Optimal. Leaf size=138 \[ a^8 \log (x)+\frac {8 a^7 b x^n}{n}+\frac {14 a^6 b^2 x^{2 n}}{n}+\frac {56 a^5 b^3 x^{3 n}}{3 n}+\frac {35 a^4 b^4 x^{4 n}}{2 n}+\frac {56 a^3 b^5 x^{5 n}}{5 n}+\frac {14 a^2 b^6 x^{6 n}}{3 n}+\frac {8 a b^7 x^{7 n}}{7 n}+\frac {b^8 x^{8 n}}{8 n} \]
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Rubi [A] time = 0.05, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac {14 a^6 b^2 x^{2 n}}{n}+\frac {56 a^5 b^3 x^{3 n}}{3 n}+\frac {35 a^4 b^4 x^{4 n}}{2 n}+\frac {56 a^3 b^5 x^{5 n}}{5 n}+\frac {14 a^2 b^6 x^{6 n}}{3 n}+\frac {8 a^7 b x^n}{n}+a^8 \log (x)+\frac {8 a b^7 x^{7 n}}{7 n}+\frac {b^8 x^{8 n}}{8 n} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b x^n\right )^8}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^8}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (8 a^7 b+\frac {a^8}{x}+28 a^6 b^2 x+56 a^5 b^3 x^2+70 a^4 b^4 x^3+56 a^3 b^5 x^4+28 a^2 b^6 x^5+8 a b^7 x^6+b^8 x^7\right ) \, dx,x,x^n\right )}{n}\\ &=\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}+a^8 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.08, size = 119, normalized size = 0.86 \[ \frac {a^8 n \log (x)+8 a^7 b x^n+14 a^6 b^2 x^{2 n}+\frac {56}{3} a^5 b^3 x^{3 n}+\frac {35}{2} a^4 b^4 x^{4 n}+\frac {56}{5} a^3 b^5 x^{5 n}+\frac {14}{3} a^2 b^6 x^{6 n}+\frac {8}{7} a b^7 x^{7 n}+\frac {1}{8} b^8 x^{8 n}}{n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 109, normalized size = 0.79 \[ \frac {840 \, a^{8} n \log \relax (x) + 105 \, b^{8} x^{8 \, n} + 960 \, a b^{7} x^{7 \, n} + 3920 \, a^{2} b^{6} x^{6 \, n} + 9408 \, a^{3} b^{5} x^{5 \, n} + 14700 \, a^{4} b^{4} x^{4 \, n} + 15680 \, a^{5} b^{3} x^{3 \, n} + 11760 \, a^{6} b^{2} x^{2 \, n} + 6720 \, a^{7} b x^{n}}{840 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{n} + a\right )}^{8}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 132, normalized size = 0.96 \[ \frac {a^{8} \ln \left (x^{n}\right )}{n}+\frac {8 a^{7} b \,x^{n}}{n}+\frac {14 a^{6} b^{2} x^{2 n}}{n}+\frac {56 a^{5} b^{3} x^{3 n}}{3 n}+\frac {35 a^{4} b^{4} x^{4 n}}{2 n}+\frac {56 a^{3} b^{5} x^{5 n}}{5 n}+\frac {14 a^{2} b^{6} x^{6 n}}{3 n}+\frac {8 a \,b^{7} x^{7 n}}{7 n}+\frac {b^{8} x^{8 n}}{8 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 113, normalized size = 0.82 \[ \frac {a^{8} \log \left (x^{n}\right )}{n} + \frac {105 \, b^{8} x^{8 \, n} + 960 \, a b^{7} x^{7 \, n} + 3920 \, a^{2} b^{6} x^{6 \, n} + 9408 \, a^{3} b^{5} x^{5 \, n} + 14700 \, a^{4} b^{4} x^{4 \, n} + 15680 \, a^{5} b^{3} x^{3 \, n} + 11760 \, a^{6} b^{2} x^{2 \, n} + 6720 \, a^{7} b x^{n}}{840 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.29, size = 126, normalized size = 0.91 \[ a^8\,\ln \relax (x)+\frac {b^8\,x^{8\,n}}{8\,n}+\frac {14\,a^6\,b^2\,x^{2\,n}}{n}+\frac {56\,a^5\,b^3\,x^{3\,n}}{3\,n}+\frac {35\,a^4\,b^4\,x^{4\,n}}{2\,n}+\frac {56\,a^3\,b^5\,x^{5\,n}}{5\,n}+\frac {14\,a^2\,b^6\,x^{6\,n}}{3\,n}+\frac {8\,a^7\,b\,x^n}{n}+\frac {8\,a\,b^7\,x^{7\,n}}{7\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.05, size = 136, normalized size = 0.99 \[ \begin {cases} a^{8} \log {\relax (x )} + \frac {8 a^{7} b x^{n}}{n} + \frac {14 a^{6} b^{2} x^{2 n}}{n} + \frac {56 a^{5} b^{3} x^{3 n}}{3 n} + \frac {35 a^{4} b^{4} x^{4 n}}{2 n} + \frac {56 a^{3} b^{5} x^{5 n}}{5 n} + \frac {14 a^{2} b^{6} x^{6 n}}{3 n} + \frac {8 a b^{7} x^{7 n}}{7 n} + \frac {b^{8} x^{8 n}}{8 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{8} \log {\relax (x )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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