3.16 \(\int \log ^p(c (d+e x)) \, dx\)

Optimal. Leaf size=45 \[ \frac{(-\log (c (d+e x)))^{-p} \log ^p(c (d+e x)) \text{Gamma}(p+1,-\log (c (d+e x)))}{c e} \]

[Out]

(Gamma[1 + p, -Log[c*(d + e*x)]]*Log[c*(d + e*x)]^p)/(c*e*(-Log[c*(d + e*x)])^p)

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Rubi [A]  time = 0.0283106, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2389, 2299, 2181} \[ \frac{(-\log (c (d+e x)))^{-p} \log ^p(c (d+e x)) \text{Gamma}(p+1,-\log (c (d+e x)))}{c e} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e*x)]^p,x]

[Out]

(Gamma[1 + p, -Log[c*(d + e*x)]]*Log[c*(d + e*x)]^p)/(c*e*(-Log[c*(d + e*x)])^p)

Rule 2389

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

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

Mathematica [A]  time = 0.0155742, size = 45, normalized size = 1. \[ \frac{(-\log (c (d+e x)))^{-p} \log ^p(c (d+e x)) \text{Gamma}(p+1,-\log (c (d+e x)))}{c e} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(d + e*x)]^p,x]

[Out]

(Gamma[1 + p, -Log[c*(d + e*x)]]*Log[c*(d + e*x)]^p)/(c*e*(-Log[c*(d + e*x)])^p)

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Maple [F]  time = 0.224, size = 0, normalized size = 0. \begin{align*} \int \left ( \ln \left ( c \left ( ex+d \right ) \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(e*x+d))^p,x)

[Out]

int(ln(c*(e*x+d))^p,x)

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Maxima [A]  time = 1.25065, size = 72, normalized size = 1.6 \begin{align*} -\frac{\left (-\log \left (c e x + c d\right )\right )^{-p - 1} \log \left (c e x + c d\right )^{p + 1} \Gamma \left (p + 1, -\log \left (c e x + c d\right )\right )}{c e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(e*x+d))^p,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.15524, size = 66, normalized size = 1.47 \begin{align*} \frac{\cos \left (\pi p\right ) \Gamma \left (p + 1, -\log \left (c e x + c d\right )\right )}{c e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(e*x+d))^p,x, algorithm="fricas")

[Out]

cos(pi*p)*gamma(p + 1, -log(c*e*x + c*d))/(c*e)

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Sympy [A]  time = 42.0275, size = 54, normalized size = 1.2 \begin{align*} \begin{cases} \tilde{\infty }^{p} x & \text{for}\: c = 0 \\x \log{\left (c d \right )}^{p} & \text{for}\: e = 0 \\\frac{\left (- \log{\left (c d + c e x \right )}\right )^{- p} \log{\left (c d + c e x \right )}^{p} \Gamma \left (p + 1, - \log{\left (c d + c e x \right )}\right )}{c e} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(e*x+d))**p,x)

[Out]

Piecewise((zoo**p*x, Eq(c, 0)), (x*log(c*d)**p, Eq(e, 0)), ((-log(c*d + c*e*x))**(-p)*log(c*d + c*e*x)**p*uppe
rgamma(p + 1, -log(c*d + c*e*x))/(c*e), True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(e*x+d))^p,x, algorithm="giac")

[Out]

integrate(log((e*x + d)*c)^p, x)