\(\int \frac {F^{e (f+d x)} H^{t (r+s x)}}{a+b F^{e (c+d x)}} \, dx\) [124]

Optimal result

Integrand size = 34, antiderivative size = 85 \[ \int \frac {F^{e (f+d x)} H^{t (r+s x)}}{a+b F^{e (c+d x)}} \, dx=\frac {F^{-e (c-f)} H^{t (r+s x)} \operatorname {Hypergeometric2F1}\left (1,-\frac {s t \log (H)}{d e \log (F)},1-\frac {s t \log (H)}{d e \log (F)},-\frac {a F^{-e (c+d x)}}{b}\right )}{b s t \log (H)} \] Output:

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/(F^(e*(c-f)))/s/t/ln(H)
 

Mathematica [A] (verified)

Time = 0.85 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.99 \[ \int \frac {F^{e (f+d x)} H^{t (r+s x)}}{a+b F^{e (c+d x)}} \, dx=-\frac {F^{e (-c+f)} H^{t (r+s x)} \left (-1+\operatorname {Hypergeometric2F1}\left (1,\frac {s t \log (H)}{d e \log (F)},1+\frac {s t \log (H)}{d e \log (F)},-\frac {b F^{e (c+d x)}}{a}\right )\right )}{b s t \log (H)} \] Input:

Integrate[(F^(e*(f + d*x))*H^(t*(r + s*x)))/(a + b*F^(e*(c + d*x))),x]
 

Output:

-((F^(e*(-c + f))*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]))
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 85, 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 = {2686, 2681}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2686

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

\(\Big \downarrow \) 2681

\(\displaystyle \frac {F^{-e (c-f)} H^{t (r+s x)} \operatorname {Hypergeometric2F1}\left (1,-\frac {s t \log (H)}{d e \log (F)},1-\frac {s t \log (H)}{d e \log (F)},-\frac {a F^{-e (c+d x)}}{b}\right )}{b s t \log (H)}\)

Input:

Int[(F^(e*(f + d*x))*H^(t*(r + s*x)))/(a + b*F^(e*(c + d*x))),x]
 

Output:

(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*F^(e*(c - f))*s*t*L 
og[H])
 

Defintions of rubi rules used

rule 2681

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]))*Hype 
rgeometric2F1[-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] || GtQ[a, 0])
 

rule 2686

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_.)*(G_)^((h_.)*((f_ 
.) + (g_.)*(x_)))*(H_)^((t_.)*((r_.) + (s_.)*(x_))), x_Symbol] :> Simp[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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {F^{e (f+d x)} H^{t (r+s x)}}{a+b F^{e (c+d x)}} \, dx=\frac {h^{r t} f^{e f} \left (h^{s t x}-\left (\int \frac {h^{s t x}}{f^{d e x +c e} b +a}d x \right ) \mathrm {log}\left (h \right ) a s t \right )}{f^{c e} \mathrm {log}\left (h \right ) b s t} \] Input:

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

Output:

(h**(r*t)*f**(e*f)*(h**(s*t*x) - int(h**(s*t*x)/(f**(c*e + d*e*x)*b + a),x 
)*log(h)*a*s*t))/(f**(c*e)*log(h)*b*s*t)