Optimal. Leaf size=24 \[ -\frac{x^{-3 n} \left (a+b x^n\right )^3}{3 a n} \]
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Rubi [A] time = 0.0210002, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{x^{-3 n} \left (a+b x^n\right )^3}{3 a n} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 - 3*n)*(a + b*x^n)^2,x]
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Rubi in Sympy [A] time = 3.11272, size = 19, normalized size = 0.79 \[ - \frac{x^{- 3 n} \left (a + b x^{n}\right )^{3}}{3 a n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1-3*n)*(a+b*x**n)**2,x)
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Mathematica [A] time = 0.0219348, size = 33, normalized size = 1.38 \[ -\frac{x^{-3 n} \left (a^2+3 a b x^n+3 b^2 x^{2 n}\right )}{3 n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 - 3*n)*(a + b*x^n)^2,x]
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Maple [A] time = 0.023, size = 45, normalized size = 1.9 \[{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}} \left ( -{\frac{{a}^{2}}{3\,n}}-{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{n}}-{\frac{a{{\rm e}^{n\ln \left ( x \right ) }}b}{n}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1-3*n)*(a+b*x^n)^2,x)
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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)^2*x^(-3*n - 1),x, algorithm="maxima")
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Fricas [A] time = 0.225579, size = 45, normalized size = 1.88 \[ -\frac{3 \, b^{2} x^{2 \, n} + 3 \, a b x^{n} + a^{2}}{3 \, n x^{3 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^2*x^(-3*n - 1),x, algorithm="fricas")
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Sympy [A] time = 38.6669, size = 39, normalized size = 1.62 \[ \begin{cases} - \frac{a^{2} x^{- 3 n}}{3 n} - \frac{a b x^{- 2 n}}{n} - \frac{b^{2} x^{- n}}{n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{2} \log{\left (x \right )} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(-1-3*n)*(a+b*x**n)**2,x)
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GIAC/XCAS [A] time = 0.219769, size = 47, normalized size = 1.96 \[ -\frac{{\left (3 \, b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 3 \, a b e^{\left (n{\rm ln}\left (x\right )\right )} + a^{2}\right )} e^{\left (-3 \, n{\rm ln}\left (x\right )\right )}}{3 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^2*x^(-3*n - 1),x, algorithm="giac")
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