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 = 0.07, antiderivative size = 69, normalized size of antiderivative = 1.00, 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 29
Rule 31
Rule 36
Rule 2191
Rule 2282
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*}
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Mathematica [A] time = 0.07, 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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fricas [A] time = 0.42, size = 74, normalized size = 1.07 \[ \frac {F^{d x + c} b d x \log \relax (F) - {\left (F^{d x + c} b + a\right )} \log \left (F^{d x + c} b + a\right )}{F^{d x + c} a b^{2} d^{2} \log \relax (F)^{2} + a^{2} b d^{2} \log \relax (F)^{2}} \]
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 \[ \int \frac {F^{d x + c} x}{{\left (F^{d x + c} b + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 67, normalized size = 0.97 \[ \frac {x \,{\mathrm e}^{\left (d x +c \right ) \ln \relax (F )}}{\left (b \,{\mathrm e}^{\left (d x +c \right ) \ln \relax (F )}+a \right ) a d \ln \relax (F )}-\frac {\ln \left (b \,{\mathrm e}^{\left (d x +c \right ) \ln \relax (F )}+a \right )}{a b \,d^{2} \ln \relax (F )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 72, normalized size = 1.04 \[ \frac {F^{d x} F^{c} x}{F^{d x} F^{c} a b d \log \relax (F) + a^{2} d \log \relax (F)} - \frac {\log \left (\frac {F^{d x} F^{c} b + a}{F^{c} b}\right )}{a b d^{2} \log \relax (F)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.62, size = 63, normalized size = 0.91 \[ \frac {F^c\,F^{d\,x}\,x}{a\,d\,\ln \relax (F)\,\left (a+F^c\,F^{d\,x}\,b\right )}-\frac {\ln \left (a+F^c\,F^{d\,x}\,b\right )}{a\,b\,d^2\,{\ln \relax (F)}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 58, normalized size = 0.84 \[ - \frac {x}{F^{c + d x} b^{2} d \log {\relax (F )} + a b d \log {\relax (F )}} + \frac {x}{a b d \log {\relax (F )}} - \frac {\log {\left (F^{c + d x} + \frac {a}{b} \right )}}{a b d^{2} \log {\relax (F )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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