Optimal. Leaf size=61 \[ -\frac {\text {Int}\left (\frac {1}{x^2 \left (a+b F^{c+d x}\right )},x\right )}{b d \log (F)}-\frac {1}{b d x \log (F) \left (a+b F^{c+d x}\right )} \]
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Rubi [A] time = 0.12, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^2 x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^2 x} \, dx &=-\frac {1}{b d \left (a+b F^{c+d x}\right ) x \log (F)}-\frac {\int \frac {1}{\left (a+b F^{c+d x}\right ) x^2} \, dx}{b d \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^2 x} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {F^{d x + c}}{2 \, F^{d x + c} a b x + F^{2 \, d x + 2 \, c} b^{2} x + a^{2} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{d x + c}}{{\left (F^{d x + c} b + a\right )}^{2} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {F^{d x +c}}{\left (b \,F^{d x +c}+a \right )^{2} x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{F^{d x} F^{c} b^{2} d x \log \relax (F) + a b d x \log \relax (F)} - \int \frac {1}{F^{d x} F^{c} b^{2} d x^{2} \log \relax (F) + a b d x^{2} \log \relax (F)}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {F^{c+d\,x}}{x\,{\left (a+F^{c+d\,x}\,b\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {1}{F^{c + d x} b^{2} d x \log {\relax (F )} + a b d x \log {\relax (F )}} - \frac {\int \frac {1}{a x^{2} + b x^{2} e^{c \log {\relax (F )}} e^{d x \log {\relax (F )}}}\, dx}{b d \log {\relax (F )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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