Integrand size = 9, antiderivative size = 99 \[ \int \left (a+b x^3\right )^8 \, dx=a^8 x+2 a^7 b x^4+4 a^6 b^2 x^7+\frac {28}{5} a^5 b^3 x^{10}+\frac {70}{13} a^4 b^4 x^{13}+\frac {7}{2} a^3 b^5 x^{16}+\frac {28}{19} a^2 b^6 x^{19}+\frac {4}{11} a b^7 x^{22}+\frac {b^8 x^{25}}{25} \]
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Time = 0.02 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {200} \[ \int \left (a+b x^3\right )^8 \, dx=a^8 x+2 a^7 b x^4+4 a^6 b^2 x^7+\frac {28}{5} a^5 b^3 x^{10}+\frac {70}{13} a^4 b^4 x^{13}+\frac {7}{2} a^3 b^5 x^{16}+\frac {28}{19} a^2 b^6 x^{19}+\frac {4}{11} a b^7 x^{22}+\frac {b^8 x^{25}}{25} \]
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Rule 200
Rubi steps \begin{align*} \text {integral}& = \int \left (a^8+8 a^7 b x^3+28 a^6 b^2 x^6+56 a^5 b^3 x^9+70 a^4 b^4 x^{12}+56 a^3 b^5 x^{15}+28 a^2 b^6 x^{18}+8 a b^7 x^{21}+b^8 x^{24}\right ) \, dx \\ & = a^8 x+2 a^7 b x^4+4 a^6 b^2 x^7+\frac {28}{5} a^5 b^3 x^{10}+\frac {70}{13} a^4 b^4 x^{13}+\frac {7}{2} a^3 b^5 x^{16}+\frac {28}{19} a^2 b^6 x^{19}+\frac {4}{11} a b^7 x^{22}+\frac {b^8 x^{25}}{25} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^3\right )^8 \, dx=a^8 x+2 a^7 b x^4+4 a^6 b^2 x^7+\frac {28}{5} a^5 b^3 x^{10}+\frac {70}{13} a^4 b^4 x^{13}+\frac {7}{2} a^3 b^5 x^{16}+\frac {28}{19} a^2 b^6 x^{19}+\frac {4}{11} a b^7 x^{22}+\frac {b^8 x^{25}}{25} \]
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Time = 3.52 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.89
method | result | size |
gosper | \(a^{8} x +2 a^{7} b \,x^{4}+4 x^{7} b^{2} a^{6}+\frac {28}{5} a^{5} b^{3} x^{10}+\frac {70}{13} x^{13} b^{4} a^{4}+\frac {7}{2} a^{3} b^{5} x^{16}+\frac {28}{19} a^{2} b^{6} x^{19}+\frac {4}{11} a \,b^{7} x^{22}+\frac {1}{25} b^{8} x^{25}\) | \(88\) |
default | \(a^{8} x +2 a^{7} b \,x^{4}+4 x^{7} b^{2} a^{6}+\frac {28}{5} a^{5} b^{3} x^{10}+\frac {70}{13} x^{13} b^{4} a^{4}+\frac {7}{2} a^{3} b^{5} x^{16}+\frac {28}{19} a^{2} b^{6} x^{19}+\frac {4}{11} a \,b^{7} x^{22}+\frac {1}{25} b^{8} x^{25}\) | \(88\) |
norman | \(a^{8} x +2 a^{7} b \,x^{4}+4 x^{7} b^{2} a^{6}+\frac {28}{5} a^{5} b^{3} x^{10}+\frac {70}{13} x^{13} b^{4} a^{4}+\frac {7}{2} a^{3} b^{5} x^{16}+\frac {28}{19} a^{2} b^{6} x^{19}+\frac {4}{11} a \,b^{7} x^{22}+\frac {1}{25} b^{8} x^{25}\) | \(88\) |
risch | \(a^{8} x +2 a^{7} b \,x^{4}+4 x^{7} b^{2} a^{6}+\frac {28}{5} a^{5} b^{3} x^{10}+\frac {70}{13} x^{13} b^{4} a^{4}+\frac {7}{2} a^{3} b^{5} x^{16}+\frac {28}{19} a^{2} b^{6} x^{19}+\frac {4}{11} a \,b^{7} x^{22}+\frac {1}{25} b^{8} x^{25}\) | \(88\) |
