3.4.7 \(\int \frac {F^{a+\frac {b}{c+d x}}}{c+d x} \, dx\) [307]

Optimal. Leaf size=20 \[ -\frac {F^a \text {Ei}\left (\frac {b \log (F)}{c+d x}\right )}{d} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2241} \begin {gather*} -\frac {F^a \text {Ei}\left (\frac {b \log (F)}{c+d x}\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[F^(a + b/(c + d*x))/(c + d*x),x]

[Out]

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

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

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Mathematica [A]
time = 0.00, size = 20, normalized size = 1.00 \begin {gather*} -\frac {F^a \text {Ei}\left (\frac {b \log (F)}{c+d x}\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.09, size = 22, normalized size = 1.10

method	result	size
risch	\(\frac {F^{a} \expIntegral \left (1, -\frac {b \ln \left (F \right )}{d x +c}\right )}{d}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.36, size = 20, normalized size = 1.00 \begin {gather*} -\frac {F^{a} {\rm Ei}\left (\frac {b \log \left (F\right )}{d x + c}\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-F^a*Ei(b*log(F)/(d*x + c))/d

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {F^{a + \frac {b}{c + d x}}}{c + d x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 3.76, size = 20, normalized size = 1.00 \begin {gather*} -\frac {F^a\,\mathrm {ei}\left (\frac {b\,\ln \left (F\right )}{c+d\,x}\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a + b/(c + d*x))/(c + d*x),x)

[Out]

-(F^a*ei((b*log(F))/(c + d*x)))/d

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