Integrand size = 22, antiderivative size = 69 \[ \int \frac {F^{c+d x} x}{\left (a+b F^{c+d x}\right )^2} \, dx=\frac {x}{a b d \log (F)}-\frac {x}{b d \left (a+b F^{c+d x}\right ) \log (F)}-\frac {\log \left (a+b F^{c+d x}\right )}{a b d^2 \log ^2(F)} \]
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Time = 0.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2222, 2320, 36, 29, 31} \[ \int \frac {F^{c+d x} x}{\left (a+b F^{c+d x}\right )^2} \, dx=-\frac {\log \left (a+b F^{c+d x}\right )}{a b d^2 \log ^2(F)}-\frac {x}{b d \log (F) \left (a+b F^{c+d x}\right )}+\frac {x}{a b d \log (F)} \]
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Rule 29
Rule 31
Rule 36
Rule 2222
Rule 2320
Rubi steps \begin{align*} \text {integral}& = -\frac {x}{b d \left (a+b F^{c+d x}\right ) \log (F)}+\frac {\int \frac {1}{a+b F^{c+d x}} \, dx}{b d \log (F)} \\ & = -\frac {x}{b d \left (a+b F^{c+d x}\right ) \log (F)}+\frac {\text {Subst}\left (\int \frac {1}{x (a+b x)} \, dx,x,F^{c+d x}\right )}{b d^2 \log ^2(F)} \\ & = -\frac {x}{b d \left (a+b F^{c+d x}\right ) \log (F)}-\frac {\text {Subst}\left (\int \frac {1}{a+b x} \, dx,x,F^{c+d x}\right )}{a d^2 \log ^2(F)}+\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,F^{c+d x}\right )}{a b d^2 \log ^2(F)} \\ & = \frac {x}{a b d \log (F)}-\frac {x}{b d \left (a+b F^{c+d x}\right ) \log (F)}-\frac {\log \left (a+b F^{c+d x}\right )}{a b d^2 \log ^2(F)} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.78 \[ \int \frac {F^{c+d x} x}{\left (a+b F^{c+d x}\right )^2} \, dx=\frac {\frac {d F^{c+d x} x \log (F)}{a+b F^{c+d x}}-\frac {\log \left (a+b F^{c+d x}\right )}{b}}{a d^2 \log ^2(F)} \]
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Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.97
method | result | size |
norman | \(\frac {x \,{\mathrm e}^{\left (d x +c \right ) \ln \left (F \right )}}{d \ln \left (F \right ) a \left (a +b \,{\mathrm e}^{\left (d x +c \right ) \ln \left (F \right )}\right )}-\frac {\ln \left (a +b \,{\mathrm e}^{\left (d x +c \right ) \ln \left (F \right )}\right )}{\ln \left (F \right )^{2} b \,d^{2} a}\) | \(67\) |
parallelrisch | \(\frac {F^{d x +c} x b d \ln \left (F \right )-\ln \left (a +b \,F^{d x +c}\right ) F^{d x +c} b -\ln \left (a +b \,F^{d x +c}\right ) a}{\ln \left (F \right )^{2} a b \,d^{2} \left (a +b \,F^{d x +c}\right )}\) | \(79\) |
risch | \(\frac {x}{a b d \ln \left (F \right )}+\frac {c}{\ln \left (F \right ) b \,d^{2} a}-\frac {x}{b d \left (a +b \,F^{d x +c}\right ) \ln \left (F \right )}-\frac {\ln \left (F^{d x +c}+\frac {a}{b}\right )}{\ln \left (F \right )^{2} b \,d^{2} a}\) | \(87\) |
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Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.07 \[ \int \frac {F^{c+d x} x}{\left (a+b F^{c+d x}\right )^2} \, dx=\frac {F^{d x + c} b d x \log \left (F\right ) - {\left (F^{d x + c} b + a\right )} \log \left (F^{d x + c} b + a\right )}{F^{d x + c} a b^{2} d^{2} \log \left (F\right )^{2} + a^{2} b d^{2} \log \left (F\right )^{2}} \]
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Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.84 \[ \int \frac {F^{c+d x} x}{\left (a+b F^{c+d x}\right )^2} \, dx=- \frac {x}{F^{c + d x} b^{2} d \log {\left (F \right )} + a b d \log {\left (F \right )}} + \frac {x}{a b d \log {\left (F \right )}} - \frac {\log {\left (F^{c + d x} + \frac {a}{b} \right )}}{a b d^{2} \log {\left (F \right )}^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.04 \[ \int \frac {F^{c+d x} x}{\left (a+b F^{c+d x}\right )^2} \, dx=\frac {F^{d x} F^{c} x}{F^{d x} F^{c} a b d \log \left (F\right ) + a^{2} d \log \left (F\right )} - \frac {\log \left (\frac {F^{d x} F^{c} b + a}{F^{c} b}\right )}{a b d^{2} \log \left (F\right )^{2}} \]
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\[ \int \frac {F^{c+d x} x}{\left (a+b F^{c+d x}\right )^2} \, dx=\int { \frac {F^{d x + c} x}{{\left (F^{d x + c} b + a\right )}^{2}} \,d x } \]
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Time = 0.26 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.91 \[ \int \frac {F^{c+d x} x}{\left (a+b F^{c+d x}\right )^2} \, dx=\frac {F^c\,F^{d\,x}\,x}{a\,d\,\ln \left (F\right )\,\left (a+F^c\,F^{d\,x}\,b\right )}-\frac {\ln \left (a+F^c\,F^{d\,x}\,b\right )}{a\,b\,d^2\,{\ln \left (F\right )}^2} \]
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