3.167 \(\int \frac{b+2 c x^n}{b x+c x^{1+n}} \, dx\)

Optimal. Leaf size=15 \[ \frac{\log \left (b+c x^n\right )}{n}+\log (x) \]

[Out]

Log[x] + Log[b + c*x^n]/n

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Rubi [A]  time = 0.0623564, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{\log \left (b+c x^n\right )}{n}+\log (x) \]

Antiderivative was successfully verified.

[In]  Int[(b + 2*c*x^n)/(b*x + c*x^(1 + n)),x]

[Out]

Log[x] + Log[b + c*x^n]/n

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Rubi in Sympy [A]  time = 11.0648, size = 15, normalized size = 1. \[ \frac{\log{\left (x^{n} \right )}}{n} + \frac{\log{\left (b + c x^{n} \right )}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b+2*c*x**n)/(b*x+c*x**(1+n)),x)

[Out]

log(x**n)/n + log(b + c*x**n)/n

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Mathematica [A]  time = 0.0167562, size = 15, normalized size = 1. \[ \frac{\log \left (b+c x^n\right )}{n}+\log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(b + 2*c*x^n)/(b*x + c*x^(1 + n)),x]

[Out]

Log[x] + Log[b + c*x^n]/n

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Maple [A]  time = 0.022, size = 18, normalized size = 1.2 \[ \ln \left ( x \right ) +{\frac{\ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }}+b \right ) }{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b+2*c*x^n)/(b*x+c*x^(1+n)),x)

[Out]

ln(x)+1/n*ln(c*exp(n*ln(x))+b)

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Maxima [A]  time = 0.817206, size = 63, normalized size = 4.2 \[ b{\left (\frac{\log \left (x\right )}{b} - \frac{\log \left (\frac{c x^{n} + b}{c}\right )}{b n}\right )} + \frac{2 \, \log \left (\frac{c x^{n} + b}{c}\right )}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^n + b)/(b*x + c*x^(n + 1)),x, algorithm="maxima")

[Out]

b*(log(x)/b - log((c*x^n + b)/c)/(b*n)) + 2*log((c*x^n + b)/c)/n

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Fricas [A]  time = 0.267056, size = 31, normalized size = 2.07 \[ \frac{{\left (n - 1\right )} \log \left (x\right ) + \log \left (b x + c x^{n + 1}\right )}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^n + b)/(b*x + c*x^(n + 1)),x, algorithm="fricas")

[Out]

((n - 1)*log(x) + log(b*x + c*x^(n + 1)))/n

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Sympy [A]  time = 4.51914, size = 29, normalized size = 1.93 \[ \begin{cases} \log{\left (x \right )} & \text{for}\: c = 0 \wedge \left (c = 0 \vee n = 0\right ) \\\frac{\left (b + 2 c\right ) \log{\left (x \right )}}{b + c} & \text{for}\: n = 0 \\\log{\left (x \right )} + \frac{\log{\left (\frac{b}{c} + x^{n} \right )}}{n} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b+2*c*x**n)/(b*x+c*x**(1+n)),x)

[Out]

Piecewise((log(x), Eq(c, 0) & (Eq(c, 0) | Eq(n, 0))), ((b + 2*c)*log(x)/(b + c),
Eq(n, 0)), (log(x) + log(b/c + x**n)/n, True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{2 \, c x^{n} + b}{b x + c x^{n + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^n + b)/(b*x + c*x^(n + 1)),x, algorithm="giac")

[Out]

integrate((2*c*x^n + b)/(b*x + c*x^(n + 1)), x)