3.25 \(\int (a+\frac {2 b n \log (c x^n)}{x}) \, dx\)

Optimal. Leaf size=14 \[ a x+b \log ^2\left (c x^n\right ) \]

[Out]

a*x+b*ln(c*x^n)^2

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2301} \[ a x+b \log ^2\left (c x^n\right ) \]

Antiderivative was successfully verified.

[In]

Int[a + (2*b*n*Log[c*x^n])/x,x]

[Out]

a*x + b*Log[c*x^n]^2

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

\begin {align*} \int \left (a+\frac {2 b n \log \left (c x^n\right )}{x}\right ) \, dx &=a x+(2 b n) \int \frac {\log \left (c x^n\right )}{x} \, dx\\ &=a x+b \log ^2\left (c x^n\right )\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \[ a x+b \log ^2\left (c x^n\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[a + (2*b*n*Log[c*x^n])/x,x]

[Out]

a*x + b*Log[c*x^n]^2

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fricas [A]  time = 0.43, size = 21, normalized size = 1.50 \[ b n^{2} \log \relax (x)^{2} + 2 \, b n \log \relax (c) \log \relax (x) + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+2*b*n*log(c*x^n)/x,x, algorithm="fricas")

[Out]

b*n^2*log(x)^2 + 2*b*n*log(c)*log(x) + a*x

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giac [A]  time = 0.17, size = 20, normalized size = 1.43 \[ {\left (n \log \relax (x)^{2} + 2 \, \log \relax (c) \log \relax (x)\right )} b n + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+2*b*n*log(c*x^n)/x,x, algorithm="giac")

[Out]

(n*log(x)^2 + 2*log(c)*log(x))*b*n + a*x

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maple [A]  time = 0.07, size = 15, normalized size = 1.07 \[ b \ln \left (c \,x^{n}\right )^{2}+a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a+2*b*n*ln(c*x^n)/x,x)

[Out]

a*x+b*ln(c*x^n)^2

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maxima [A]  time = 0.45, size = 14, normalized size = 1.00 \[ b \log \left (c x^{n}\right )^{2} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+2*b*n*log(c*x^n)/x,x, algorithm="maxima")

[Out]

b*log(c*x^n)^2 + a*x

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mupad [B]  time = 0.26, size = 14, normalized size = 1.00 \[ b\,{\ln \left (c\,x^n\right )}^2+a\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + (2*b*n*log(c*x^n))/x,x)

[Out]

a*x + b*log(c*x^n)^2

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sympy [A]  time = 1.72, size = 60, normalized size = 4.29 \[ a x + 2 b n \left (\begin {cases} \frac {\log {\left (c x^{n} \right )}^{2}}{2 n} & \text {for}\: \left |{c x^{n}}\right | < 1 \\\frac {\log {\left (\frac {x^{- n}}{c} \right )}^{2}}{2 n} & \text {for}\: \frac {1}{\left |{c x^{n}}\right |} < 1 \\\frac {{G_{3, 3}^{3, 0}\left (\begin {matrix} & 1, 1, 1 \\0, 0, 0 & \end {matrix} \middle | {c x^{n}} \right )}}{n} + \frac {{G_{3, 3}^{0, 3}\left (\begin {matrix} 1, 1, 1 & \\ & 0, 0, 0 \end {matrix} \middle | {c x^{n}} \right )}}{n} & \text {otherwise} \end {cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+2*b*n*ln(c*x**n)/x,x)

[Out]

a*x + 2*b*n*Piecewise((log(c*x**n)**2/(2*n), Abs(c*x**n) < 1), (log(x**(-n)/c)**2/(2*n), 1/Abs(c*x**n) < 1), (
meijerg(((), (1, 1, 1)), ((0, 0, 0), ()), c*x**n)/n + meijerg(((1, 1, 1), ()), ((), (0, 0, 0)), c*x**n)/n, Tru
e))

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