3.46 \(\int (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=18 \[ a x+b x \log \left (c x^n\right )-b n x \]

[Out]

a*x - b*n*x + b*x*Log[c*x^n]

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Rubi [A]  time = 0.005548, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2295} \[ a x+b x \log \left (c x^n\right )-b n x \]

Antiderivative was successfully verified.

[In]

Int[a + b*Log[c*x^n],x]

[Out]

a*x - b*n*x + b*x*Log[c*x^n]

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 \left (a+b \log \left (c x^n\right )\right ) \, dx &=a x+b \int \log \left (c x^n\right ) \, dx\\ &=a x-b n x+b x \log \left (c x^n\right )\\ \end{align*}

Mathematica [A]  time = 0.000665, size = 18, normalized size = 1. \[ a x+b x \log \left (c x^n\right )-b n x \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*Log[c*x^n],x]

[Out]

a*x - b*n*x + b*x*Log[c*x^n]

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Maple [A]  time = 0.038, size = 19, normalized size = 1.1 \begin{align*} ax-bnx+bx\ln \left ( c{x}^{n} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x-b*n*x+b*x*ln(c*x^n)

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Maxima [A]  time = 1.10327, size = 24, normalized size = 1.33 \begin{align*} -b n x + b x \log \left (c x^{n}\right ) + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*n*x + b*x*log(c*x^n) + a*x

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Fricas [A]  time = 0.793783, size = 55, normalized size = 3.06 \begin{align*} b n x \log \left (x\right ) + b x \log \left (c\right ) -{\left (b n - a\right )} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*n*x*log(x) + b*x*log(c) - (b*n - a)*x

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Sympy [A]  time = 0.239296, size = 19, normalized size = 1.06 \begin{align*} a x + b \left (n x \log{\left (x \right )} - n x + x \log{\left (c \right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + b*(n*x*log(x) - n*x + x*log(c))

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Giac [A]  time = 1.17849, size = 27, normalized size = 1.5 \begin{align*}{\left (n x \log \left (x\right ) - n x + x \log \left (c\right )\right )} b + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(n*x*log(x) - n*x + x*log(c))*b + a*x