3.3.21 \(\int \frac {f^{\frac {c}{a+b x}}}{x} \, dx\) [221]

Optimal. Leaf size=41 \[ -\text {Ei}\left (\frac {c \log (f)}{a+b x}\right )+f^{\frac {c}{a}} \text {Ei}\left (-\frac {b c x \log (f)}{a (a+b x)}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.10, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2254, 2241, 2260, 2209} \begin {gather*} f^{\frac {c}{a}} \text {Ei}\left (-\frac {b c x \log (f)}{a (a+b x)}\right )-\text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[f^(c/(a + b*x))/x,x]

[Out]

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

Rule 2209

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

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]

Rule 2254

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

Rule 2260

Int[(F_)^((a_.) + (b_.)/((c_.) + (d_.)*(x_)))/(((e_.) + (f_.)*(x_))*((g_.) + (h_.)*(x_))), x_Symbol] :> Dist[-
d/(f*(d*g - c*h)), Subst[Int[F^(a - b*(h/(d*g - c*h)) + d*b*(x/(d*g - c*h)))/x, x], x, (g + h*x)/(c + d*x)], x
] /; FreeQ[{F, a, b, c, d, e, f}, x] && EqQ[d*e - c*f, 0]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 41, normalized size = 1.00 \begin {gather*} -\text {Ei}\left (\frac {c \log (f)}{a+b x}\right )+f^{\frac {c}{a}} \text {Ei}\left (-\frac {b c x \log (f)}{a^2+a b x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[f^(c/(a + b*x))/x,x]

[Out]

-ExpIntegralEi[(c*Log[f])/(a + b*x)] + f^(c/a)*ExpIntegralEi[-((b*c*x*Log[f])/(a^2 + a*b*x))]

________________________________________________________________________________________

Maple [A]
time = 0.09, size = 47, normalized size = 1.15

method	result	size
risch	\(-f^{\frac {c}{a}} \expIntegral \left (1, -\frac {c \ln \left (f \right )}{b x +a}+\frac {c \ln \left (f \right )}{a}\right )+\expIntegral \left (1, -\frac {c \ln \left (f \right )}{b x +a}\right )\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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^(c/(b*x+a))/x,x, algorithm="maxima")

[Out]

integrate(f^(c/(b*x + a))/x, x)

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 41, normalized size = 1.00 \begin {gather*} f^{\frac {c}{a}} {\rm Ei}\left (-\frac {b c x \log \left (f\right )}{a b x + a^{2}}\right ) - {\rm Ei}\left (\frac {c \log \left (f\right )}{b x + a}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

f^(c/a)*Ei(-b*c*x*log(f)/(a*b*x + a^2)) - Ei(c*log(f)/(b*x + a))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(c/(b*x+a))/x,x)

[Out]

Integral(f**(c/(a + b*x))/x, x)

________________________________________________________________________________________

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^(c/(b*x+a))/x,x, algorithm="giac")

[Out]

integrate(f^(c/(b*x + a))/x, x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {f^{\frac {c}{a+b\,x}}}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(c/(a + b*x))/x,x)

[Out]

int(f^(c/(a + b*x))/x, x)

________________________________________________________________________________________