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

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

[Out]

(a + b*Log[x])^(1 + n)/(b*(1 + n))

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Rubi [A]  time = 0.0467822, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{(a+b \log (x))^{n+1}}{b (n+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Log[x])^n/x,x]

[Out]

(a + b*Log[x])^(1 + n)/(b*(1 + n))

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Rubi in Sympy [A]  time = 2.35429, size = 14, normalized size = 0.74 \[ \frac{\left (a + b \log{\left (x \right )}\right )^{n + 1}}{b \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*ln(x))**n/x,x)

[Out]

(a + b*log(x))**(n + 1)/(b*(n + 1))

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Mathematica [A]  time = 0.0152306, size = 18, normalized size = 0.95 \[ \frac{(a+b \log (x))^{n+1}}{b n+b} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Log[x])^n/x,x]

[Out]

(a + b*Log[x])^(1 + n)/(b + b*n)

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Maple [A]  time = 0.003, size = 20, normalized size = 1.1 \[{\frac{ \left ( a+b\ln \left ( x \right ) \right ) ^{1+n}}{b \left ( 1+n \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*ln(x))^n/x,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220611, size = 30, normalized size = 1.58 \[ \frac{{\left (b \log \left (x\right ) + a\right )}{\left (b \log \left (x\right ) + a\right )}^{n}}{b n + b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.75653, size = 31, normalized size = 1.63 \[ \begin{cases} \frac{\begin{cases} \frac{\left (a + b \log{\left (x \right )}\right )^{n + 1}}{n + 1} & \text{for}\: n \neq -1 \\\log{\left (a + b \log{\left (x \right )} \right )} & \text{otherwise} \end{cases}}{b} & \text{for}\: b \neq 0 \\a^{n} \log{\left (x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*ln(x))**n/x,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.208722, size = 26, normalized size = 1.37 \[ \frac{{\left (b{\rm ln}\left (x\right ) + a\right )}^{n + 1}}{b{\left (n + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*ln(x) + a)^(n + 1)/(b*(n + 1))