Optimal. Leaf size=25 \[ -\frac{a x^{-2 n}}{2 n}-\frac{b x^{-n}}{n} \]
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Rubi [A] time = 0.0222497, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{a x^{-2 n}}{2 n}-\frac{b x^{-n}}{n} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 - 2*n)*(a + b*x^n),x]
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Rubi in Sympy [A] time = 3.95813, size = 17, normalized size = 0.68 \[ - \frac{a x^{- 2 n}}{2 n} - \frac{b x^{- n}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1-2*n)*(a+b*x**n),x)
[Out]
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Mathematica [A] time = 0.0106446, size = 20, normalized size = 0.8 \[ -\frac{x^{-2 n} \left (a+2 b x^n\right )}{2 n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 - 2*n)*(a + b*x^n),x]
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Maple [A] time = 0.023, size = 27, normalized size = 1.1 \[{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ( -{\frac{a}{2\,n}}-{\frac{b{{\rm e}^{n\ln \left ( x \right ) }}}{n}} \right ) } \]
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 [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)*x^(-2*n - 1),x, algorithm="maxima")
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Fricas [A] time = 0.221604, size = 27, normalized size = 1.08 \[ -\frac{2 \, b x^{n} + a}{2 \, n x^{2 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)*x^(-2*n - 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 16.2041, size = 24, normalized size = 0.96 \[ \begin{cases} - \frac{a x^{- 2 n}}{2 n} - \frac{b x^{- n}}{n} & \text{for}\: n \neq 0 \\\left (a + b\right ) \log{\left (x \right )} & \text{otherwise} \end{cases} \]
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 [A] time = 0.214343, size = 28, normalized size = 1.12 \[ -\frac{{\left (2 \, b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right )} e^{\left (-2 \, n{\rm ln}\left (x\right )\right )}}{2 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)*x^(-2*n - 1),x, algorithm="giac")
[Out]