Optimal. Leaf size=53 \[ \frac{b F^a \log (F) \text{Ei}\left (b (c+d x)^2 \log (F)\right )}{2 d}-\frac{F^{a+b (c+d x)^2}}{2 d (c+d x)^2} \]
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Rubi [A] time = 0.132028, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2214, 2210} \[ \frac{b F^a \log (F) \text{Ei}\left (b (c+d x)^2 \log (F)\right )}{2 d}-\frac{F^{a+b (c+d x)^2}}{2 d (c+d x)^2} \]
Antiderivative was successfully verified.
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Rule 2214
Rule 2210
Rubi steps
\begin{align*} \int \frac{F^{a+b (c+d x)^2}}{(c+d x)^3} \, dx &=-\frac{F^{a+b (c+d x)^2}}{2 d (c+d x)^2}+(b \log (F)) \int \frac{F^{a+b (c+d x)^2}}{c+d x} \, dx\\ &=-\frac{F^{a+b (c+d x)^2}}{2 d (c+d x)^2}+\frac{b F^a \text{Ei}\left (b (c+d x)^2 \log (F)\right ) \log (F)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0393825, size = 47, normalized size = 0.89 \[ \frac{F^a \left (b \log (F) \text{Ei}\left (b (c+d x)^2 \log (F)\right )-\frac{F^{b (c+d x)^2}}{(c+d x)^2}\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 53, normalized size = 1. \begin{align*} -{\frac{{F}^{b \left ( dx+c \right ) ^{2}}{F}^{a}}{2\,d \left ( dx+c \right ) ^{2}}}-{\frac{b\ln \left ( F \right ){F}^{a}{\it Ei} \left ( 1,-b \left ( dx+c \right ) ^{2}\ln \left ( F \right ) \right ) }{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.53328, size = 220, normalized size = 4.15 \begin{align*} \frac{{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} F^{a}{\rm Ei}\left ({\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \left (F\right )\right ) \log \left (F\right ) - F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{2 \,{\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{a + b \left (c + d x\right )^{2}}}{\left (c + d x\right )^{3}}\, dx \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^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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