3.494 \(\int \frac{x^{-1+3 n}}{b x^n+c x^{2 n}} \, dx\)

Optimal. Leaf size=28 \[ \frac{x^n}{c n}-\frac{b \log \left (b+c x^n\right )}{c^2 n} \]

[Out]

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

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Rubi [A]  time = 0.0479923, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{x^n}{c n}-\frac{b \log \left (b+c x^n\right )}{c^2 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 3*n)/(b*x^n + c*x^(2*n)),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{b \log{\left (b + c x^{n} \right )}}{c^{2} n} + \frac{\int ^{x^{n}} \frac{1}{c}\, dx}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+3*n)/(b*x**n+c*x**(2*n)),x)

[Out]

-b*log(b + c*x**n)/(c**2*n) + Integral(1/c, (x, x**n))/n

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Mathematica [A]  time = 0.017544, size = 24, normalized size = 0.86 \[ \frac{c x^n-b \log \left (b+c x^n\right )}{c^2 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 3*n)/(b*x^n + c*x^(2*n)),x]

[Out]

(c*x^n - b*Log[b + c*x^n])/(c^2*n)

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Maple [A]  time = 0.032, size = 33, normalized size = 1.2 \[{\frac{{{\rm e}^{n\ln \left ( x \right ) }}}{cn}}-{\frac{b\ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }}+b \right ) }{{c}^{2}n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+3*n)/(b*x^n+c*x^(2*n)),x)

[Out]

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

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Maxima [A]  time = 0.745154, size = 43, normalized size = 1.54 \[ \frac{x^{n}}{c n} - \frac{b \log \left (\frac{c x^{n} + b}{c}\right )}{c^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3*n - 1)/(c*x^(2*n) + b*x^n),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.298037, size = 32, normalized size = 1.14 \[ \frac{c x^{n} - b \log \left (c x^{n} + b\right )}{c^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3*n - 1)/(c*x^(2*n) + b*x^n),x, algorithm="fricas")

[Out]

(c*x^n - b*log(c*x^n + b))/(c^2*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+3*n)/(b*x**n+c*x**(2*n)),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3 \, n - 1}}{c x^{2 \, n} + b x^{n}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3*n - 1)/(c*x^(2*n) + b*x^n),x, algorithm="giac")

[Out]

integrate(x^(3*n - 1)/(c*x^(2*n) + b*x^n), x)