Optimal. Leaf size=56 \[ -\frac{8 \log \left (b x^n+2\right )}{b^4 n}+\frac{4 x^n}{b^3 n}-\frac{x^{2 n}}{b^2 n}+\frac{x^{3 n}}{3 b n} \]
[Out]
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Rubi [A] time = 0.0726141, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{8 \log \left (b x^n+2\right )}{b^4 n}+\frac{4 x^n}{b^3 n}-\frac{x^{2 n}}{b^2 n}+\frac{x^{3 n}}{3 b n} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 + 4*n)/(2 + b*x^n),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{x^{3 n}}{3 b n} - \frac{2 \int ^{x^{n}} x\, dx}{b^{2} n} + \frac{4 x^{n}}{b^{3} n} - \frac{8 \log{\left (b x^{n} + 2 \right )}}{b^{4} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1+4*n)/(2+b*x**n),x)
[Out]
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Mathematica [A] time = 0.0279009, size = 43, normalized size = 0.77 \[ \frac{b x^n \left (b^2 x^{2 n}-3 b x^n+12\right )-24 \log \left (b x^n+2\right )}{3 b^4 n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 + 4*n)/(2 + b*x^n),x]
[Out]
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Maple [A] time = 0.034, size = 63, normalized size = 1.1 \[ 4\,{\frac{{{\rm e}^{n\ln \left ( x \right ) }}}{{b}^{3}n}}-{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{b}^{2}n}}+{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{3\,bn}}-8\,{\frac{\ln \left ( 2+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{b}^{4}n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1+4*n)/(2+b*x^n),x)
[Out]
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Maxima [A] time = 1.44734, size = 70, normalized size = 1.25 \[ \frac{b^{2} x^{3 \, n} - 3 \, b x^{2 \, n} + 12 \, x^{n}}{3 \, b^{3} n} - \frac{8 \, \log \left (\frac{b x^{n} + 2}{b}\right )}{b^{4} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(4*n - 1)/(b*x^n + 2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.22594, size = 59, normalized size = 1.05 \[ \frac{b^{3} x^{3 \, n} - 3 \, b^{2} x^{2 \, n} + 12 \, b x^{n} - 24 \, \log \left (b x^{n} + 2\right )}{3 \, b^{4} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(4*n - 1)/(b*x^n + 2),x, algorithm="fricas")
[Out]
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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)/(2+b*x**n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4 \, n - 1}}{b x^{n} + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(4*n - 1)/(b*x^n + 2),x, algorithm="giac")
[Out]