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

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Rubi [A]  time = 0.0519797, antiderivative size = 46, normalized size of antiderivative = 1., 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 = 0.0245606, 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

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Maple [A]  time = 0.076, size = 61, normalized size = 1.3 \begin{align*}{F}^{a}{F}^{{\frac{b}{dx+c}}}x+{\frac{{F}^{a}c}{d}{F}^{{\frac{b}{dx+c}}}}+{\frac{b\ln \left ( F \right ){F}^{a}}{d}{\it Ei} \left ( 1,-{\frac{b\ln \left ( F \right ) }{dx+c}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 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 \left (F\right ) + F^{a} F^{\frac{b}{d x + c}} x \end{align*}

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

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Fricas [A]  time = 1.53923, size = 116, normalized size = 2.52 \begin{align*} -\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} \end{align*}

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int F^{a + \frac{b}{c + d x}}\, dx \end{align*}

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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int F^{a + \frac{b}{d x + c}}\,{d x} \end{align*}

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)