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

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2436, 2336, 2212} \begin {gather*} \frac {(-\log (c (d+e x)))^{-p} \log ^p(c (d+e x)) \text {Gamma}(p+1,-\log (c (d+e x)))}{c e} \end {gather*}

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 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*} \int \log ^p(c (d+e x)) \, dx &=\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*}

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Mathematica [A]
time = 0.02, size = 45, normalized size = 1.00 \begin {gather*} \frac {\Gamma (1+p,-\log (c (d+e x))) (-\log (c (d+e x)))^{-p} \log ^p(c (d+e x))}{c e} \end {gather*}

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 = 0.02, size = 0, normalized size = 0.00 \[\int \ln \left (c \left (e x +d \right )\right )^{p}\, dx\]

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 = 0.08, size = 55, normalized size = 1.22 \begin {gather*} -\frac {\left (-\log \left (c x e + c d\right )\right )^{-p - 1} \log \left (c x e + c d\right )^{p + 1} e^{\left (-1\right )} \Gamma \left (p + 1, -\log \left (c x e + c d\right )\right )}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.11, size = 26, normalized size = 0.58 \begin {gather*} \frac {\cos \left (\pi p\right ) e^{\left (-1\right )} \Gamma \left (p + 1, -\log \left (c x e + c d\right )\right )}{c} \end {gather*}

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

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Sympy [A]
time = 3.95, size = 54, normalized size = 1.20 \begin {gather*} \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 {gather*}

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log((x*e + d)*c)^p, x)

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Mupad [B]
time = 0.18, size = 45, normalized size = 1.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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