3.493 \(\int \frac{x^{-1+4 n}}{b x^n+c x^{2 n}} \, dx\)

Optimal. Leaf size=46 \[ \frac{b^2 \log \left (b+c x^n\right )}{c^3 n}-\frac{b x^n}{c^2 n}+\frac{x^{2 n}}{2 c n} \]

[Out]

-((b*x^n)/(c^2*n)) + x^(2*n)/(2*c*n) + (b^2*Log[b + c*x^n])/(c^3*n)

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Rubi [A]  time = 0.0716772, antiderivative size = 46, 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 \[ \frac{b^2 \log \left (b+c x^n\right )}{c^3 n}-\frac{b x^n}{c^2 n}+\frac{x^{2 n}}{2 c n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 4*n)/(b*x^n + c*x^(2*n)),x]

[Out]

-((b*x^n)/(c^2*n)) + x^(2*n)/(2*c*n) + (b^2*Log[b + c*x^n])/(c^3*n)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b^{2} \log{\left (b + c x^{n} \right )}}{c^{3} n} + \frac{\int ^{x^{n}} x\, dx}{c n} - \frac{\int ^{x^{n}} b\, dx}{c^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+4*n)/(b*x**n+c*x**(2*n)),x)

[Out]

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

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Mathematica [A]  time = 0.0326373, size = 38, normalized size = 0.83 \[ \frac{2 b^2 \log \left (b+c x^n\right )+c x^n \left (c x^n-2 b\right )}{2 c^3 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 4*n)/(b*x^n + c*x^(2*n)),x]

[Out]

(c*x^n*(-2*b + c*x^n) + 2*b^2*Log[b + c*x^n])/(2*c^3*n)

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Maple [A]  time = 0.04, size = 62, normalized size = 1.4 \[{\frac{1}{{{\rm e}^{n\ln \left ( x \right ) }}} \left ({\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{2\,cn}}-{\frac{b \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{c}^{2}n}} \right ) }+{\frac{{b}^{2}\ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }}+b \right ) }{{c}^{3}n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+4*n)/(b*x^n+c*x^(2*n)),x)

[Out]

(1/2/c/n*exp(n*ln(x))^3-b/c^2/n*exp(n*ln(x))^2)/exp(n*ln(x))+b^2/c^3/n*ln(c*exp(
n*ln(x))+b)

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Maxima [A]  time = 0.754667, size = 61, normalized size = 1.33 \[ \frac{b^{2} \log \left (\frac{c x^{n} + b}{c}\right )}{c^{3} n} + \frac{c x^{2 \, n} - 2 \, b x^{n}}{2 \, c^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(4*n - 1)/(c*x^(2*n) + b*x^n),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.303133, size = 51, normalized size = 1.11 \[ \frac{c^{2} x^{2 \, n} - 2 \, b c x^{n} + 2 \, b^{2} \log \left (c x^{n} + b\right )}{2 \, c^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(4*n - 1)/(c*x^(2*n) + b*x^n),x, algorithm="fricas")

[Out]

1/2*(c^2*x^(2*n) - 2*b*c*x^n + 2*b^2*log(c*x^n + b))/(c^3*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+4*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. \[ \int \frac{x^{4 \, n - 1}}{c x^{2 \, n} + b x^{n}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(4*n - 1)/(c*x^(2*n) + b*x^n),x, algorithm="giac")

[Out]

integrate(x^(4*n - 1)/(c*x^(2*n) + b*x^n), x)