\(\int \log ^p(c (d+e x)) \, dx\) [16]

Optimal result

Integrand size = 10, antiderivative size = 45 \[ \int \log ^p(c (d+e x)) \, dx=\frac {\Gamma (1+p,-\log (c (d+e x))) (-\log (c (d+e x)))^{-p} \log ^p(c (d+e x))}{c e} \]

[Out]

GAMMA(p+1,-ln(c*(e*x+d)))*ln(c*(e*x+d))^p/c/e/((-ln(c*(e*x+d)))^p)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2436, 2336, 2212} \[ \int \log ^p(c (d+e x)) \, dx=\frac {(-\log (c (d+e x)))^{-p} \log ^p(c (d+e x)) \Gamma (p+1,-\log (c (d+e x)))}{c e} \]

[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 2212

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

Rule 2336

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 2436

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \log ^p(c x) \, dx,x,d+e x\right )}{e} \\ & = \frac {\text {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] (verified)

Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \log ^p(c (d+e x)) \, dx=\frac {\Gamma (1+p,-\log (c (d+e x))) (-\log (c (d+e x)))^{-p} \log ^p(c (d+e x))}{c e} \]

[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)

Maple [F]

\[\int \ln \left (c \left (e x +d \right )\right )^{p}d x\]

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.60 \[ \int \log ^p(c (d+e x)) \, dx=\frac {e^{\left (-i \, \pi p\right )} \Gamma \left (p + 1, -\log \left (c e x + c d\right )\right )}{c e} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 3.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.20 \[ \int \log ^p(c (d+e x)) \, dx=\begin {cases} \tilde {\infty }^{p} x & \text {for}\: c = 0 \\x \log {\left (c d \right )}^{p} & \text {for}\: e = 0 \\\frac {\left (- \log {\left (c d + c e x \right )}\right )^{- p} \log {\left (c d + c e x \right )}^{p} \Gamma \left (p + 1, - \log {\left (c d + c e x \right )}\right )}{c e} & \text {otherwise} \end {cases} \]

[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*uppergamma(p + 1, -log(c*d + c
*e*x))/(c*e*(-log(c*d + c*e*x))**p), True))

Maxima [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.18 \[ \int \log ^p(c (d+e x)) \, dx=-\frac {\left (-\log \left (c e x + c d\right )\right )^{-p - 1} \log \left (c e x + c d\right )^{p + 1} \Gamma \left (p + 1, -\log \left (c e x + c d\right )\right )}{c e} \]

[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)

Giac [F]

\[ \int \log ^p(c (d+e x)) \, dx=\int { \log \left ({\left (e x + d\right )} c\right )^{p} \,d x } \]

[In]

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

[Out]

integrate(log((e*x + d)*c)^p, x)

Mupad [B] (verification not implemented)

Time = 1.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \log ^p(c (d+e x)) \, dx=\frac {{\ln \left (c\,\left (d+e\,x\right )\right )}^p\,\Gamma \left (p+1,-\ln \left (c\,\left (d+e\,x\right )\right )\right )}{c\,e\,{\left (-\ln \left (c\,\left (d+e\,x\right )\right )\right )}^p} \]

[In]

int(log(c*(d + e*x))^p,x)

[Out]

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