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

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

[Out]

Log[b + c*x^n]/(c*n)

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Rubi [A]  time = 0.0230653, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{\log \left (b+c x^n\right )}{c n} \]

Antiderivative was successfully verified.

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

[Out]

Log[b + c*x^n]/(c*n)

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Rubi in Sympy [A]  time = 4.48543, size = 10, normalized size = 0.67 \[ \frac{\log{\left (b + c x^{n} \right )}}{c n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(b + c*x**n)/(c*n)

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Mathematica [A]  time = 0.00442025, size = 15, normalized size = 1. \[ \frac{\log \left (b+c x^n\right )}{c n} \]

Antiderivative was successfully verified.

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

[Out]

Log[b + c*x^n]/(c*n)

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Maple [A]  time = 0.029, size = 18, normalized size = 1.2 \[{\frac{\ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }}+b \right ) }{cn}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.747025, size = 26, normalized size = 1.73 \[ \frac{\log \left (\frac{c x^{n} + b}{c}\right )}{c n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log((c*x^n + b)/c)/(c*n)

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Fricas [A]  time = 0.286718, size = 20, normalized size = 1.33 \[ \frac{\log \left (c x^{n} + b\right )}{c n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(c*x^n + b)/(c*n)

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Sympy [A]  time = 58.4706, size = 37, normalized size = 2.47 \[ \begin{cases} \frac{\log{\left (x \right )}}{b} & \text{for}\: c = 0 \wedge n = 0 \\\frac{x^{n}}{b n} & \text{for}\: c = 0 \\\frac{\log{\left (x \right )}}{b + c} & \text{for}\: n = 0 \\- \frac{\log{\left (x \right )}}{c} + \frac{\log{\left (\frac{b x^{n}}{c} + x^{2 n} \right )}}{c n} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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