Optimal. Leaf size=69 \[ -\frac{\log \left (a+b F^{c+d x}\right )}{a b d^2 \log ^2(F)}-\frac{x}{b d \log (F) \left (a+b F^{c+d x}\right )}+\frac{x}{a b d \log (F)} \]
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Rubi [A] time = 0.0738341, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {2191, 2282, 36, 29, 31} \[ -\frac{\log \left (a+b F^{c+d x}\right )}{a b d^2 \log ^2(F)}-\frac{x}{b d \log (F) \left (a+b F^{c+d x}\right )}+\frac{x}{a b d \log (F)} \]
Antiderivative was successfully verified.
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Rule 2191
Rule 2282
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{F^{c+d x} x}{\left (a+b F^{c+d x}\right )^2} \, dx &=-\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{\operatorname{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{\operatorname{Subst}\left (\int \frac{1}{a+b x} \, dx,x,F^{c+d x}\right )}{a d^2 \log ^2(F)}+\frac{\operatorname{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*}
Mathematica [A] time = 0.0701456, size = 54, normalized size = 0.78 \[ \frac{\frac{d x \log (F) F^{c+d x}}{a+b F^{c+d x}}-\frac{\log \left (a+b F^{c+d x}\right )}{b}}{a d^2 \log ^2(F)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 67, normalized size = 1. \begin{align*}{\frac{x{{\rm e}^{ \left ( dx+c \right ) \ln \left ( F \right ) }}}{\ln \left ( F \right ) ad \left ( a+b{{\rm e}^{ \left ( dx+c \right ) \ln \left ( F \right ) }} \right ) }}-{\frac{\ln \left ( a+b{{\rm e}^{ \left ( dx+c \right ) \ln \left ( F \right ) }} \right ) }{ \left ( \ln \left ( F \right ) \right ) ^{2}ab{d}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0704, size = 97, normalized size = 1.41 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51882, size = 171, normalized size = 2.48 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.169933, size = 58, normalized size = 0.84 \begin{align*} - \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}} \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 \frac{F^{d x + c} x}{{\left (F^{d x + c} b + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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