Optimal. Leaf size=138 \[ a^8 \log (x)+\frac{8 a^7 b x^n}{n}+\frac{14 a^6 b^2 x^{2 n}}{n}+\frac{56 a^5 b^3 x^{3 n}}{3 n}+\frac{35 a^4 b^4 x^{4 n}}{2 n}+\frac{56 a^3 b^5 x^{5 n}}{5 n}+\frac{14 a^2 b^6 x^{6 n}}{3 n}+\frac{8 a b^7 x^{7 n}}{7 n}+\frac{b^8 x^{8 n}}{8 n} \]
[Out]
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Rubi [A] time = 0.140255, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ a^8 \log (x)+\frac{8 a^7 b x^n}{n}+\frac{14 a^6 b^2 x^{2 n}}{n}+\frac{56 a^5 b^3 x^{3 n}}{3 n}+\frac{35 a^4 b^4 x^{4 n}}{2 n}+\frac{56 a^3 b^5 x^{5 n}}{5 n}+\frac{14 a^2 b^6 x^{6 n}}{3 n}+\frac{8 a b^7 x^{7 n}}{7 n}+\frac{b^8 x^{8 n}}{8 n} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^n)^8/x,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{a^{8} \log{\left (x^{n} \right )}}{n} + \frac{8 a^{7} b x^{n}}{n} + \frac{28 a^{6} b^{2} \int ^{x^{n}} x\, dx}{n} + \frac{56 a^{5} b^{3} x^{3 n}}{3 n} + \frac{35 a^{4} b^{4} x^{4 n}}{2 n} + \frac{56 a^{3} b^{5} x^{5 n}}{5 n} + \frac{14 a^{2} b^{6} x^{6 n}}{3 n} + \frac{8 a b^{7} x^{7 n}}{7 n} + \frac{b^{8} x^{8 n}}{8 n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*x**n)**8/x,x)
[Out]
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Mathematica [A] time = 0.0838256, size = 106, normalized size = 0.77 \[ a^8 \log (x)+\frac{b x^n \left (6720 a^7+11760 a^6 b x^n+15680 a^5 b^2 x^{2 n}+14700 a^4 b^3 x^{3 n}+9408 a^3 b^4 x^{4 n}+3920 a^2 b^5 x^{5 n}+960 a b^6 x^{6 n}+105 b^7 x^{7 n}\right )}{840 n} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^n)^8/x,x]
[Out]
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Maple [A] time = 0.003, size = 132, normalized size = 1. \[{\frac{{b}^{8} \left ({x}^{n} \right ) ^{8}}{8\,n}}+{\frac{8\,a{b}^{7} \left ({x}^{n} \right ) ^{7}}{7\,n}}+{\frac{14\,{a}^{2}{b}^{6} \left ({x}^{n} \right ) ^{6}}{3\,n}}+{\frac{56\,{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{5}}{5\,n}}+{\frac{35\,{a}^{4}{b}^{4} \left ({x}^{n} \right ) ^{4}}{2\,n}}+{\frac{56\,{a}^{5}{b}^{3} \left ({x}^{n} \right ) ^{3}}{3\,n}}+14\,{\frac{{a}^{6}{b}^{2} \left ({x}^{n} \right ) ^{2}}{n}}+8\,{\frac{b{a}^{7}{x}^{n}}{n}}+{\frac{{a}^{8}\ln \left ({x}^{n} \right ) }{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*x^n)^8/x,x)
[Out]
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Maxima [A] time = 1.44968, size = 153, normalized size = 1.11 \[ \frac{a^{8} \log \left (x^{n}\right )}{n} + \frac{105 \, b^{8} x^{8 \, n} + 960 \, a b^{7} x^{7 \, n} + 3920 \, a^{2} b^{6} x^{6 \, n} + 9408 \, a^{3} b^{5} x^{5 \, n} + 14700 \, a^{4} b^{4} x^{4 \, n} + 15680 \, a^{5} b^{3} x^{3 \, n} + 11760 \, a^{6} b^{2} x^{2 \, n} + 6720 \, a^{7} b x^{n}}{840 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^8/x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.226953, size = 147, normalized size = 1.07 \[ \frac{840 \, a^{8} n \log \left (x\right ) + 105 \, b^{8} x^{8 \, n} + 960 \, a b^{7} x^{7 \, n} + 3920 \, a^{2} b^{6} x^{6 \, n} + 9408 \, a^{3} b^{5} x^{5 \, n} + 14700 \, a^{4} b^{4} x^{4 \, n} + 15680 \, a^{5} b^{3} x^{3 \, n} + 11760 \, a^{6} b^{2} x^{2 \, n} + 6720 \, a^{7} b x^{n}}{840 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^8/x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.77068, size = 136, normalized size = 0.99 \[ \begin{cases} a^{8} \log{\left (x \right )} + \frac{8 a^{7} b x^{n}}{n} + \frac{14 a^{6} b^{2} x^{2 n}}{n} + \frac{56 a^{5} b^{3} x^{3 n}}{3 n} + \frac{35 a^{4} b^{4} x^{4 n}}{2 n} + \frac{56 a^{3} b^{5} x^{5 n}}{5 n} + \frac{14 a^{2} b^{6} x^{6 n}}{3 n} + \frac{8 a b^{7} x^{7 n}}{7 n} + \frac{b^{8} x^{8 n}}{8 n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{8} \log{\left (x \right )} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*x**n)**8/x,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{n} + a\right )}^{8}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^8/x,x, algorithm="giac")
[Out]