Optimal. Leaf size=83 \[ -\frac{a^5 x^{-n}}{n}+5 a^4 b \log (x)+\frac{10 a^3 b^2 x^n}{n}+\frac{5 a^2 b^3 x^{2 n}}{n}+\frac{5 a b^4 x^{3 n}}{3 n}+\frac{b^5 x^{4 n}}{4 n} \]
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Rubi [A] time = 0.0962486, antiderivative size = 83, 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 \[ -\frac{a^5 x^{-n}}{n}+5 a^4 b \log (x)+\frac{10 a^3 b^2 x^n}{n}+\frac{5 a^2 b^3 x^{2 n}}{n}+\frac{5 a b^4 x^{3 n}}{3 n}+\frac{b^5 x^{4 n}}{4 n} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 - n)*(a + b*x^n)^5,x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a^{5} x^{- n}}{n} + \frac{5 a^{4} b \log{\left (x^{n} \right )}}{n} + \frac{10 a^{3} b^{2} x^{n}}{n} + \frac{10 a^{2} b^{3} \int ^{x^{n}} x\, dx}{n} + \frac{5 a b^{4} x^{3 n}}{3 n} + \frac{b^{5} x^{4 n}}{4 n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1-n)*(a+b*x**n)**5,x)
[Out]
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Mathematica [A] time = 0.0424755, size = 72, normalized size = 0.87 \[ -\frac{12 a^5 x^{-n}-60 a^4 b n \log (x)-120 a^3 b^2 x^n-60 a^2 b^3 x^{2 n}-20 a b^4 x^{3 n}-3 b^5 x^{4 n}}{12 n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 - n)*(a + b*x^n)^5,x]
[Out]
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Maple [A] time = 0.027, size = 98, normalized size = 1.2 \[{\frac{1}{{{\rm e}^{n\ln \left ( x \right ) }}} \left ( 5\,{a}^{4}b\ln \left ( x \right ){{\rm e}^{n\ln \left ( x \right ) }}-{\frac{{a}^{5}}{n}}+{\frac{{b}^{5} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{5}}{4\,n}}+{\frac{5\,a{b}^{4} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{3\,n}}+5\,{\frac{{a}^{2}{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{n}}+10\,{\frac{{a}^{3}{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{n}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1-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^(-n - 1),x, algorithm="maxima")
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Fricas [A] time = 0.226039, size = 101, normalized size = 1.22 \[ \frac{60 \, a^{4} b n x^{n} \log \left (x\right ) + 3 \, b^{5} x^{5 \, n} + 20 \, a b^{4} x^{4 \, n} + 60 \, a^{2} b^{3} x^{3 \, n} + 120 \, a^{3} b^{2} x^{2 \, n} - 12 \, a^{5}}{12 \, n x^{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^5*x^(-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-n)*(a+b*x**n)**5,x)
[Out]
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GIAC/XCAS [A] time = 0.229502, size = 111, normalized size = 1.34 \[ \frac{{\left (60 \, a^{4} b n e^{\left (n{\rm ln}\left (x\right )\right )}{\rm ln}\left (x\right ) + 3 \, b^{5} e^{\left (5 \, n{\rm ln}\left (x\right )\right )} + 20 \, a b^{4} e^{\left (4 \, n{\rm ln}\left (x\right )\right )} + 60 \, a^{2} b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 120 \, a^{3} b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} - 12 \, a^{5}\right )} e^{\left (-n{\rm ln}\left (x\right )\right )}}{12 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^5*x^(-n - 1),x, algorithm="giac")
[Out]