Optimal. Leaf size=75 \[ \frac{H^{t (r+s x)} \, _2F_1\left (1,-\frac{s t \log (H)}{d e \log (F)};1-\frac{s t \log (H)}{d e \log (F)};-\frac{a F^{-e (c+d x)}}{b}\right )}{b s t \log (H)} \]
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Rubi [A] time = 0.134529, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2256, 2251} \[ \frac{H^{t (r+s x)} \, _2F_1\left (1,-\frac{s t \log (H)}{d e \log (F)};1-\frac{s t \log (H)}{d e \log (F)};-\frac{a F^{-e (c+d x)}}{b}\right )}{b s t \log (H)} \]
Antiderivative was successfully verified.
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Rule 2256
Rule 2251
Rubi steps
\begin{align*} \int \frac{F^{e (c+d x)} H^{t (r+s x)}}{a+b F^{e (c+d x)}} \, dx &=\int \frac{H^{t (r+s x)}}{b+a F^{-e (c+d x)}} \, dx\\ &=\frac{H^{t (r+s x)} \, _2F_1\left (1,-\frac{s t \log (H)}{d e \log (F)};1-\frac{s t \log (H)}{d e \log (F)};-\frac{a F^{-e (c+d x)}}{b}\right )}{b s t \log (H)}\\ \end{align*}
Mathematica [A] time = 0.16019, size = 75, normalized size = 1. \[ -\frac{H^{t (r+s x)} \left (\, _2F_1\left (1,\frac{s t \log (H)}{d e \log (F)};\frac{s t \log (H)}{d e \log (F)}+1;-\frac{b F^{e (c+d x)}}{a}\right )-1\right )}{b s t \log (H)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.025, size = 0, normalized size = 0. \begin{align*} \int{\frac{{F}^{e \left ( dx+c \right ) }{H}^{t \left ( sx+r \right ) }}{a+b{F}^{e \left ( dx+c \right ) }}}\, dx \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*} -H^{r t} a^{2} d e \int \frac{H^{s t x}}{a^{2} b d e \log \left (F\right ) - a^{2} b s t \log \left (H\right ) +{\left (F^{2 \, c e} b^{3} d e \log \left (F\right ) - F^{2 \, c e} b^{3} s t \log \left (H\right )\right )} F^{2 \, d e x} + 2 \,{\left (F^{c e} a b^{2} d e \log \left (F\right ) - F^{c e} a b^{2} s t \log \left (H\right )\right )} F^{d e x}}\,{d x} \log \left (F\right ) + \frac{{\left (H^{r t} a d e \log \left (F\right ) +{\left (F^{c e} H^{r t} b d e \log \left (F\right ) - F^{c e} H^{r t} b s t \log \left (H\right )\right )} F^{d e x}\right )} H^{s t x}}{a b d e s t \log \left (F\right ) \log \left (H\right ) - a b s^{2} t^{2} \log \left (H\right )^{2} +{\left (F^{c e} b^{2} d e s t \log \left (F\right ) \log \left (H\right ) - F^{c e} b^{2} s^{2} t^{2} \log \left (H\right )^{2}\right )} F^{d e x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{F^{d e x + c e} H^{s t x + r t}}{F^{d e x + c e} b + a}, x\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^{e \left (c + d x\right )} H^{t \left (r + s x\right )}}{F^{c e} F^{d e x} b + a}\, 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 )} e} H^{{\left (s x + r\right )} t}}{F^{{\left (d x + c\right )} e} b + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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