Optimal. Leaf size=82 \[ -\frac{a^5 x^{-4 n}}{4 n}-\frac{5 a^4 b x^{-3 n}}{3 n}-\frac{5 a^3 b^2 x^{-2 n}}{n}-\frac{10 a^2 b^3 x^{-n}}{n}+5 a b^4 \log (x)+\frac{b^5 x^n}{n} \]
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Rubi [A] time = 0.0974764, antiderivative size = 82, 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 \[ -\frac{a^5 x^{-4 n}}{4 n}-\frac{5 a^4 b x^{-3 n}}{3 n}-\frac{5 a^3 b^2 x^{-2 n}}{n}-\frac{10 a^2 b^3 x^{-n}}{n}+5 a b^4 \log (x)+\frac{b^5 x^n}{n} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 - 4*n)*(a + b*x^n)^5,x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a^{5} x^{- 4 n}}{4 n} - \frac{5 a^{4} b x^{- 3 n}}{3 n} - \frac{5 a^{3} b^{2} x^{- 2 n}}{n} - \frac{10 a^{2} b^{3} x^{- n}}{n} + \frac{5 a b^{4} \log{\left (x^{n} \right )}}{n} + \frac{\int ^{x^{n}} b^{5}\, dx}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1-4*n)*(a+b*x**n)**5,x)
[Out]
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Mathematica [A] time = 0.101281, size = 72, normalized size = 0.88 \[ 5 a b^4 \log (x)-\frac{x^{-4 n} \left (3 a^5+20 a^4 b x^n+60 a^3 b^2 x^{2 n}+120 a^2 b^3 x^{3 n}-12 b^5 x^{5 n}\right )}{12 n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 - 4*n)*(a + b*x^n)^5,x]
[Out]
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Maple [A] time = 0.028, size = 97, normalized size = 1.2 \[{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}} \left ({\frac{{b}^{5} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{5}}{n}}+5\,a{b}^{4}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}-{\frac{{a}^{5}}{4\,n}}-10\,{\frac{{a}^{2}{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{n}}-5\,{\frac{{a}^{3}{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{n}}-{\frac{5\,{a}^{4}b{{\rm e}^{n\ln \left ( x \right ) }}}{3\,n}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1-4*n)*(a+b*x^n)^5,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)^5*x^(-4*n - 1),x, algorithm="maxima")
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Fricas [A] time = 0.226463, size = 104, normalized size = 1.27 \[ \frac{60 \, a b^{4} n x^{4 \, n} \log \left (x\right ) + 12 \, b^{5} x^{5 \, n} - 120 \, a^{2} b^{3} x^{3 \, n} - 60 \, a^{3} b^{2} x^{2 \, n} - 20 \, a^{4} b x^{n} - 3 \, a^{5}}{12 \, n x^{4 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^5*x^(-4*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**(-1-4*n)*(a+b*x**n)**5,x)
[Out]
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GIAC/XCAS [A] time = 0.228652, size = 111, normalized size = 1.35 \[ \frac{{\left (60 \, a b^{4} n e^{\left (4 \, n{\rm ln}\left (x\right )\right )}{\rm ln}\left (x\right ) + 12 \, b^{5} e^{\left (5 \, n{\rm ln}\left (x\right )\right )} - 120 \, a^{2} b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} - 60 \, a^{3} b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} - 20 \, a^{4} b e^{\left (n{\rm ln}\left (x\right )\right )} - 3 \, a^{5}\right )} e^{\left (-4 \, n{\rm ln}\left (x\right )\right )}}{12 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^5*x^(-4*n - 1),x, algorithm="giac")
[Out]