Integrand size = 13, antiderivative size = 104 \[ \int \frac {\left (a+b x^3\right )^8}{x} \, dx=\frac {8}{3} a^7 b x^3+\frac {14}{3} a^6 b^2 x^6+\frac {56}{9} a^5 b^3 x^9+\frac {35}{6} a^4 b^4 x^{12}+\frac {56}{15} a^3 b^5 x^{15}+\frac {14}{9} a^2 b^6 x^{18}+\frac {8}{21} a b^7 x^{21}+\frac {b^8 x^{24}}{24}+a^8 \log (x) \]
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Time = 0.04 (sec) , antiderivative size = 104, 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} \[ \int \frac {\left (a+b x^3\right )^8}{x} \, dx=a^8 \log (x)+\frac {8}{3} a^7 b x^3+\frac {14}{3} a^6 b^2 x^6+\frac {56}{9} a^5 b^3 x^9+\frac {35}{6} a^4 b^4 x^{12}+\frac {56}{15} a^3 b^5 x^{15}+\frac {14}{9} a^2 b^6 x^{18}+\frac {8}{21} a b^7 x^{21}+\frac {b^8 x^{24}}{24} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {(a+b x)^8}{x} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \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^3\right ) \\ & = \frac {8}{3} a^7 b x^3+\frac {14}{3} a^6 b^2 x^6+\frac {56}{9} a^5 b^3 x^9+\frac {35}{6} a^4 b^4 x^{12}+\frac {56}{15} a^3 b^5 x^{15}+\frac {14}{9} a^2 b^6 x^{18}+\frac {8}{21} a b^7 x^{21}+\frac {b^8 x^{24}}{24}+a^8 \log (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^8}{x} \, dx=\frac {8}{3} a^7 b x^3+\frac {14}{3} a^6 b^2 x^6+\frac {56}{9} a^5 b^3 x^9+\frac {35}{6} a^4 b^4 x^{12}+\frac {56}{15} a^3 b^5 x^{15}+\frac {14}{9} a^2 b^6 x^{18}+\frac {8}{21} a b^7 x^{21}+\frac {b^8 x^{24}}{24}+a^8 \log (x) \]
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Time = 3.74 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.86
method | result | size |
default | \(\frac {8 x^{3} b \,a^{7}}{3}+\frac {14 a^{6} b^{2} x^{6}}{3}+\frac {56 x^{9} b^{3} a^{5}}{9}+\frac {35 a^{4} b^{4} x^{12}}{6}+\frac {56 a^{3} b^{5} x^{15}}{15}+\frac {14 a^{2} b^{6} x^{18}}{9}+\frac {8 a \,b^{7} x^{21}}{21}+\frac {b^{8} x^{24}}{24}+a^{8} \ln \left (x \right )\) | \(89\) |
norman | \(\frac {8 x^{3} b \,a^{7}}{3}+\frac {14 a^{6} b^{2} x^{6}}{3}+\frac {56 x^{9} b^{3} a^{5}}{9}+\frac {35 a^{4} b^{4} x^{12}}{6}+\frac {56 a^{3} b^{5} x^{15}}{15}+\frac {14 a^{2} b^{6} x^{18}}{9}+\frac {8 a \,b^{7} x^{21}}{21}+\frac {b^{8} x^{24}}{24}+a^{8} \ln \left (x \right )\) | \(89\) |
parallelrisch | \(\frac {8 x^{3} b \,a^{7}}{3}+\frac {14 a^{6} b^{2} x^{6}}{3}+\frac {56 x^{9} b^{3} a^{5}}{9}+\frac {35 a^{4} b^{4} x^{12}}{6}+\frac {56 a^{3} b^{5} x^{15}}{15}+\frac {14 a^{2} b^{6} x^{18}}{9}+\frac {8 a \,b^{7} x^{21}}{21}+\frac {b^{8} x^{24}}{24}+a^{8} \ln \left (x \right )\) | \(89\) |
risch | \(\frac {352 a^{8}}{315}+\frac {b^{8} x^{24}}{24}+\frac {8 x^{3} b \,a^{7}}{3}+\frac {8 a \,b^{7} x^{21}}{21}+\frac {56 a^{3} b^{5} x^{15}}{15}+\frac {14 a^{2} b^{6} x^{18}}{9}+\frac {35 a^{4} b^{4} x^{12}}{6}+a^{8} \ln \left (x \right )+\frac {56 x^{9} b^{3} a^{5}}{9}+\frac {14 a^{6} b^{2} x^{6}}{3}\) | \(94\) |
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Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^3\right )^8}{x} \, dx=\frac {1}{24} \, b^{8} x^{24} + \frac {8}{21} \, a b^{7} x^{21} + \frac {14}{9} \, a^{2} b^{6} x^{18} + \frac {56}{15} \, a^{3} b^{5} x^{15} + \frac {35}{6} \, a^{4} b^{4} x^{12} + \frac {56}{9} \, a^{5} b^{3} x^{9} + \frac {14}{3} \, a^{6} b^{2} x^{6} + \frac {8}{3} \, a^{7} b x^{3} + a^{8} \log \left (x\right ) \]
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Time = 0.06 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x^3\right )^8}{x} \, dx=a^{8} \log {\left (x \right )} + \frac {8 a^{7} b x^{3}}{3} + \frac {14 a^{6} b^{2} x^{6}}{3} + \frac {56 a^{5} b^{3} x^{9}}{9} + \frac {35 a^{4} b^{4} x^{12}}{6} + \frac {56 a^{3} b^{5} x^{15}}{15} + \frac {14 a^{2} b^{6} x^{18}}{9} + \frac {8 a b^{7} x^{21}}{21} + \frac {b^{8} x^{24}}{24} \]
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Time = 0.19 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^3\right )^8}{x} \, dx=\frac {1}{24} \, b^{8} x^{24} + \frac {8}{21} \, a b^{7} x^{21} + \frac {14}{9} \, a^{2} b^{6} x^{18} + \frac {56}{15} \, a^{3} b^{5} x^{15} + \frac {35}{6} \, a^{4} b^{4} x^{12} + \frac {56}{9} \, a^{5} b^{3} x^{9} + \frac {14}{3} \, a^{6} b^{2} x^{6} + \frac {8}{3} \, a^{7} b x^{3} + \frac {1}{3} \, a^{8} \log \left (x^{3}\right ) \]
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Time = 0.29 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b x^3\right )^8}{x} \, dx=\frac {1}{24} \, b^{8} x^{24} + \frac {8}{21} \, a b^{7} x^{21} + \frac {14}{9} \, a^{2} b^{6} x^{18} + \frac {56}{15} \, a^{3} b^{5} x^{15} + \frac {35}{6} \, a^{4} b^{4} x^{12} + \frac {56}{9} \, a^{5} b^{3} x^{9} + \frac {14}{3} \, a^{6} b^{2} x^{6} + \frac {8}{3} \, a^{7} b x^{3} + a^{8} \log \left ({\left | x \right |}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^3\right )^8}{x} \, dx=a^8\,\ln \left (x\right )+\frac {b^8\,x^{24}}{24}+\frac {8\,a^7\,b\,x^3}{3}+\frac {8\,a\,b^7\,x^{21}}{21}+\frac {14\,a^6\,b^2\,x^6}{3}+\frac {56\,a^5\,b^3\,x^9}{9}+\frac {35\,a^4\,b^4\,x^{12}}{6}+\frac {56\,a^3\,b^5\,x^{15}}{15}+\frac {14\,a^2\,b^6\,x^{18}}{9} \]
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