3.450 \(\int \frac{a+b n x^{-1+n}}{a x+b x^n} \, dx\)

Optimal. Leaf size=10 \[ \log \left (a x+b x^n\right ) \]

[Out]

Log[a*x + b*x^n]

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Rubi [A]  time = 0.140964, antiderivative size = 17, normalized size of antiderivative = 1.7, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \log \left (a x^{1-n}+b\right )+n \log (x) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*n*x^(-1 + n))/(a*x + b*x^n),x]

[Out]

n*Log[x] + Log[b + a*x^(1 - n)]

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Rubi in Sympy [A]  time = 8.96809, size = 20, normalized size = 2. \[ \frac{n \log{\left (x^{- n + 1} \right )}}{- n + 1} + \log{\left (a x^{- n + 1} + b \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

n*log(x**(-n + 1))/(-n + 1) + log(a*x**(-n + 1) + b)

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Mathematica [A]  time = 0.018744, size = 10, normalized size = 1. \[ \log \left (a x+b x^n\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*n*x^(-1 + n))/(a*x + b*x^n),x]

[Out]

Log[a*x + b*x^n]

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Maple [A]  time = 0.024, size = 13, normalized size = 1.3 \[ \ln \left ( ax+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*n*x^(-1+n))/(a*x+b*x^n),x)

[Out]

ln(a*x+b*exp(n*ln(x)))

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Maxima [A]  time = 0.686981, size = 14, normalized size = 1.4 \[ \log \left (a x + b x^{n}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a*x + b*x^n)

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Fricas [A]  time = 0.308787, size = 14, normalized size = 1.4 \[ \log \left (a x + b x^{n}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a*x + b*x^n)

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Sympy [A]  time = 30.6255, size = 32, normalized size = 3.2 \[ \begin{cases} \log{\left (x + \frac{b x^{n}}{a} \right )} & \text{for}\: a \neq 0 \\n \left (\frac{n^{2} \log{\left (x \right )}}{n^{2} - n} - \frac{n \log{\left (x \right )}}{n^{2} - n}\right ) & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((log(x + b*x**n/a), Ne(a, 0)), (n*(n**2*log(x)/(n**2 - n) - n*log(x)/(
n**2 - n)), True))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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