Integrand size = 13, antiderivative size = 138 \[ \int \frac {\left (a+b x^n\right )^8}{x} \, dx=\frac {8 a^7 b x^n}{n}+\frac {14 a^6 b^2 x^{2 n}}{n}+\frac {56 a^5 b^3 x^{3 n}}{3 n}+\frac {35 a^4 b^4 x^{4 n}}{2 n}+\frac {56 a^3 b^5 x^{5 n}}{5 n}+\frac {14 a^2 b^6 x^{6 n}}{3 n}+\frac {8 a b^7 x^{7 n}}{7 n}+\frac {b^8 x^{8 n}}{8 n}+a^8 \log (x) \]
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Time = 0.03 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \[ \int \frac {\left (a+b x^n\right )^8}{x} \, dx=a^8 \log (x)+\frac {8 a^7 b x^n}{n}+\frac {14 a^6 b^2 x^{2 n}}{n}+\frac {56 a^5 b^3 x^{3 n}}{3 n}+\frac {35 a^4 b^4 x^{4 n}}{2 n}+\frac {56 a^3 b^5 x^{5 n}}{5 n}+\frac {14 a^2 b^6 x^{6 n}}{3 n}+\frac {8 a b^7 x^{7 n}}{7 n}+\frac {b^8 x^{8 n}}{8 n} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b x)^8}{x} \, dx,x,x^n\right )}{n} \\ & = \frac {\text {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*}
Time = 0.10 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b x^n\right )^8}{x} \, dx=\frac {b x^n \left (6720 a^7+11760 a^6 b x^n+15680 a^5 b^2 x^{2 n}+14700 a^4 b^3 x^{3 n}+9408 a^3 b^4 x^{4 n}+3920 a^2 b^5 x^{5 n}+960 a b^6 x^{6 n}+105 b^7 x^{7 n}\right )}{840 n}+\frac {a^8 \log \left (x^n\right )}{n} \]
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Time = 6.38 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.79
method | result | size |
derivativedivides | \(\frac {\frac {b^{8} x^{8 n}}{8}+\frac {8 a \,b^{7} x^{7 n}}{7}+\frac {14 a^{2} b^{6} x^{6 n}}{3}+\frac {56 a^{3} b^{5} x^{5 n}}{5}+\frac {35 a^{4} b^{4} x^{4 n}}{2}+\frac {56 a^{5} b^{3} x^{3 n}}{3}+14 a^{6} b^{2} x^{2 n}+8 a^{7} b \,x^{n}+a^{8} \ln \left (x^{n}\right )}{n}\) | \(109\) |
default | \(\frac {\frac {b^{8} x^{8 n}}{8}+\frac {8 a \,b^{7} x^{7 n}}{7}+\frac {14 a^{2} b^{6} x^{6 n}}{3}+\frac {56 a^{3} b^{5} x^{5 n}}{5}+\frac {35 a^{4} b^{4} x^{4 n}}{2}+\frac {56 a^{5} b^{3} x^{3 n}}{3}+14 a^{6} b^{2} x^{2 n}+8 a^{7} b \,x^{n}+a^{8} \ln \left (x^{n}\right )}{n}\) | \(109\) |
parallelrisch | \(\frac {105 b^{8} x^{8 n}+960 a \,b^{7} x^{7 n}+3920 a^{2} b^{6} x^{6 n}+9408 a^{3} b^{5} x^{5 n}+14700 a^{4} b^{4} x^{4 n}+15680 a^{5} b^{3} x^{3 n}+840 a^{8} \ln \left (x \right ) n +11760 a^{6} b^{2} x^{2 n}+6720 a^{7} b \,x^{n}}{840 n}\) | \(110\) |
risch | \(\frac {8 a^{7} b \,x^{n}}{n}+\frac {14 a^{6} b^{2} x^{2 n}}{n}+\frac {56 a^{5} b^{3} x^{3 n}}{3 n}+\frac {35 a^{4} b^{4} x^{4 n}}{2 n}+\frac {56 a^{3} b^{5} x^{5 n}}{5 n}+\frac {14 a^{2} b^{6} x^{6 n}}{3 n}+\frac {8 a \,b^{7} x^{7 n}}{7 n}+\frac {b^{8} x^{8 n}}{8 n}+a^{8} \ln \left (x \right )\) | \(127\) |
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Time = 0.46 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b x^n\right )^8}{x} \, dx=\frac {840 \, a^{8} n \log \left (x\right ) + 105 \, b^{8} x^{8 \, n} + 960 \, a b^{7} x^{7 \, n} + 3920 \, a^{2} b^{6} x^{6 \, n} + 9408 \, a^{3} b^{5} x^{5 \, n} + 14700 \, a^{4} b^{4} x^{4 \, n} + 15680 \, a^{5} b^{3} x^{3 \, n} + 11760 \, a^{6} b^{2} x^{2 \, n} + 6720 \, a^{7} b x^{n}}{840 \, n} \]
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Time = 0.28 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^n\right )^8}{x} \, dx=\begin {cases} a^{8} \log {\left (x \right )} + \frac {8 a^{7} b x^{n}}{n} + \frac {14 a^{6} b^{2} x^{2 n}}{n} + \frac {56 a^{5} b^{3} x^{3 n}}{3 n} + \frac {35 a^{4} b^{4} x^{4 n}}{2 n} + \frac {56 a^{3} b^{5} x^{5 n}}{5 n} + \frac {14 a^{2} b^{6} x^{6 n}}{3 n} + \frac {8 a b^{7} x^{7 n}}{7 n} + \frac {b^{8} x^{8 n}}{8 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{8} \log {\left (x \right )} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+b x^n\right )^8}{x} \, dx=\frac {a^{8} \log \left (x^{n}\right )}{n} + \frac {105 \, b^{8} x^{8 \, n} + 960 \, a b^{7} x^{7 \, n} + 3920 \, a^{2} b^{6} x^{6 \, n} + 9408 \, a^{3} b^{5} x^{5 \, n} + 14700 \, a^{4} b^{4} x^{4 \, n} + 15680 \, a^{5} b^{3} x^{3 \, n} + 11760 \, a^{6} b^{2} x^{2 \, n} + 6720 \, a^{7} b x^{n}}{840 \, n} \]
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\[ \int \frac {\left (a+b x^n\right )^8}{x} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{8}}{x} \,d x } \]
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Time = 5.69 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b x^n\right )^8}{x} \, dx=a^8\,\ln \left (x\right )+\frac {b^8\,x^{8\,n}}{8\,n}+\frac {14\,a^6\,b^2\,x^{2\,n}}{n}+\frac {56\,a^5\,b^3\,x^{3\,n}}{3\,n}+\frac {35\,a^4\,b^4\,x^{4\,n}}{2\,n}+\frac {56\,a^3\,b^5\,x^{5\,n}}{5\,n}+\frac {14\,a^2\,b^6\,x^{6\,n}}{3\,n}+\frac {8\,a^7\,b\,x^n}{n}+\frac {8\,a\,b^7\,x^{7\,n}}{7\,n} \]
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