \left (f \right )^{-m -1} \left (x^{m} \left (-c b \right )^{m} \ln \left (f \right )^{m} m \Gamma \left (m \right ) \left (-b c x \ln \left (f \right )\right )^{-m}-x^{m} \left (-c b \right )^{m} \ln \left (f \right )^{m} {\mathrm e}^{b c x \ln \left (f \right )}-x^{m} \left (-c b \right )^{m} \ln \left (f \right )^{m} m \left (-b c x \ln \left (f \right )\right )^{-m} \Gamma \left (m , -b c x \ln \left (f \right )\right )\right )}{c b}\) | \(117\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.07, size = 36, normalized size = 0.88 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.09, size = 39, normalized size = 0.95 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int f^{c \left (a + b x\right )} x^{m}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.50, size = 41, normalized size = 1.00 \begin {gather*} \frac {f^{a\,c}\,x^m\,\Gamma \left (m+1,-b\,c\,x\,\ln \left (f\right )\right )}{b\,c\,\ln \left (f\right )\,{\left (-b\,c\,x\,\ln \left (f\right )\right )}^m} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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