Optimal. Leaf size=135 \[ -\frac{a^8 x^{-6 n}}{6 n}-\frac{8 a^7 b x^{-5 n}}{5 n}-\frac{7 a^6 b^2 x^{-4 n}}{n}-\frac{56 a^5 b^3 x^{-3 n}}{3 n}-\frac{35 a^4 b^4 x^{-2 n}}{n}-\frac{56 a^3 b^5 x^{-n}}{n}+28 a^2 b^6 \log (x)+\frac{8 a b^7 x^n}{n}+\frac{b^8 x^{2 n}}{2 n} \]
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Rubi [A] time = 0.158556, antiderivative size = 135, 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 \[ -\frac{a^8 x^{-6 n}}{6 n}-\frac{8 a^7 b x^{-5 n}}{5 n}-\frac{7 a^6 b^2 x^{-4 n}}{n}-\frac{56 a^5 b^3 x^{-3 n}}{3 n}-\frac{35 a^4 b^4 x^{-2 n}}{n}-\frac{56 a^3 b^5 x^{-n}}{n}+28 a^2 b^6 \log (x)+\frac{8 a b^7 x^n}{n}+\frac{b^8 x^{2 n}}{2 n} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 - 6*n)*(a + b*x^n)^8,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a^{8} x^{- 6 n}}{6 n} - \frac{8 a^{7} b x^{- 5 n}}{5 n} - \frac{7 a^{6} b^{2} x^{- 4 n}}{n} - \frac{56 a^{5} b^{3} x^{- 3 n}}{3 n} - \frac{35 a^{4} b^{4} x^{- 2 n}}{n} - \frac{56 a^{3} b^{5} x^{- n}}{n} + \frac{28 a^{2} b^{6} \log{\left (x^{n} \right )}}{n} + \frac{8 a b^{7} x^{n}}{n} + \frac{b^{8} \int ^{x^{n}} x\, dx}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1-6*n)*(a+b*x**n)**8,x)
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Mathematica [A] time = 0.155288, size = 111, normalized size = 0.82 \[ 28 a^2 b^6 \log (x)-\frac{x^{-6 n} \left (5 a^8+48 a^7 b x^n+210 a^6 b^2 x^{2 n}+560 a^5 b^3 x^{3 n}+1050 a^4 b^4 x^{4 n}+1680 a^3 b^5 x^{5 n}-240 a b^7 x^{7 n}-15 b^8 x^{8 n}\right )}{30 n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 - 6*n)*(a + b*x^n)^8,x]
[Out]
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Maple [A] time = 0.043, size = 128, normalized size = 1. \[ 28\,{a}^{2}{b}^{6}\ln \left ( x \right ) +{\frac{{b}^{8} \left ({x}^{n} \right ) ^{2}}{2\,n}}+8\,{\frac{a{b}^{7}{x}^{n}}{n}}-56\,{\frac{{a}^{3}{b}^{5}}{n{x}^{n}}}-35\,{\frac{{a}^{4}{b}^{4}}{n \left ({x}^{n} \right ) ^{2}}}-{\frac{56\,{a}^{5}{b}^{3}}{3\,n \left ({x}^{n} \right ) ^{3}}}-7\,{\frac{{a}^{6}{b}^{2}}{n \left ({x}^{n} \right ) ^{4}}}-{\frac{8\,b{a}^{7}}{5\,n \left ({x}^{n} \right ) ^{5}}}-{\frac{{a}^{8}}{6\,n \left ({x}^{n} \right ) ^{6}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1-6*n)*(a+b*x^n)^8,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)^8*x^(-6*n - 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.228656, size = 157, normalized size = 1.16 \[ \frac{840 \, a^{2} b^{6} n x^{6 \, n} \log \left (x\right ) + 15 \, b^{8} x^{8 \, n} + 240 \, a b^{7} x^{7 \, n} - 1680 \, a^{3} b^{5} x^{5 \, n} - 1050 \, a^{4} b^{4} x^{4 \, n} - 560 \, a^{5} b^{3} x^{3 \, n} - 210 \, a^{6} b^{2} x^{2 \, n} - 48 \, a^{7} b x^{n} - 5 \, a^{8}}{30 \, n x^{6 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^8*x^(-6*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**(-1-6*n)*(a+b*x**n)**8,x)
[Out]
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GIAC/XCAS [A] time = 0.236683, size = 167, normalized size = 1.24 \[ \frac{{\left (840 \, a^{2} b^{6} n e^{\left (6 \, n{\rm ln}\left (x\right )\right )}{\rm ln}\left (x\right ) + 15 \, b^{8} e^{\left (8 \, n{\rm ln}\left (x\right )\right )} + 240 \, a b^{7} e^{\left (7 \, n{\rm ln}\left (x\right )\right )} - 1680 \, a^{3} b^{5} e^{\left (5 \, n{\rm ln}\left (x\right )\right )} - 1050 \, a^{4} b^{4} e^{\left (4 \, n{\rm ln}\left (x\right )\right )} - 560 \, a^{5} b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} - 210 \, a^{6} b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} - 48 \, a^{7} b e^{\left (n{\rm ln}\left (x\right )\right )} - 5 \, a^{8}\right )} e^{\left (-6 \, n{\rm ln}\left (x\right )\right )}}{30 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^8*x^(-6*n - 1),x, algorithm="giac")
[Out]