Optimal. Leaf size=20 \[ e x \log ^{n+1}(d x) F^{c (a+b x)} \]
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Rubi [A] time = 0.0464877, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029, Rules used = {2201} \[ e x \log ^{n+1}(d x) F^{c (a+b x)} \]
Antiderivative was successfully verified.
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Rule 2201
Rubi steps
\begin{align*} \int F^{c (a+b x)} \log ^n(d x) (e+e n+e (1+b c x \log (F)) \log (d x)) \, dx &=e F^{c (a+b x)} x \log ^{1+n}(d x)\\ \end{align*}
Mathematica [A] time = 0.17145, size = 21, normalized size = 1.05 \[ e x \log ^{n+1}(d x) F^{a c+b c x} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.108, size = 186, normalized size = 9.3 \begin{align*} \left ( -{\frac{i}{2}}x{F}^{c \left ( bx+a \right ) }e\pi \,{\it csgn} \left ( id \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( idx \right ) +{\frac{i}{2}}x{F}^{c \left ( bx+a \right ) }e\pi \,{\it csgn} \left ( id \right ) \left ({\it csgn} \left ( idx \right ) \right ) ^{2}+{\frac{i}{2}}x{F}^{c \left ( bx+a \right ) }e\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( idx \right ) \right ) ^{2}-{\frac{i}{2}}x{F}^{c \left ( bx+a \right ) }e\pi \, \left ({\it csgn} \left ( idx \right ) \right ) ^{3}+x{F}^{c \left ( bx+a \right ) }e\ln \left ( d \right ) +ex{F}^{c \left ( bx+a \right ) }\ln \left ( x \right ) \right ) \left ( \ln \left ( d \right ) +\ln \left ( x \right ) -{\frac{i}{2}}\pi \,{\it csgn} \left ( idx \right ) \left ( -{\it csgn} \left ( idx \right ) +{\it csgn} \left ( id \right ) \right ) \left ( -{\it csgn} \left ( idx \right ) +{\it csgn} \left ( ix \right ) \right ) \right ) ^{n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.3577, size = 51, normalized size = 2.55 \begin{align*}{\left (F^{a c} e x \log \left (d\right ) + F^{a c} e x \log \left (x\right )\right )} e^{\left (b c x \log \left (F\right ) + n \log \left (\log \left (d\right ) + \log \left (x\right )\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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