\(\int \frac {F^{e (c+d x)} H^{t (r+s x)}}{a+b F^{e (c+d x)}} \, dx\) [67]

Optimal result

Integrand size = 34, antiderivative size = 75 \[ \int \frac {F^{e (c+d x)} H^{t (r+s x)}}{a+b F^{e (c+d x)}} \, dx=\frac {H^{t (r+s x)} \operatorname {Hypergeometric2F1}\left (1,-\frac {s t \log (H)}{d e \log (F)},1-\frac {s t \log (H)}{d e \log (F)},-\frac {a F^{-e (c+d x)}}{b}\right )}{b s t \log (H)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2288, 2283} \[ \int \frac {F^{e (c+d x)} H^{t (r+s x)}}{a+b F^{e (c+d x)}} \, dx=\frac {H^{t (r+s x)} \operatorname {Hypergeometric2F1}\left (1,-\frac {s t \log (H)}{d e \log (F)},1-\frac {s t \log (H)}{d e \log (F)},-\frac {a F^{-e (c+d x)}}{b}\right )}{b s t \log (H)} \]

[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])

Rule 2283

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Simp
[a^p*(G^(h*(f + g*x))/(g*h*Log[G]))*Hypergeometric2F1[-p, g*h*(Log[G]/(d*e*Log[F])), g*h*(Log[G]/(d*e*Log[F]))
+ 1, Simplify[(-b/a)*F^(e*(c + d*x))]], x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x] && (ILtQ[p, 0] || G
tQ[a, 0])

Rule 2288

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_.)*(G_)^((h_.)*((f_.) + (g_.)*(x_)))*(H_)^((t_.)*((r_.
) + (s_.)*(x_))), x_Symbol] :> Dist[G^((f - c*(g/d))*h), Int[H^(t*(r + s*x))*(b + a/F^(e*(c + d*x)))^p, x], x]
/; FreeQ[{F, G, H, a, b, c, d, e, f, g, h, r, s, t}, x] && EqQ[d*e*p*Log[F] + g*h*Log[G], 0] && IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00 \[ \int \frac {F^{e (c+d x)} H^{t (r+s x)}}{a+b F^{e (c+d x)}} \, dx=-\frac {H^{t (r+s x)} \left (-1+\operatorname {Hypergeometric2F1}\left (1,\frac {s t \log (H)}{d e \log (F)},1+\frac {s t \log (H)}{d e \log (F)},-\frac {b F^{e (c+d x)}}{a}\right )\right )}{b s t \log (H)} \]

[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]))

Maple [F]

\[\int \frac {F^{e \left (d x +c \right )} H^{t \left (s x +r \right )}}{a +b \,F^{e \left (d x +c \right )}}d x\]

[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)

Fricas [F]

\[ \int \frac {F^{e (c+d x)} H^{t (r+s x)}}{a+b F^{e (c+d x)}} \, dx=\int { \frac {F^{{\left (d x + c\right )} e} H^{{\left (s x + r\right )} t}}{F^{{\left (d x + c\right )} e} b + a} \,d x } \]

[In]

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

Sympy [F]

\[ \int \frac {F^{e (c+d x)} H^{t (r+s x)}}{a+b F^{e (c+d x)}} \, dx=\int \frac {F^{e \left (c + d x\right )} H^{t \left (r + s x\right )}}{F^{c e + d e x} b + a}\, dx \]

[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 + d*e*x)*b + a), x)

Maxima [F]

\[ \int \frac {F^{e (c+d x)} H^{t (r+s x)}}{a+b F^{e (c+d x)}} \, dx=\int { \frac {F^{{\left (d x + c\right )} e} H^{{\left (s x + r\right )} t}}{F^{{\left (d x + c\right )} e} b + a} \,d x } \]

[In]

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

Giac [F]

\[ \int \frac {F^{e (c+d x)} H^{t (r+s x)}}{a+b F^{e (c+d x)}} \, dx=\int { \frac {F^{{\left (d x + c\right )} e} H^{{\left (s x + r\right )} t}}{F^{{\left (d x + c\right )} e} b + a} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {F^{e (c+d x)} H^{t (r+s x)}}{a+b F^{e (c+d x)}} \, dx=\int \frac {F^{e\,\left (c+d\,x\right )}\,H^{t\,\left (r+s\,x\right )}}{a+F^{e\,\left (c+d\,x\right )}\,b} \,d x \]

[In]

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

[Out]

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