3.178 \(\int (a+b \log (c x))^p \, dx\)

Optimal. Leaf size=56 \[ \frac{e^{-\frac{a}{b}} (a+b \log (c x))^p \left (-\frac{a+b \log (c x)}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log (c x)}{b}\right )}{c} \]

[Out]

(Gamma[1 + p, -((a + b*Log[c*x])/b)]*(a + b*Log[c*x])^p)/(c*E^(a/b)*(-((a + b*Log[c*x])/b))^p)

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Rubi [A]  time = 0.0329044, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2299, 2181} \[ \frac{e^{-\frac{a}{b}} (a+b \log (c x))^p \left (-\frac{a+b \log (c x)}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log (c x)}{b}\right )}{c} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x])^p,x]

[Out]

(Gamma[1 + p, -((a + b*Log[c*x])/b)]*(a + b*Log[c*x])^p)/(c*E^(a/b)*(-((a + b*Log[c*x])/b))^p)

Rule 2299

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[1/(n*c^(1/n)), Subst[Int[E^(x/n)*(a + b*x)^p
, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[1/n]

Rule 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +
d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*
Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rubi steps

\begin{align*} \int (a+b \log (c x))^p \, dx &=\frac{\operatorname{Subst}\left (\int e^x (a+b x)^p \, dx,x,\log (c x)\right )}{c}\\ &=\frac{e^{-\frac{a}{b}} \Gamma \left (1+p,-\frac{a+b \log (c x)}{b}\right ) (a+b \log (c x))^p \left (-\frac{a+b \log (c x)}{b}\right )^{-p}}{c}\\ \end{align*}

Mathematica [A]  time = 0.0260325, size = 56, normalized size = 1. \[ \frac{e^{-\frac{a}{b}} (a+b \log (c x))^p \left (-\frac{a+b \log (c x)}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log (c x)}{b}\right )}{c} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x])^p,x]

[Out]

(Gamma[1 + p, -((a + b*Log[c*x])/b)]*(a + b*Log[c*x])^p)/(c*E^(a/b)*(-((a + b*Log[c*x])/b))^p)

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Maple [F]  time = 0.048, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( cx \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x))^p,x)

[Out]

int((a+b*ln(c*x))^p,x)

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Maxima [A]  time = 1.24848, size = 59, normalized size = 1.05 \begin{align*} -\frac{{\left (b \log \left (c x\right ) + a\right )}^{p + 1} e^{\left (-\frac{a}{b}\right )} E_{-p}\left (-\frac{b \log \left (c x\right ) + a}{b}\right )}{b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x))^p,x, algorithm="maxima")

[Out]

-(b*log(c*x) + a)^(p + 1)*e^(-a/b)*exp_integral_e(-p, -(b*log(c*x) + a)/b)/(b*c)

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Fricas [A]  time = 1.05313, size = 86, normalized size = 1.54 \begin{align*} \frac{e^{\left (-\frac{b p \log \left (-\frac{1}{b}\right ) + a}{b}\right )} \Gamma \left (p + 1, -\frac{b \log \left (c x\right ) + a}{b}\right )}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x))^p,x, algorithm="fricas")

[Out]

e^(-(b*p*log(-1/b) + a)/b)*gamma(p + 1, -(b*log(c*x) + a)/b)/c

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x))**p,x)

[Out]

Integral((a + b*log(c*x))**p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x))^p,x, algorithm="giac")

[Out]

integrate((b*log(c*x) + a)^p, x)