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

Optimal. Leaf size=85 \[ \frac{F^{-e (c-f)} H^{t (r+s x)} \, _2F_1\left (1,-\frac{s t \log (H)}{d e \log (F)};1-\frac{s t \log (H)}{d e \log (F)};-\frac{a F^{-e (c+d x)}}{b}\right )}{b s t \log (H)} \]

[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*F^(e*(c - f))*s*t*Log[H])

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Rubi [A]  time = 0.132062, antiderivative size = 85, 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, 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.

[In]

Int[(F^(e*(f + 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*F^(e*(c - f))*s*t*Log[H])

Rule 2256

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]

Rule 2251

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

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*}

Mathematica [A]  time = 0.143593, 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.

[In]

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

[Out]

-((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]))

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \begin{align*} \int{\frac{{F}^{e \left ( dx+f \right ) }{H}^{t \left ( sx+r \right ) }}{a+b{F}^{e \left ( dx+c \right ) }}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(F^(e*(d*x+f))*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. \begin{align*} -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 \left (F\right ) - F^{c e} a^{2} b s t \log \left (H\right ) +{\left (F^{3 \, c e} b^{3} d e \log \left (F\right ) - F^{3 \, c e} b^{3} s t \log \left (H\right )\right )} F^{2 \, d e x} + 2 \,{\left (F^{2 \, c e} a b^{2} d e \log \left (F\right ) - F^{2 \, c e} a b^{2} s t \log \left (H\right )\right )} F^{d e x}}\,{d x} \log \left (F\right ) + \frac{{\left (F^{e f} H^{r t} a d e \log \left (F\right ) +{\left (F^{c e + e f} H^{r t} b d e \log \left (F\right ) - F^{c e + e f} H^{r t} b s t \log \left (H\right )\right )} F^{d e x}\right )} H^{s t x}}{F^{c e} a b d e s t \log \left (F\right ) \log \left (H\right ) - F^{c e} a b s^{2} t^{2} \log \left (H\right )^{2} +{\left (F^{2 \, c e} b^{2} d e s t \log \left (F\right ) \log \left (H\right ) - F^{2 \, c e} b^{2} s^{2} t^{2} \log \left (H\right )^{2}\right )} F^{d e x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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*log(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*
log(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))

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(F^(d*e*x + e*f)*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. \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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