Optimal. Leaf size=85 \[ \frac {F^{-e (c-f)} H^{t (r+s x)} \, _2F_1\left (1,-\frac {s t \log (H)}{d e \log (F)};1-\frac {s t \log (H)}{d e \log (F)};-\frac {a F^{-e (c+d x)}}{b}\right )}{b s t \log (H)} \]
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Rubi [A] time = 0.13, antiderivative size = 85, normalized size of antiderivative = 1.00, 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} \[ \frac {F^{-e (c-f)} H^{t (r+s x)} \, _2F_1\left (1,-\frac {s t \log (H)}{d e \log (F)};1-\frac {s t \log (H)}{d e \log (F)};-\frac {a F^{-e (c+d x)}}{b}\right )}{b s t \log (H)} \]
Antiderivative was successfully verified.
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Rule 2251
Rule 2256
Rubi steps
\begin {align*} \int \frac {F^{e (f+d x)} H^{t (r+s x)}}{a+b F^{e (c+d x)}} \, dx &=F^{-e (c-f)} \int \frac {H^{t (r+s x)}}{b+a F^{-e (c+d x)}} \, dx\\ &=\frac {F^{-e (c-f)} 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*}
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Mathematica [A] time = 0.15, size = 84, normalized size = 0.99 \[ -\frac {F^{e (f-c)} H^{t (r+s x)} \left (\, _2F_1\left (1,\frac {s t \log (H)}{d e \log (F)};\frac {s t \log (H)}{d e \log (F)}+1;-\frac {b F^{e (c+d x)}}{a}\right )-1\right )}{b s t \log (H)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {F^{d e x + e f} H^{s t x + r t}}{F^{d e x + c e} b + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{{\left (d x + f\right )} e} H^{{\left (s x + r\right )} t}}{F^{{\left (d x + c\right )} e} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {F^{\left (d x +f \right ) e} H^{\left (s x +r \right ) t}}{b \,F^{\left (d x +c \right ) e}+a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -F^{e f} H^{r t} a^{2} d e \int \frac {H^{s t x}}{F^{c e} a^{2} b d e \log \relax (F) - F^{c e} a^{2} b s t \log \relax (H) + {\left (F^{3 \, c e} b^{3} d e \log \relax (F) - F^{3 \, c e} b^{3} s t \log \relax (H)\right )} F^{2 \, d e x} + 2 \, {\left (F^{2 \, c e} a b^{2} d e \log \relax (F) - F^{2 \, c e} a b^{2} s t \log \relax (H)\right )} F^{d e x}}\,{d x} \log \relax (F) + \frac {{\left (F^{e f} H^{r t} a d e \log \relax (F) + {\left (F^{c e + e f} H^{r t} b d e \log \relax (F) - F^{c e + e f} H^{r t} b s t \log \relax (H)\right )} F^{d e x}\right )} H^{s t x}}{F^{c e} a b d e s t \log \relax (F) \log \relax (H) - F^{c e} a b s^{2} t^{2} \log \relax (H)^{2} + {\left (F^{2 \, c e} b^{2} d e s t \log \relax (F) \log \relax (H) - F^{2 \, c e} b^{2} s^{2} t^{2} \log \relax (H)^{2}\right )} F^{d e x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {F^{e\,\left (f+d\,x\right )}\,H^{t\,\left (r+s\,x\right )}}{a+F^{e\,\left (c+d\,x\right )}\,b} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{e \left (d x + f\right )} H^{t \left (r + s x\right )}}{F^{c e} F^{d e x} b + a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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