Optimal. Leaf size=52 \[ -\frac{a^3 x^{-3 n}}{3 n}-\frac{3 a^2 b x^{-2 n}}{2 n}-\frac{3 a b^2 x^{-n}}{n}+b^3 \log (x) \]
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Rubi [A] time = 0.0628968, antiderivative size = 52, 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 \[ -\frac{a^3 x^{-3 n}}{3 n}-\frac{3 a^2 b x^{-2 n}}{2 n}-\frac{3 a b^2 x^{-n}}{n}+b^3 \log (x) \]
Antiderivative was successfully verified.
[In] Int[x^(-1 - 3*n)*(a + b*x^n)^3,x]
[Out]
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Rubi in Sympy [A] time = 10.6021, size = 48, normalized size = 0.92 \[ - \frac{a^{3} x^{- 3 n}}{3 n} - \frac{3 a^{2} b x^{- 2 n}}{2 n} - \frac{3 a b^{2} x^{- n}}{n} + \frac{b^{3} \log{\left (x^{n} \right )}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1-3*n)*(a+b*x**n)**3,x)
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Mathematica [A] time = 0.0474944, size = 43, normalized size = 0.83 \[ b^3 \log (x)-\frac{a x^{-3 n} \left (2 a^2+9 a b x^n+18 b^2 x^{2 n}\right )}{6 n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 - 3*n)*(a + b*x^n)^3,x]
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Maple [A] time = 0.024, size = 61, normalized size = 1.2 \[{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}} \left ({b}^{3}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}-{\frac{{a}^{3}}{3\,n}}-3\,{\frac{a{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{n}}-{\frac{3\,{a}^{2}b{{\rm e}^{n\ln \left ( x \right ) }}}{2\,n}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1-3*n)*(a+b*x^n)^3,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)^3*x^(-3*n - 1),x, algorithm="maxima")
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Fricas [A] time = 0.225226, size = 69, normalized size = 1.33 \[ \frac{6 \, b^{3} n x^{3 \, n} \log \left (x\right ) - 18 \, a b^{2} x^{2 \, n} - 9 \, a^{2} b x^{n} - 2 \, a^{3}}{6 \, n x^{3 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^3*x^(-3*n - 1),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**(-1-3*n)*(a+b*x**n)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.220839, size = 73, normalized size = 1.4 \[ \frac{{\left (6 \, b^{3} n e^{\left (3 \, n{\rm ln}\left (x\right )\right )}{\rm ln}\left (x\right ) - 18 \, a b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} - 9 \, a^{2} b e^{\left (n{\rm ln}\left (x\right )\right )} - 2 \, a^{3}\right )} e^{\left (-3 \, n{\rm ln}\left (x\right )\right )}}{6 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^3*x^(-3*n - 1),x, algorithm="giac")
[Out]