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

Optimal. Leaf size=21 \[ \frac{\log \left (a-b x^n-c x^{2 n}\right )}{n} \]

[Out]

Log[a - b*x^n - c*x^(2*n)]/n

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Rubi [A]  time = 0.0732985, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{\log \left (a-b x^n-c x^{2 n}\right )}{n} \]

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 = 12.9037, size = 15, normalized size = 0.71 \[ \frac{\log{\left (- a + b x^{n} + c x^{2 n} \right )}}{n} \]

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 = 0.0479568, size = 29, normalized size = 1.38 \[ \frac{\log \left (a x^{-2 n}-b x^{-n}-c\right )}{n}+2 \log (x) \]

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 = 26, normalized size = 1.2 \[{\frac{\ln \left ( -c \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}-b{{\rm e}^{n\ln \left ( x \right ) }}+a \right ) }{n}} \]

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

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Maxima [A]  time = 0.852579, size = 34, normalized size = 1.62 \[ \frac{\log \left (\frac{c x^{2 \, n} + b x^{n} - a}{c}\right )}{n} \]

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 = 0.311909, size = 28, normalized size = 1.33 \[ \frac{\log \left (c x^{2 \, n} + b x^{n} - a\right )}{n} \]

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 = 0.26687, size = 28, normalized size = 1.33 \[ \frac{{\rm ln}\left (c x^{2 \, n} + b x^{n} - a\right )}{n} \]

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