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

Optimal. Leaf size=46 \[ \frac {(c+d x) F^{a+\frac {b}{c+d x}}}{d}-\frac {b F^a \log (F) \text {Ei}\left (\frac {b \log (F)}{c+d x}\right )}{d} \]

[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]  time = 0.05, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2206, 2210} \[ \frac {(c+d x) F^{a+\frac {b}{c+d x}}}{d}-\frac {b F^a \log (F) \text {Ei}\left (\frac {b \log (F)}{c+d x}\right )}{d} \]

Antiderivative was successfully verified.

[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 2206

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 2210

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*} \int F^{a+\frac {b}{c+d x}} \, dx &=\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]  time = 0.03, size = 42, normalized size = 0.91 \[ \frac {F^a \left ((c+d x) F^{\frac {b}{c+d x}}-b \log (F) \text {Ei}\left (\frac {b \log (F)}{c+d x}\right )\right )}{d} \]

Antiderivative was successfully verified.

[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

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 51, normalized size = 1.11 \[ -\frac {F^{a} b {\rm Ei}\left (\frac {b \log \relax (F)}{d x + c}\right ) \log \relax (F) - {\left (d x + c\right )} F^{\frac {a d x + a c + b}{d x + c}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{a + \frac {b}{d x + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(F^(a + b/(d*x + c)), x)

________________________________________________________________________________________

maple [A]  time = 0.12, size = 61, normalized size = 1.33 \[ \frac {b \,F^{a} \Ei \left (1, -\frac {b \ln \relax (F )}{d x +c}\right ) \ln \relax (F )}{d}+x \,F^{a} F^{\frac {b}{d x +c}}+\frac {c \,F^{a} F^{\frac {b}{d x +c}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+1/(d*x+c)*b),x)

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ F^{a} b d \int \frac {F^{\frac {b}{d x + c}} x}{d^{2} x^{2} + 2 \, c d x + c^{2}}\,{d x} \log \relax (F) + F^{a} F^{\frac {b}{d x + c}} x \]

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

mupad [B]  time = 4.40, size = 47, normalized size = 1.02 \[ \frac {F^a\,F^{\frac {b}{c+d\,x}}\,\left (c+d\,x\right )}{d}+\frac {F^a\,b\,\ln \relax (F)\,\mathrm {expint}\left (-\frac {b\,\ln \relax (F)}{c+d\,x}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{a + \frac {b}{c + d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(F**(a + b/(c + d*x)), x)

________________________________________________________________________________________