Optimal. Leaf size=28 \[ \frac{x^n}{b n}-\frac{a \log \left (a+b x^n\right )}{b^2 n} \]
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Rubi [A] time = 0.049351, antiderivative size = 28, 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 \[ \frac{x^n}{b n}-\frac{a \log \left (a+b x^n\right )}{b^2 n} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 + 2*n)/(a + b*x^n),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a \log{\left (a + b x^{n} \right )}}{b^{2} n} + \frac{\int ^{x^{n}} \frac{1}{b}\, dx}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1+2*n)/(a+b*x**n),x)
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Mathematica [A] time = 0.0177155, size = 24, normalized size = 0.86 \[ \frac{b x^n-a \log \left (a+b x^n\right )}{b^2 n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 + 2*n)/(a + b*x^n),x]
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Maple [A] time = 0.026, size = 33, normalized size = 1.2 \[{\frac{{{\rm e}^{n\ln \left ( x \right ) }}}{bn}}-{\frac{a\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{b}^{2}n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1+2*n)/(a+b*x^n),x)
[Out]
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Maxima [A] time = 1.4294, size = 43, normalized size = 1.54 \[ \frac{x^{n}}{b n} - \frac{a \log \left (\frac{b x^{n} + a}{b}\right )}{b^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(2*n - 1)/(b*x^n + a),x, algorithm="maxima")
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Fricas [A] time = 0.226178, size = 32, normalized size = 1.14 \[ \frac{b x^{n} - a \log \left (b x^{n} + a\right )}{b^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(2*n - 1)/(b*x^n + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 46.084, size = 41, normalized size = 1.46 \[ \begin{cases} \frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \wedge n = 0 \\\frac{\log{\left (x \right )}}{a + b} & \text{for}\: n = 0 \\\frac{x^{2 n}}{2 a n} & \text{for}\: b = 0 \\- \frac{a \log{\left (\frac{a}{b} + x^{n} \right )}}{b^{2} n} + \frac{x^{n}}{b n} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(-1+2*n)/(a+b*x**n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2 \, n - 1}}{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(2*n - 1)/(b*x^n + a),x, algorithm="giac")
[Out]