Integrand size = 13, antiderivative size = 98 \[ \int \frac {\left (a+b x^3\right )^8}{x^3} \, dx=-\frac {a^8}{2 x^2}+8 a^7 b x+7 a^6 b^2 x^4+8 a^5 b^3 x^7+7 a^4 b^4 x^{10}+\frac {56}{13} a^3 b^5 x^{13}+\frac {7}{4} a^2 b^6 x^{16}+\frac {8}{19} a b^7 x^{19}+\frac {b^8 x^{22}}{22} \]
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Time = 0.03 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {276} \[ \int \frac {\left (a+b x^3\right )^8}{x^3} \, dx=-\frac {a^8}{2 x^2}+8 a^7 b x+7 a^6 b^2 x^4+8 a^5 b^3 x^7+7 a^4 b^4 x^{10}+\frac {56}{13} a^3 b^5 x^{13}+\frac {7}{4} a^2 b^6 x^{16}+\frac {8}{19} a b^7 x^{19}+\frac {b^8 x^{22}}{22} \]
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Rule 276
Rubi steps \begin{align*} \text {integral}& = \int \left (8 a^7 b+\frac {a^8}{x^3}+28 a^6 b^2 x^3+56 a^5 b^3 x^6+70 a^4 b^4 x^9+56 a^3 b^5 x^{12}+28 a^2 b^6 x^{15}+8 a b^7 x^{18}+b^8 x^{21}\right ) \, dx \\ & = -\frac {a^8}{2 x^2}+8 a^7 b x+7 a^6 b^2 x^4+8 a^5 b^3 x^7+7 a^4 b^4 x^{10}+\frac {56}{13} a^3 b^5 x^{13}+\frac {7}{4} a^2 b^6 x^{16}+\frac {8}{19} a b^7 x^{19}+\frac {b^8 x^{22}}{22} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^8}{x^3} \, dx=-\frac {a^8}{2 x^2}+8 a^7 b x+7 a^6 b^2 x^4+8 a^5 b^3 x^7+7 a^4 b^4 x^{10}+\frac {56}{13} a^3 b^5 x^{13}+\frac {7}{4} a^2 b^6 x^{16}+\frac {8}{19} a b^7 x^{19}+\frac {b^8 x^{22}}{22} \]
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Time = 3.78 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.91
method | result | size |
default | \(-\frac {a^{8}}{2 x^{2}}+8 a^{7} x b +7 x^{4} b^{2} a^{6}+8 x^{7} b^{3} a^{5}+7 a^{4} b^{4} x^{10}+\frac {56 x^{13} b^{5} a^{3}}{13}+\frac {7 a^{2} b^{6} x^{16}}{4}+\frac {8 a \,b^{7} x^{19}}{19}+\frac {b^{8} x^{22}}{22}\) | \(89\) |
risch | \(-\frac {a^{8}}{2 x^{2}}+8 a^{7} x b +7 x^{4} b^{2} a^{6}+8 x^{7} b^{3} a^{5}+7 a^{4} b^{4} x^{10}+\frac {56 x^{13} b^{5} a^{3}}{13}+\frac {7 a^{2} b^{6} x^{16}}{4}+\frac {8 a \,b^{7} x^{19}}{19}+\frac {b^{8} x^{22}}{22}\) | \(89\) |
norman | \(\frac {\frac {56}{13} a^{3} b^{5} x^{15}-\frac {1}{2} a^{8}+\frac {8}{19} a \,b^{7} x^{21}+7 a^{6} b^{2} x^{6}+8 x^{9} b^{3} a^{5}+7 a^{4} b^{4} x^{12}+\frac {7}{4} a^{2} b^{6} x^{18}+8 x^{3} b \,a^{7}+\frac {1}{22} b^{8} x^{24}}{x^{2}}\) | \(92\) |
gosper | \(-\frac {-494 b^{8} x^{24}-4576 a \,b^{7} x^{21}-19019 a^{2} b^{6} x^{18}-46816 a^{3} b^{5} x^{15}-76076 a^{4} b^{4} x^{12}-86944 x^{9} b^{3} a^{5}-76076 a^{6} b^{2} x^{6}-86944 x^{3} b \,a^{7}+5434 a^{8}}{10868 x^{2}}\) | \(93\) |
parallelrisch | \(\frac {494 b^{8} x^{24}+4576 a \,b^{7} x^{21}+19019 a^{2} b^{6} x^{18}+46816 a^{3} b^{5} x^{15}+76076 a^{4} b^{4} x^{12}+86944 x^{9} b^{3} a^{5}+76076 a^{6} b^{2} x^{6}+86944 x^{3} b \,a^{7}-5434 a^{8}}{10868 x^{2}}\) | \(93\) |
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Time = 0.27 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^3\right )^8}{x^3} \, dx=\frac {494 \, b^{8} x^{24} + 4576 \, a b^{7} x^{21} + 19019 \, a^{2} b^{6} x^{18} + 46816 \, a^{3} b^{5} x^{15} + 76076 \, a^{4} b^{4} x^{12} + 86944 \, a^{5} b^{3} x^{9} + 76076 \, a^{6} b^{2} x^{6} + 86944 \, a^{7} b x^{3} - 5434 \, a^{8}}{10868 \, x^{2}} \]
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Time = 0.07 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x^3\right )^8}{x^3} \, dx=- \frac {a^{8}}{2 x^{2}} + 8 a^{7} b x + 7 a^{6} b^{2} x^{4} + 8 a^{5} b^{3} x^{7} + 7 a^{4} b^{4} x^{10} + \frac {56 a^{3} b^{5} x^{13}}{13} + \frac {7 a^{2} b^{6} x^{16}}{4} + \frac {8 a b^{7} x^{19}}{19} + \frac {b^{8} x^{22}}{22} \]
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Time = 0.19 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^3\right )^8}{x^3} \, dx=\frac {1}{22} \, b^{8} x^{22} + \frac {8}{19} \, a b^{7} x^{19} + \frac {7}{4} \, a^{2} b^{6} x^{16} + \frac {56}{13} \, a^{3} b^{5} x^{13} + 7 \, a^{4} b^{4} x^{10} + 8 \, a^{5} b^{3} x^{7} + 7 \, a^{6} b^{2} x^{4} + 8 \, a^{7} b x - \frac {a^{8}}{2 \, x^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^3\right )^8}{x^3} \, dx=\frac {1}{22} \, b^{8} x^{22} + \frac {8}{19} \, a b^{7} x^{19} + \frac {7}{4} \, a^{2} b^{6} x^{16} + \frac {56}{13} \, a^{3} b^{5} x^{13} + 7 \, a^{4} b^{4} x^{10} + 8 \, a^{5} b^{3} x^{7} + 7 \, a^{6} b^{2} x^{4} + 8 \, a^{7} b x - \frac {a^{8}}{2 \, x^{2}} \]
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Time = 0.05 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^3\right )^8}{x^3} \, dx=\frac {b^8\,x^{22}}{22}-\frac {a^8}{2\,x^2}+\frac {8\,a\,b^7\,x^{19}}{19}+7\,a^6\,b^2\,x^4+8\,a^5\,b^3\,x^7+7\,a^4\,b^4\,x^{10}+\frac {56\,a^3\,b^5\,x^{13}}{13}+\frac {7\,a^2\,b^6\,x^{16}}{4}+8\,a^7\,b\,x \]
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