3.2467 \(\int \frac{1}{x \left (a+b x^n\right )} \, dx\)

Optimal. Leaf size=23 \[ \frac{\log (x)}{a}-\frac{\log \left (a+b x^n\right )}{a n} \]

[Out]

Log[x]/a - Log[a + b*x^n]/(a*n)

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Rubi [A]  time = 0.0363581, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{\log (x)}{a}-\frac{\log \left (a+b x^n\right )}{a n} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x^n)),x]

[Out]

Log[x]/a - Log[a + b*x^n]/(a*n)

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Rubi in Sympy [A]  time = 6.36424, size = 19, normalized size = 0.83 \[ \frac{\log{\left (x^{n} \right )}}{a n} - \frac{\log{\left (a + b x^{n} \right )}}{a n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(a+b*x**n),x)

[Out]

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

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Mathematica [A]  time = 0.0163694, size = 22, normalized size = 0.96 \[ \frac{n \log (x)-\log \left (a+b x^n\right )}{a n} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a + b*x^n)),x]

[Out]

(n*Log[x] - Log[a + b*x^n])/(a*n)

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Maple [A]  time = 0.003, size = 29, normalized size = 1.3 \[{\frac{\ln \left ({x}^{n} \right ) }{na}}-{\frac{\ln \left ( a+b{x}^{n} \right ) }{na}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a+b*x^n),x)

[Out]

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

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Maxima [A]  time = 1.44482, size = 38, normalized size = 1.65 \[ -\frac{\log \left (b x^{n} + a\right )}{a n} + \frac{\log \left (x^{n}\right )}{a n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^n + a)*x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.245461, size = 30, normalized size = 1.3 \[ \frac{n \log \left (x\right ) - \log \left (b x^{n} + a\right )}{a n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^n + a)*x),x, algorithm="fricas")

[Out]

(n*log(x) - log(b*x^n + a))/(a*n)

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Sympy [A]  time = 2.20038, size = 41, normalized size = 1.78 \[ \begin{cases} \tilde{\infty } \log{\left (x \right )} & \text{for}\: a = 0 \wedge b = 0 \wedge n = 0 \\\frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \\\frac{\log{\left (x \right )}}{a + b} & \text{for}\: n = 0 \\- \frac{x^{- n}}{b n} & \text{for}\: a = 0 \\\frac{\log{\left (x \right )}}{a} - \frac{\log{\left (\frac{a}{b} + x^{n} \right )}}{a n} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(a+b*x**n),x)

[Out]

Piecewise((zoo*log(x), Eq(a, 0) & Eq(b, 0) & Eq(n, 0)), (log(x)/a, Eq(b, 0)), (l
og(x)/(a + b), Eq(n, 0)), (-x**(-n)/(b*n), Eq(a, 0)), (log(x)/a - log(a/b + x**n
)/(a*n), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{n} + a\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^n + a)*x),x, algorithm="giac")

[Out]

integrate(1/((b*x^n + a)*x), x)