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

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

[Out]

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

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Rubi [A]  time = 0.0292191, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2206, 2210} \[ \frac{(a+b x) f^{\frac{c}{a+b x}}}{b}-\frac{c \log (f) \text{Ei}\left (\frac{c \log (f)}{a+b x}\right )}{b} \]

Antiderivative was successfully verified.

[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 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^{\frac{c}{a+b x}} \, dx &=\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]  time = 0.0152805, size = 41, normalized size = 1. \[ \frac{(a+b x) f^{\frac{c}{a+b x}}}{b}-\frac{c \log (f) \text{Ei}\left (\frac{c \log (f)}{a+b x}\right )}{b} \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.06, size = 52, normalized size = 1.3 \begin{align*}{f}^{{\frac{c}{bx+a}}}x+{\frac{a}{b}{f}^{{\frac{c}{bx+a}}}}+{\frac{c\ln \left ( f \right ) }{b}{\it Ei} \left ( 1,-{\frac{c\ln \left ( f \right ) }{bx+a}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} b c \int \frac{f^{\frac{c}{b x + a}} x}{b^{2} x^{2} + 2 \, a b x + a^{2}}\,{d x} \log \left (f\right ) + f^{\frac{c}{b x + a}} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.51698, size = 89, normalized size = 2.17 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(c/(a + b*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(c/(b*x + a)), x)