parallelrisch | \(a^{8} x +2 a^{7} b \,x^{4}+4 x^{7} b^{2} a^{6}+\frac {28}{5} a^{5} b^{3} x^{10}+\frac {70}{13} x^{13} b^{4} a^{4}+\frac {7}{2} a^{3} b^{5} x^{16}+\frac {28}{19} a^{2} b^{6} x^{19}+\frac {4}{11} a \,b^{7} x^{22}+\frac {1}{25} b^{8} x^{25}\) | \(88\) |
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Time = 0.30 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.88 \[ \int \left (a+b x^3\right )^8 \, dx=\frac {1}{25} \, b^{8} x^{25} + \frac {4}{11} \, a b^{7} x^{22} + \frac {28}{19} \, a^{2} b^{6} x^{19} + \frac {7}{2} \, a^{3} b^{5} x^{16} + \frac {70}{13} \, a^{4} b^{4} x^{13} + \frac {28}{5} \, a^{5} b^{3} x^{10} + 4 \, a^{6} b^{2} x^{7} + 2 \, a^{7} b x^{4} + a^{8} x \]
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Time = 0.02 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.01 \[ \int \left (a+b x^3\right )^8 \, dx=a^{8} x + 2 a^{7} b x^{4} + 4 a^{6} b^{2} x^{7} + \frac {28 a^{5} b^{3} x^{10}}{5} + \frac {70 a^{4} b^{4} x^{13}}{13} + \frac {7 a^{3} b^{5} x^{16}}{2} + \frac {28 a^{2} b^{6} x^{19}}{19} + \frac {4 a b^{7} x^{22}}{11} + \frac {b^{8} x^{25}}{25} \]
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Time = 0.32 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.88 \[ \int \left (a+b x^3\right )^8 \, dx=\frac {1}{25} \, b^{8} x^{25} + \frac {4}{11} \, a b^{7} x^{22} + \frac {28}{19} \, a^{2} b^{6} x^{19} + \frac {7}{2} \, a^{3} b^{5} x^{16} + \frac {70}{13} \, a^{4} b^{4} x^{13} + \frac {28}{5} \, a^{5} b^{3} x^{10} + 4 \, a^{6} b^{2} x^{7} + 2 \, a^{7} b x^{4} + a^{8} x \]
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Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.88 \[ \int \left (a+b x^3\right )^8 \, dx=\frac {1}{25} \, b^{8} x^{25} + \frac {4}{11} \, a b^{7} x^{22} + \frac {28}{19} \, a^{2} b^{6} x^{19} + \frac {7}{2} \, a^{3} b^{5} x^{16} + \frac {70}{13} \, a^{4} b^{4} x^{13} + \frac {28}{5} \, a^{5} b^{3} x^{10} + 4 \, a^{6} b^{2} x^{7} + 2 \, a^{7} b x^{4} + a^{8} x \]
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Time = 0.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.88 \[ \int \left (a+b x^3\right )^8 \, dx=a^8\,x+2\,a^7\,b\,x^4+4\,a^6\,b^2\,x^7+\frac {28\,a^5\,b^3\,x^{10}}{5}+\frac {70\,a^4\,b^4\,x^{13}}{13}+\frac {7\,a^3\,b^5\,x^{16}}{2}+\frac {28\,a^2\,b^6\,x^{19}}{19}+\frac {4\,a\,b^7\,x^{22}}{11}+\frac {b^8\,x^{25}}{25} \]
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