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

Optimal. Leaf size=19 \[ \frac {(a+b \log (x))^{n+1}}{b (n+1)} \]

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Rubi [A]  time = 0.03, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2302, 30} \[ \frac {(a+b \log (x))^{n+1}}{b (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[x])^n/x,x]

[Out]

(a + b*Log[x])^(1 + n)/(b*(1 + n))

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rubi steps

\begin {align*} \int \frac {(a+b \log (x))^n}{x} \, dx &=\frac {\operatorname {Subst}\left (\int x^n \, dx,x,a+b \log (x)\right )}{b}\\ &=\frac {(a+b \log (x))^{1+n}}{b (1+n)}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \[ \frac {(a+b \log (x))^{n+1}}{b (n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[x])^n/x,x]

[Out]

(a + b*Log[x])^(1 + n)/(b*(1 + n))

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {(a+b \log (x))^n}{x} \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*Log[x])^n/x,x]

[Out]

Could not integrate

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fricas [A]  time = 0.96, size = 22, normalized size = 1.16 \[ \frac {{\left (b \log \relax (x) + a\right )} {\left (b \log \relax (x) + a\right )}^{n}}{b n + b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*log(x) + a)*(b*log(x) + a)^n/(b*n + b)

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giac [A]  time = 0.61, size = 19, normalized size = 1.00 \[ \frac {{\left (b \log \relax (x) + a\right )}^{n + 1}}{b {\left (n + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*log(x) + a)^(n + 1)/(b*(n + 1))

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maple [A]  time = 0.04, size = 20, normalized size = 1.05

method	result	size
derivativedivides	\(\frac {\left (a +b \ln \relax (x )\right )^{1+n}}{b \left (1+n \right )}\)	\(20\)
default	\(\frac {\left (a +b \ln \relax (x )\right )^{1+n}}{b \left (1+n \right )}\)	\(20\)
risch	\(\frac {\left (a +b \ln \relax (x )\right ) \left (a +b \ln \relax (x )\right )^{n}}{b \left (1+n \right )}\)	\(24\)
norman	\(\frac {\ln \relax (x ) {\mathrm e}^{n \ln \left (a +b \ln \relax (x )\right )}}{1+n}+\frac {a \,{\mathrm e}^{n \ln \left (a +b \ln \relax (x )\right )}}{b \left (1+n \right )}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a+b*ln(x))^(1+n)/b/(1+n)

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maxima [A]  time = 0.43, size = 19, normalized size = 1.00 \[ \frac {{\left (b \log \relax (x) + a\right )}^{n + 1}}{b {\left (n + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*log(x) + a)^(n + 1)/(b*(n + 1))

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mupad [B]  time = 0.45, size = 19, normalized size = 1.00 \[ \frac {{\left (a+b\,\ln \relax (x)\right )}^{n+1}}{b\,\left (n+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(x))^n/x,x)

[Out]

(a + b*log(x))^(n + 1)/(b*(n + 1))

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sympy [A]  time = 1.27, size = 36, normalized size = 1.89 \[ - \begin {cases} - a^{n} \log {\relax (x )} & \text {for}\: b = 0 \\- \frac {\begin {cases} \frac {\left (a + b \log {\relax (x )}\right )^{n + 1}}{n + 1} & \text {for}\: n \neq -1 \\\log {\left (a + b \log {\relax (x )} \right )} & \text {otherwise} \end {cases}}{b} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Piecewise((-a**n*log(x), Eq(b, 0)), (-Piecewise(((a + b*log(x))**(n + 1)/(n + 1), Ne(n, -1)), (log(a + b*log(
x)), True))/b, True))

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