\(\int F^{a+\frac {b}{c+d x}} \, dx\) [306]

Optimal result

Integrand size = 13, antiderivative size = 46 \[ \int F^{a+\frac {b}{c+d x}} \, dx=\frac {F^{a+\frac {b}{c+d x}} (c+d x)}{d}-\frac {b F^a \operatorname {ExpIntegralEi}\left (\frac {b \log (F)}{c+d x}\right ) \log (F)}{d} \]

[Out]

F^(a+b/(d*x+c))*(d*x+c)/d-b*F^a*Ei(b*ln(F)/(d*x+c))*ln(F)/d

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2237, 2241} \[ \int F^{a+\frac {b}{c+d x}} \, dx=\frac {(c+d x) F^{a+\frac {b}{c+d x}}}{d}-\frac {b F^a \log (F) \operatorname {ExpIntegralEi}\left (\frac {b \log (F)}{c+d x}\right )}{d} \]

[In]

Int[F^(a + b/(c + d*x)),x]

[Out]

(F^(a + b/(c + d*x))*(c + d*x))/d - (b*F^a*ExpIntegralEi[(b*Log[F])/(c + d*x)]*Log[F])/d

Rule 2237

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[(c + d*x)*(F^(a + b*(c + d*x)^n)/d), x]
- Dist[b*n*Log[F], Int[(c + d*x)^n*F^(a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2/n]
&& ILtQ[n, 0]

Rule 2241

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[F^a*(ExpIntegralEi[
b*(c + d*x)^n*Log[F]]/(f*n)), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {F^{a+\frac {b}{c+d x}} (c+d x)}{d}+(b \log (F)) \int \frac {F^{a+\frac {b}{c+d x}}}{c+d x} \, dx \\ & = \frac {F^{a+\frac {b}{c+d x}} (c+d x)}{d}-\frac {b F^a \text {Ei}\left (\frac {b \log (F)}{c+d x}\right ) \log (F)}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.91 \[ \int F^{a+\frac {b}{c+d x}} \, dx=\frac {F^a \left (F^{\frac {b}{c+d x}} (c+d x)-b \operatorname {ExpIntegralEi}\left (\frac {b \log (F)}{c+d x}\right ) \log (F)\right )}{d} \]

[In]

Integrate[F^(a + b/(c + d*x)),x]

[Out]

(F^a*(F^(b/(c + d*x))*(c + d*x) - b*ExpIntegralEi[(b*Log[F])/(c + d*x)]*Log[F]))/d

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.33

method	result	size
risch	\(F^{a} F^{\frac {b}{d x +c}} x +\frac {F^{a} F^{\frac {b}{d x +c}} c}{d}+\frac {b \ln \left (F \right ) F^{a} \operatorname {Ei}_{1}\left (-\frac {b \ln \left (F \right )}{d x +c}\right )}{d}\)	\(61\)

[In]

int(F^(a+b/(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

F^a*F^(b/(d*x+c))*x+1/d*F^a*F^(b/(d*x+c))*c+b/d*ln(F)*F^a*Ei(1,-b*ln(F)/(d*x+c))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.11 \[ \int F^{a+\frac {b}{c+d x}} \, dx=-\frac {F^{a} b {\rm Ei}\left (\frac {b \log \left (F\right )}{d x + c}\right ) \log \left (F\right ) - {\left (d x + c\right )} F^{\frac {a d x + a c + b}{d x + c}}}{d} \]

[In]

integrate(F^(a+b/(d*x+c)),x, algorithm="fricas")

[Out]

-(F^a*b*Ei(b*log(F)/(d*x + c))*log(F) - (d*x + c)*F^((a*d*x + a*c + b)/(d*x + c)))/d

Sympy [F]

\[ \int F^{a+\frac {b}{c+d x}} \, dx=\int F^{a + \frac {b}{c + d x}}\, dx \]

[In]

integrate(F**(a+b/(d*x+c)),x)

[Out]

Integral(F**(a + b/(c + d*x)), x)

Maxima [F]

\[ \int F^{a+\frac {b}{c+d x}} \, dx=\int { F^{a + \frac {b}{d x + c}} \,d x } \]

[In]

integrate(F^(a+b/(d*x+c)),x, algorithm="maxima")

[Out]

F^a*b*d*integrate(F^(b/(d*x + c))*x/(d^2*x^2 + 2*c*d*x + c^2), x)*log(F) + F^a*F^(b/(d*x + c))*x

Giac [F]

\[ \int F^{a+\frac {b}{c+d x}} \, dx=\int { F^{a + \frac {b}{d x + c}} \,d x } \]

[In]

integrate(F^(a+b/(d*x+c)),x, algorithm="giac")

[Out]

integrate(F^(a + b/(d*x + c)), x)

Mupad [B] (verification not implemented)

Time = 0.91 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.02 \[ \int F^{a+\frac {b}{c+d x}} \, dx=\frac {F^a\,F^{\frac {b}{c+d\,x}}\,\left (c+d\,x\right )}{d}+\frac {F^a\,b\,\ln \left (F\right )\,\mathrm {expint}\left (-\frac {b\,\ln \left (F\right )}{c+d\,x}\right )}{d} \]

[In]

int(F^(a + b/(c + d*x)),x)

[Out]

(F^a*F^(b/(c + d*x))*(c + d*x))/d + (F^a*b*log(F)*expint(-(b*log(F))/(c + d*x)))/d