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*ln(c*x^n)

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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*}

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.00 \[ a x+b x \log \left (c x^n\right )-b n x \]

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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fricas [A]  time = 0.44, size = 22, normalized size = 1.22 \[ b n x \log \relax (x) + b x \log \relax (c) - {\left (b n - a\right )} x \]

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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giac [A]  time = 0.27, size = 20, normalized size = 1.11 \[ {\left (n x \log \relax (x) - n x + x \log \relax (c)\right )} b + a x \]

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

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maple [A]  time = 0.03, size = 19, normalized size = 1.06 \[ -b n x +b x \ln \left (c \,x^{n}\right )+a x \]

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 = 0.59, size = 18, normalized size = 1.00 \[ -b n x + b x \log \left (c x^{n}\right ) + a x \]

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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mupad [B]  time = 3.50, size = 18, normalized size = 1.00 \[ x\,\left (a-b\,n\right )+b\,x\,\ln \left (c\,x^n\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*log(c*x^n),x)

[Out]

x*(a - b*n) + b*x*log(c*x^n)

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sympy [A]  time = 0.25, size = 19, normalized size = 1.06 \[ a x + b \left (n x \log {\relax (x )} - n x + x \log {\relax (c )}\right ) \]

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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