Integrand size = 15, antiderivative size = 59 \[ \int x^8 \left (a+b x^3\right )^{2/3} \, dx=\frac {a^2 \left (a+b x^3\right )^{5/3}}{5 b^3}-\frac {a \left (a+b x^3\right )^{8/3}}{4 b^3}+\frac {\left (a+b x^3\right )^{11/3}}{11 b^3} \]
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Time = 0.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int x^8 \left (a+b x^3\right )^{2/3} \, dx=\frac {a^2 \left (a+b x^3\right )^{5/3}}{5 b^3}+\frac {\left (a+b x^3\right )^{11/3}}{11 b^3}-\frac {a \left (a+b x^3\right )^{8/3}}{4 b^3} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int x^2 (a+b x)^{2/3} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (\frac {a^2 (a+b x)^{2/3}}{b^2}-\frac {2 a (a+b x)^{5/3}}{b^2}+\frac {(a+b x)^{8/3}}{b^2}\right ) \, dx,x,x^3\right ) \\ & = \frac {a^2 \left (a+b x^3\right )^{5/3}}{5 b^3}-\frac {a \left (a+b x^3\right )^{8/3}}{4 b^3}+\frac {\left (a+b x^3\right )^{11/3}}{11 b^3} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.66 \[ \int x^8 \left (a+b x^3\right )^{2/3} \, dx=\frac {\left (a+b x^3\right )^{5/3} \left (9 a^2-15 a b x^3+20 b^2 x^6\right )}{220 b^3} \]
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Time = 3.80 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.61
method | result | size |
gosper | \(\frac {\left (b \,x^{3}+a \right )^{\frac {5}{3}} \left (20 b^{2} x^{6}-15 a b \,x^{3}+9 a^{2}\right )}{220 b^{3}}\) | \(36\) |
pseudoelliptic | \(\frac {\left (b \,x^{3}+a \right )^{\frac {5}{3}} \left (20 b^{2} x^{6}-15 a b \,x^{3}+9 a^{2}\right )}{220 b^{3}}\) | \(36\) |
trager | \(\frac {\left (20 b^{3} x^{9}+5 a \,b^{2} x^{6}-6 a^{2} b \,x^{3}+9 a^{3}\right ) \left (b \,x^{3}+a \right )^{\frac {2}{3}}}{220 b^{3}}\) | \(47\) |
risch | \(\frac {\left (20 b^{3} x^{9}+5 a \,b^{2} x^{6}-6 a^{2} b \,x^{3}+9 a^{3}\right ) \left (b \,x^{3}+a \right )^{\frac {2}{3}}}{220 b^{3}}\) | \(47\) |
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Time = 0.25 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.78 \[ \int x^8 \left (a+b x^3\right )^{2/3} \, dx=\frac {{\left (20 \, b^{3} x^{9} + 5 \, a b^{2} x^{6} - 6 \, a^{2} b x^{3} + 9 \, a^{3}\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{220 \, b^{3}} \]
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Time = 0.35 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.47 \[ \int x^8 \left (a+b x^3\right )^{2/3} \, dx=\begin {cases} \frac {9 a^{3} \left (a + b x^{3}\right )^{\frac {2}{3}}}{220 b^{3}} - \frac {3 a^{2} x^{3} \left (a + b x^{3}\right )^{\frac {2}{3}}}{110 b^{2}} + \frac {a x^{6} \left (a + b x^{3}\right )^{\frac {2}{3}}}{44 b} + \frac {x^{9} \left (a + b x^{3}\right )^{\frac {2}{3}}}{11} & \text {for}\: b \neq 0 \\\frac {a^{\frac {2}{3}} x^{9}}{9} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.80 \[ \int x^8 \left (a+b x^3\right )^{2/3} \, dx=\frac {{\left (b x^{3} + a\right )}^{\frac {11}{3}}}{11 \, b^{3}} - \frac {{\left (b x^{3} + a\right )}^{\frac {8}{3}} a}{4 \, b^{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {5}{3}} a^{2}}{5 \, b^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.73 \[ \int x^8 \left (a+b x^3\right )^{2/3} \, dx=\frac {20 \, {\left (b x^{3} + a\right )}^{\frac {11}{3}} - 55 \, {\left (b x^{3} + a\right )}^{\frac {8}{3}} a + 44 \, {\left (b x^{3} + a\right )}^{\frac {5}{3}} a^{2}}{220 \, b^{3}} \]
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Time = 5.97 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.75 \[ \int x^8 \left (a+b x^3\right )^{2/3} \, dx={\left (b\,x^3+a\right )}^{2/3}\,\left (\frac {x^9}{11}+\frac {9\,a^3}{220\,b^3}+\frac {a\,x^6}{44\,b}-\frac {3\,a^2\,x^3}{110\,b^2}\right ) \]
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