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

Optimal. Leaf size=31 \[ \frac{F^{c \left (a-\frac{b d}{e}\right )} \text{Ei}\left (\frac{b c (d+e x) \log (F)}{e}\right )}{e} \]

[Out]

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

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Rubi [A]  time = 0.0238996, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2178} \[ \frac{F^{c \left (a-\frac{b d}{e}\right )} \text{Ei}\left (\frac{b c (d+e x) \log (F)}{e}\right )}{e} \]

Antiderivative was successfully verified.

[In]

Int[F^(c*(a + b*x))/(d + e*x),x]

[Out]

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

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rubi steps

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

Mathematica [A]  time = 0.0407943, size = 31, normalized size = 1. \[ \frac{F^{c \left (a-\frac{b d}{e}\right )} \text{Ei}\left (\frac{b c (d+e x) \log (F)}{e}\right )}{e} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(c*(a + b*x))/(d + e*x),x]

[Out]

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

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Maple [A]  time = 0.033, size = 56, normalized size = 1.8 \begin{align*} -{\frac{1}{e}{F}^{{\frac{c \left ( ae-bd \right ) }{e}}}{\it Ei} \left ( 1,-bcx\ln \left ( F \right ) -ac\ln \left ( F \right ) -{\frac{-\ln \left ( F \right ) ace+\ln \left ( F \right ) bcd}{e}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.54641, size = 78, normalized size = 2.52 \begin{align*} \frac{{\rm Ei}\left (\frac{{\left (b c e x + b c d\right )} \log \left (F\right )}{e}\right )}{F^{\frac{b c d - a c e}{e}} e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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