Optimal. Leaf size=62 \[ -\frac{\text{Unintegrable}\left (\frac{1}{x^3 \left (a+b F^{c+d x}\right )^2},x\right )}{b d \log (F)}-\frac{1}{2 b d x^2 \log (F) \left (a+b F^{c+d x}\right )^2} \]
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Rubi [A] time = 0.113154, 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 = {} \[ \int \frac{F^{c+d x}}{\left (a+b F^{c+d x}\right )^3 x^2} \, dx \]
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 )^3 x^2} \, dx &=-\frac{1}{2 b d \left (a+b F^{c+d x}\right )^2 x^2 \log (F)}-\frac{\int \frac{1}{\left (a+b F^{c+d x}\right )^2 x^3} \, dx}{b d \log (F)}\\ \end{align*}
Mathematica [A] time = 0.72888, size = 0, normalized size = 0. \[ \int \frac{F^{c+d x}}{\left (a+b F^{c+d x}\right )^3 x^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.185, size = 0, normalized size = 0. \begin{align*} \int{\frac{{F}^{dx+c}}{ \left ( a+b{F}^{dx+c} \right ) ^{3}{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{a d x \log \left (F\right ) + 2 \, F^{d x} F^{c} b + 2 \, a}{2 \,{\left (2 \, F^{d x} F^{c} a^{2} b^{2} d^{2} x^{3} \log \left (F\right )^{2} + F^{2 \, d x} F^{2 \, c} a b^{3} d^{2} x^{3} \log \left (F\right )^{2} + a^{3} b d^{2} x^{3} \log \left (F\right )^{2}\right )}} - \int \frac{d x \log \left (F\right ) + 3}{F^{d x} F^{c} a b^{2} d^{2} x^{4} \log \left (F\right )^{2} + a^{2} b d^{2} x^{4} \log \left (F\right )^{2}}\,{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}}{3 \, F^{d x + c} a^{2} b x^{2} + 3 \, F^{2 \, d x + 2 \, c} a b^{2} x^{2} + F^{3 \, d x + 3 \, c} b^{3} x^{2} + a^{3} 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{- 2 F^{c + d x} b - a d x \log{\left (F \right )} - 2 a}{4 F^{c + d x} a^{2} b^{2} d^{2} x^{3} \log{\left (F \right )}^{2} + 2 F^{2 c + 2 d x} a b^{3} d^{2} x^{3} \log{\left (F \right )}^{2} + 2 a^{3} b d^{2} x^{3} \log{\left (F \right )}^{2}} - \frac{\int \frac{d x \log{\left (F \right )}}{a x^{4} + b x^{4} e^{c \log{\left (F \right )}} e^{d x \log{\left (F \right )}}}\, dx + \int \frac{3}{a x^{4} + b x^{4} e^{c \log{\left (F \right )}} e^{d x \log{\left (F \right )}}}\, dx}{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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{d x + c}}{{\left (F^{d x + c} b + a\right )}^{3} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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