Optimal. Leaf size=23 \[ \frac {\log \left (a+b F^{c+d x}\right )}{b d \log (F)} \]
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Rubi [A]
time = 0.03, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2278, 31}
\begin {gather*} \frac {\log \left (a+b F^{c+d x}\right )}{b d \log (F)} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 2278
Rubi steps
\begin {align*} \int \frac {F^{c+d x}}{a+b F^{c+d x}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{a+b x} \, dx,x,F^{c+d x}\right )}{d \log (F)}\\ &=\frac {\log \left (a+b F^{c+d x}\right )}{b d \log (F)}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 30, normalized size = 1.30 \begin {gather*} \frac {\log \left (a d \log (F)+b d F^{c+d x} \log (F)\right )}{b d \log (F)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 24, normalized size = 1.04
method | result | size |
derivativedivides | \(\frac {\ln \left (a +b \,F^{d x +c}\right )}{b d \ln \left (F \right )}\) | \(24\) |
default | \(\frac {\ln \left (a +b \,F^{d x +c}\right )}{b d \ln \left (F \right )}\) | \(24\) |
norman | \(\frac {\ln \left (a +b \,{\mathrm e}^{\left (d x +c \right ) \ln \left (F \right )}\right )}{b \ln \left (F \right ) d}\) | \(26\) |
risch | \(-\frac {c}{b d}+\frac {\ln \left (F^{d x +c}+\frac {a}{b}\right )}{b \ln \left (F \right ) d}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.41, size = 23, normalized size = 1.00 \begin {gather*} \frac {\log \left (F^{d x + c} b + a\right )}{b d \log \left (F\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 23, normalized size = 1.00 \begin {gather*} \frac {\log \left (F^{d x + c} b + a\right )}{b d \log \left (F\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 17, normalized size = 0.74 \begin {gather*} \frac {\log {\left (F^{c + d x} + \frac {a}{b} \right )}}{b d \log {\left (F \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.70, size = 24, normalized size = 1.04 \begin {gather*} \frac {\log \left ({\left | F^{d x + c} b + a \right |}\right )}{b d \log \left (F\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.43, size = 23, normalized size = 1.00 \begin {gather*} \frac {\ln \left (a+F^{c+d\,x}\,b\right )}{b\,d\,\ln \left (F\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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