3.67 \(\int \frac{F^{e (c+d x)} H^{t (r+s x)}}{a+b F^{e (c+d x)}} \, dx\)

Optimal. Leaf size=75 \[ \frac{H^{t (r+s x)} \text{Hypergeometric2F1}\left (1,-\frac{s t \log (H)}{d e \log (F)},1-\frac{s t \log (H)}{d e \log (F)},-\frac{a F^{-e (c+d x)}}{b}\right )}{b s t \log (H)} \]

[Out]

(H^(t*(r + s*x))*Hypergeometric2F1[1, -((s*t*Log[H])/(d*e*Log[F])), 1 - (s*t*Log
[H])/(d*e*Log[F]), -(a/(b*F^(e*(c + d*x))))])/(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]

[Out]

(H^(t*(r + s*x))*Hypergeometric2F1[1, -((s*t*Log[H])/(d*e*Log[F])), 1 - (s*t*Log
[H])/(d*e*Log[F]), -(a/(b*F^(e*(c + d*x))))])/(b*s*t*Log[H])

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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)

[Out]

H**(t*(r + s*x))*hyper((1, -s*t*log(H)/(d*e*log(F))), (1 - s*t*log(H)/(d*e*log(F
)),), -F**(e*(-c - d*x))*a/b)/(b*s*t*log(H))

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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]

[Out]

-((H^(t*(r + s*x))*(-1 + Hypergeometric2F1[1, (s*t*Log[H])/(d*e*Log[F]), 1 + (s*
t*Log[H])/(d*e*Log[F]), -((b*F^(e*(c + d*x)))/a)]))/(b*s*t*Log[H]))

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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)

[Out]

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")

[Out]

-H^(r*t)*a^2*d*e*integrate(H^(s*t*x)/(a^2*b*d*e*log(F) - a^2*b*s*t*log(H) + (F^(
2*c*e)*b^3*d*e*log(F) - F^(2*c*e)*b^3*s*t*log(H))*F^(2*d*e*x) + 2*(F^(c*e)*a*b^2
*d*e*log(F) - F^(c*e)*a*b^2*s*t*log(H))*F^(d*e*x)), x)*log(F) + (H^(r*t)*a*d*e*l
og(F) + (F^(c*e)*H^(r*t)*b*d*e*log(F) - F^(c*e)*H^(r*t)*b*s*t*log(H))*F^(d*e*x))
*H^(s*t*x)/(a*b*d*e*s*t*log(F)*log(H) - a*b*s^2*t^2*log(H)^2 + (F^(c*e)*b^2*d*e*
s*t*log(F)*log(H) - F^(c*e)*b^2*s^2*t^2*log(H)^2)*F^(d*e*x))

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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")

[Out]

integral(F^(d*e*x + c*e)*H^(s*t*x + r*t)/(F^(d*e*x + c*e)*b + a), x)

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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)

[Out]

Integral(F**(e*(c + d*x))*H**(t*(r + s*x))/(F**(c*e)*F**(d*e*x)*b + a), 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")

[Out]

integrate(F^((d*x + c)*e)*H^((s*x + r)*t)/(F^((d*x + c)*e)*b + a), x)