Integrand size = 11, antiderivative size = 21 \[ \int x^{5/3} (a+b x) \, dx=\frac {3}{8} a x^{8/3}+\frac {3}{11} b x^{11/3} \]
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Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int x^{5/3} (a+b x) \, dx=\frac {3}{8} a x^{8/3}+\frac {3}{11} b x^{11/3} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (a x^{5/3}+b x^{8/3}\right ) \, dx \\ & = \frac {3}{8} a x^{8/3}+\frac {3}{11} b x^{11/3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int x^{5/3} (a+b x) \, dx=\frac {3}{88} x^{8/3} (11 a+8 b x) \]
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Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67
method | result | size |
gosper | \(\frac {3 x^{\frac {8}{3}} \left (8 b x +11 a \right )}{88}\) | \(14\) |
derivativedivides | \(\frac {3 a \,x^{\frac {8}{3}}}{8}+\frac {3 b \,x^{\frac {11}{3}}}{11}\) | \(14\) |
default | \(\frac {3 a \,x^{\frac {8}{3}}}{8}+\frac {3 b \,x^{\frac {11}{3}}}{11}\) | \(14\) |
trager | \(\frac {3 x^{\frac {8}{3}} \left (8 b x +11 a \right )}{88}\) | \(14\) |
risch | \(\frac {3 x^{\frac {8}{3}} \left (8 b x +11 a \right )}{88}\) | \(14\) |
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Time = 0.22 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int x^{5/3} (a+b x) \, dx=\frac {3}{88} \, {\left (8 \, b x^{3} + 11 \, a x^{2}\right )} x^{\frac {2}{3}} \]
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Time = 0.43 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int x^{5/3} (a+b x) \, dx=\frac {3 a x^{\frac {8}{3}}}{8} + \frac {3 b x^{\frac {11}{3}}}{11} \]
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Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int x^{5/3} (a+b x) \, dx=\frac {3}{11} \, b x^{\frac {11}{3}} + \frac {3}{8} \, a x^{\frac {8}{3}} \]
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Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int x^{5/3} (a+b x) \, dx=\frac {3}{11} \, b x^{\frac {11}{3}} + \frac {3}{8} \, a x^{\frac {8}{3}} \]
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Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int x^{5/3} (a+b x) \, dx=\frac {3\,x^{8/3}\,\left (11\,a+8\,b\,x\right )}{88} \]
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