Optimal. Leaf size=75 \[ \frac{H^{t (r+s x)} \text{Hypergeometric2F1}\left (1,-\frac{s t \log (H)}{d e \log (F)},1-\frac{s t \log (H)}{d e \log (F)},-\frac{a F^{-e (c+d x)}}{b}\right )}{b s t \log (H)} \]
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Rubi [A] time = 0.213812, 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 \[ \frac{H^{t (r+s x)} \text{Hypergeometric2F1}\left (1,-\frac{s t \log (H)}{d e \log (F)},1-\frac{s t \log (H)}{d e \log (F)},-\frac{a F^{-e (c+d x)}}{b}\right )}{b s t \log (H)} \]
Antiderivative was successfully verified.
[In] Int[(F^(e*(c + d*x))*H^(t*(r + s*x)))/(a + b*F^(e*(c + d*x))),x]
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Rubi in Sympy [A] time = 35.7703, size = 60, normalized size = 0.8 \[ \frac{H^{t \left (r + s x\right )}{{}_{2}F_{1}\left (\begin{matrix} 1, - \frac{s t \log{\left (H \right )}}{d e \log{\left (F \right )}} \\ 1 - \frac{s t \log{\left (H \right )}}{d e \log{\left (F \right )}} \end{matrix}\middle |{- \frac{F^{e \left (- c - d x\right )} a}{b}} \right )}}{b s t \log{\left (H \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(F**(e*(d*x+c))*H**(t*(s*x+r))/(a+b*F**(e*(d*x+c))),x)
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Mathematica [A] time = 0.0876424, size = 75, normalized size = 1. \[ -\frac{H^{t (r+s x)} \left (\text{Hypergeometric2F1}\left (1,\frac{s t \log (H)}{d e \log (F)},\frac{s t \log (H)}{d e \log (F)}+1,-\frac{b F^{e (c+d x)}}{a}\right )-1\right )}{b s t \log (H)} \]
Antiderivative was successfully verified.
[In] Integrate[(F^(e*(c + d*x))*H^(t*(r + s*x)))/(a + b*F^(e*(c + d*x))),x]
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Maple [F] time = 0.033, size = 0, normalized size = 0. \[ \int{\frac{{F}^{e \left ( dx+c \right ) }{H}^{t \left ( sx+r \right ) }}{a+b{F}^{e \left ( dx+c \right ) }}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(F^(e*(d*x+c))*H^(t*(s*x+r))/(a+b*F^(e*(d*x+c))),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -H^{r t} a^{2} d e \int \frac{H^{s t x}}{a^{2} b d e \log \left (F\right ) - a^{2} b s t \log \left (H\right ) +{\left (F^{2 \, c e} b^{3} d e \log \left (F\right ) - F^{2 \, c e} b^{3} s t \log \left (H\right )\right )} F^{2 \, d e x} + 2 \,{\left (F^{c e} a b^{2} d e \log \left (F\right ) - F^{c e} a b^{2} s t \log \left (H\right )\right )} F^{d e x}}\,{d x} \log \left (F\right ) + \frac{{\left (H^{r t} a d e \log \left (F\right ) +{\left (F^{c e} H^{r t} b d e \log \left (F\right ) - F^{c e} H^{r t} b s t \log \left (H\right )\right )} F^{d e x}\right )} H^{s t x}}{a b d e s t \log \left (F\right ) \log \left (H\right ) - a b s^{2} t^{2} \log \left (H\right )^{2} +{\left (F^{c e} b^{2} d e s t \log \left (F\right ) \log \left (H\right ) - F^{c e} b^{2} s^{2} t^{2} \log \left (H\right )^{2}\right )} F^{d e x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^((d*x + c)*e)*H^((s*x + r)*t)/(F^((d*x + c)*e)*b + a),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{F^{d e x + c e} H^{s t x + r t}}{F^{d e x + c e} b + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^((d*x + c)*e)*H^((s*x + r)*t)/(F^((d*x + c)*e)*b + a),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{e \left (c + d x\right )} H^{t \left (r + s x\right )}}{F^{c e} F^{d e x} b + a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F**(e*(d*x+c))*H**(t*(s*x+r))/(a+b*F**(e*(d*x+c))),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{{\left (d x + c\right )} e} H^{{\left (s x + r\right )} t}}{F^{{\left (d x + c\right )} e} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^((d*x + c)*e)*H^((s*x + r)*t)/(F^((d*x + c)*e)*b + a),x, algorithm="giac")
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