Optimal. Leaf size=60 \[ -\frac{2 \text{Unintegrable}\left (\frac{1}{x^3 \left (a+b F^{c+d x}\right )},x\right )}{b d \log (F)}-\frac{1}{b d x^2 \log (F) \left (a+b F^{c+d x}\right )} \]
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Rubi [A] time = 0.117562, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{F^{c+d x}}{\left (a+b F^{c+d x}\right )^2 x^2} \, dx \]
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^2} \, dx &=-\frac{1}{b d \left (a+b F^{c+d x}\right ) x^2 \log (F)}-\frac{2 \int \frac{1}{\left (a+b F^{c+d x}\right ) x^3} \, dx}{b d \log (F)}\\ \end{align*}
Mathematica [A] time = 0.153874, size = 0, normalized size = 0. \[ \int \frac{F^{c+d x}}{\left (a+b F^{c+d x}\right )^2 x^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.058, size = 0, normalized size = 0. \begin{align*} \int{\frac{{F}^{dx+c}}{ \left ( a+b{F}^{dx+c} \right ) ^{2}{x}^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{F^{d x} F^{c} b^{2} d x^{2} \log \left (F\right ) + a b d x^{2} \log \left (F\right )} - 2 \, \int \frac{1}{F^{d x} F^{c} b^{2} d x^{3} \log \left (F\right ) + a b d x^{3} \log \left (F\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{F^{d x + c}}{2 \, F^{d x + c} a b x^{2} + F^{2 \, d x + 2 \, c} b^{2} x^{2} + a^{2} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{1}{F^{c + d x} b^{2} d x^{2} \log{\left (F \right )} + a b d x^{2} \log{\left (F \right )}} - \frac{2 \int \frac{1}{a x^{3} + b x^{3} e^{c \log{\left (F \right )}} e^{d x \log{\left (F \right )}}}\, dx}{b d \log{\left (F \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{d x + c}}{{\left (F^{d x + c} b + a\right )}^{2} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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