\(\int f^{\frac {c}{a+b x}} \, dx\) [220]

Optimal result

Integrand size = 11, antiderivative size = 41 \[ \int f^{\frac {c}{a+b x}} \, dx=\frac {f^{\frac {c}{a+b x}} (a+b x)}{b}-\frac {c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2237, 2241} \[ \int f^{\frac {c}{a+b x}} \, dx=\frac {(a+b x) f^{\frac {c}{a+b x}}}{b}-\frac {c \log (f) \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right )}{b} \]

[In]

Int[f^(c/(a + b*x)),x]

[Out]

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

Rule 2237

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int f^{\frac {c}{a+b x}} \, dx=\frac {f^{\frac {c}{a+b x}} (a+b x)}{b}-\frac {c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{b} \]

[In]

Integrate[f^(c/(a + b*x)),x]

[Out]

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

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.27

method	result	size
risch	\(f^{\frac {c}{b x +a}} x +\frac {f^{\frac {c}{b x +a}} a}{b}+\frac {c \ln \left (f \right ) \operatorname {Ei}_{1}\left (-\frac {c \ln \left (f \right )}{b x +a}\right )}{b}\)	\(52\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.98 \[ \int f^{\frac {c}{a+b x}} \, dx=-\frac {c {\rm Ei}\left (\frac {c \log \left (f\right )}{b x + a}\right ) \log \left (f\right ) - {\left (b x + a\right )} f^{\frac {c}{b x + a}}}{b} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int f^{\frac {c}{a+b x}} \, dx=\int f^{\frac {c}{a + b x}}\, dx \]

[In]

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

[Out]

Integral(f**(c/(a + b*x)), x)

Maxima [F]

\[ \int f^{\frac {c}{a+b x}} \, dx=\int { f^{\frac {c}{b x + a}} \,d x } \]

[In]

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

[Out]

b*c*integrate(f^(c/(b*x + a))*x/(b^2*x^2 + 2*a*b*x + a^2), x)*log(f) + f^(c/(b*x + a))*x

Giac [F]

\[ \int f^{\frac {c}{a+b x}} \, dx=\int { f^{\frac {c}{b x + a}} \,d x } \]

[In]

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

[Out]

integrate(f^(c/(b*x + a)), x)

Mupad [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.22 \[ \int f^{\frac {c}{a+b x}} \, dx=f^{\frac {c}{a+b\,x}}\,x+\frac {a\,f^{\frac {c}{a+b\,x}}}{b}-\frac {c\,\mathrm {ei}\left (\frac {c\,\ln \left (f\right )}{a+b\,x}\right )\,\ln \left (f\right )}{b} \]

[In]

int(f^(c/(a + b*x)),x)

[Out]

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