3.108 \(\int \frac{x^{-1+n} \left (b+2 c x^n\right )}{a+b x^n+c x^{2 n}} \, dx\)

Optimal. Leaf size=19 \[ \frac{\log \left (a+b x^n+c x^{2 n}\right )}{n} \]

[Out]

Log[a + b*x^n + c*x^(2*n)]/n

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Rubi [A]  time = 0.0668016, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{\log \left (a+b x^n+c x^{2 n}\right )}{n} \]

Antiderivative was successfully verified.

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

[Out]

Log[a + b*x^n + c*x^(2*n)]/n

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Rubi in Sympy [A]  time = 12.1377, size = 15, normalized size = 0.79 \[ \frac{\log{\left (a + b x^{n} + c x^{2 n} \right )}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a + b*x**n + c*x**(2*n))/n

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Mathematica [A]  time = 0.0489865, size = 26, normalized size = 1.37 \[ \frac{\log \left (a x^{-2 n}+b x^{-n}+c\right )}{n}+2 \log (x) \]

Antiderivative was successfully verified.

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

[Out]

2*Log[x] + Log[c + a/x^(2*n) + b/x^n]/n

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Maple [A]  time = 0.033, size = 24, normalized size = 1.3 \[{\frac{\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }}+c \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2} \right ) }{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/n*ln(a+b*exp(n*ln(x))+c*exp(n*ln(x))^2)

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Maxima [A]  time = 0.825763, size = 31, normalized size = 1.63 \[ \frac{\log \left (\frac{c x^{2 \, n} + b x^{n} + a}{c}\right )}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log((c*x^(2*n) + b*x^n + a)/c)/n

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Fricas [A]  time = 0.297355, size = 26, normalized size = 1.37 \[ \frac{\log \left (c x^{2 \, n} + b x^{n} + a\right )}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(c*x^(2*n) + b*x^n + a)/n

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.26569, size = 26, normalized size = 1.37 \[ \frac{{\rm ln}\left (c x^{2 \, n} + b x^{n} + a\right )}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(c*x^(2*n) + b*x^n + a)/n