\(\int (a+b \log (c x^n)) \, dx\) [46]

Optimal result

Integrand size = 10, antiderivative size = 18 \[ \int \left (a+b \log \left (c x^n\right )\right ) \, dx=a x-b n x+b x \log \left (c x^n\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2332} \[ \int \left (a+b \log \left (c x^n\right )\right ) \, dx=a x+b x \log \left (c x^n\right )-b n x \]

[In]

Int[a + b*Log[c*x^n],x]

[Out]

a*x - b*n*x + b*x*Log[c*x^n]

Rule 2332

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*} \text {integral}& = 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] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \left (a+b \log \left (c x^n\right )\right ) \, dx=a x-b n x+b x \log \left (c x^n\right ) \]

[In]

Integrate[a + b*Log[c*x^n],x]

[Out]

a*x - b*n*x + b*x*Log[c*x^n]

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06

method	result	size
default	\(a x -b n x +b x \ln \left (c \,x^{n}\right )\)	\(19\)
parts	\(a x -b n x +b x \ln \left (c \,x^{n}\right )\)	\(19\)
parallelrisch	\(b \left (x \ln \left (c \,x^{n}\right )-n x \right )+a x\)	\(20\)
norman	\(\left (-b n +a \right ) x +b x \ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}\right )\)	\(21\)
risch	\(a x +b x \ln \left (x^{n}\right )+\frac {b x \left (-i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 \ln \left (c \right )-2 n \right )}{2}\)	\(102\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \left (a+b \log \left (c x^n\right )\right ) \, dx=b n x \log \left (x\right ) + b x \log \left (c\right ) - {\left (b n - a\right )} x \]

[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

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \left (a+b \log \left (c x^n\right )\right ) \, dx=a x + b \left (- n x + x \log {\left (c x^{n} \right )}\right ) \]

[In]

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

[Out]

a*x + b*(-n*x + x*log(c*x**n))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \left (a+b \log \left (c x^n\right )\right ) \, dx=-b n x + b x \log \left (c x^{n}\right ) + a x \]

[In]

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

[Out]

-b*n*x + b*x*log(c*x^n) + a*x

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \left (a+b \log \left (c x^n\right )\right ) \, dx={\left (n x \log \left (x\right ) - n x + x \log \left (c\right )\right )} b + a x \]

[In]

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

[Out]

(n*x*log(x) - n*x + x*log(c))*b + a*x

Mupad [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \left (a+b \log \left (c x^n\right )\right ) \, dx=x\,\left (a-b\,n\right )+b\,x\,\ln \left (c\,x^n\right ) \]

[In]

int(a + b*log(c*x^n),x)

[Out]

x*(a - b*n) + b*x*log(c*x^n)