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

Optimal. Leaf size=24 \[ \frac{(a+b x) \log \left (c (a+b x)^p\right )}{b}-p x \]

[Out]

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

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Rubi [A]  time = 0.0090786, antiderivative size = 24, 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 = {2389, 2295} \[ \frac{(a+b x) \log \left (c (a+b x)^p\right )}{b}-p x \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b*x)^p],x]

[Out]

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

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 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

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

Mathematica [A]  time = 0.0056086, size = 24, normalized size = 1. \[ \frac{(a+b x) \log \left (c (a+b x)^p\right )}{b}-p x \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(a + b*x)^p],x]

[Out]

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

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Maple [A]  time = 0.069, size = 30, normalized size = 1.3 \begin{align*} \ln \left ( c \left ( bx+a \right ) ^{p} \right ) x-px+{\frac{ap\ln \left ( bx+a \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(c*(b*x+a)^p)*x-p*x+1/b*p*a*ln(b*x+a)

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Maxima [A]  time = 1.0432, size = 47, normalized size = 1.96 \begin{align*} -b p{\left (\frac{x}{b} - \frac{a \log \left (b x + a\right )}{b^{2}}\right )} + x \log \left ({\left (b x + a\right )}^{p} c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*p*(x/b - a*log(b*x + a)/b^2) + x*log((b*x + a)^p*c)

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Fricas [A]  time = 1.90589, size = 73, normalized size = 3.04 \begin{align*} -\frac{b p x - b x \log \left (c\right ) -{\left (b p x + a p\right )} \log \left (b x + a\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*p*x - b*x*log(c) - (b*p*x + a*p)*log(b*x + a))/b

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Sympy [A]  time = 0.539807, size = 37, normalized size = 1.54 \begin{align*} \begin{cases} \frac{a p \log{\left (a + b x \right )}}{b} + p x \log{\left (a + b x \right )} - p x + x \log{\left (c \right )} & \text{for}\: b \neq 0 \\x \log{\left (a^{p} c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*p*log(a + b*x)/b + p*x*log(a + b*x) - p*x + x*log(c), Ne(b, 0)), (x*log(a**p*c), True))

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Giac [A]  time = 1.18071, size = 53, normalized size = 2.21 \begin{align*} \frac{{\left (b x + a\right )} p \log \left (b x + a\right )}{b} - \frac{{\left (b x + a\right )} p}{b} + \frac{{\left (b x + a\right )} \log \left (c\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x + a)*p*log(b*x + a)/b - (b*x + a)*p/b + (b*x + a)*log(c)/b