Integrand size = 13, antiderivative size = 16 \[ \int x^2 \left (a+b x^3\right )^5 \, dx=\frac {\left (a+b x^3\right )^6}{18 b} \]
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Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {267} \[ \int x^2 \left (a+b x^3\right )^5 \, dx=\frac {\left (a+b x^3\right )^6}{18 b} \]
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Rule 267
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a+b x^3\right )^6}{18 b} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(69\) vs. \(2(16)=32\).
Time = 0.00 (sec) , antiderivative size = 69, normalized size of antiderivative = 4.31 \[ \int x^2 \left (a+b x^3\right )^5 \, dx=\frac {a^5 x^3}{3}+\frac {5}{6} a^4 b x^6+\frac {10}{9} a^3 b^2 x^9+\frac {5}{6} a^2 b^3 x^{12}+\frac {1}{3} a b^4 x^{15}+\frac {b^5 x^{18}}{18} \]
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Time = 3.61 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {\left (b \,x^{3}+a \right )^{6}}{18 b}\) | \(15\) |
gosper | \(\frac {1}{3} a^{5} x^{3}+\frac {5}{6} a^{4} b \,x^{6}+\frac {10}{9} a^{3} b^{2} x^{9}+\frac {5}{6} a^{2} b^{3} x^{12}+\frac {1}{3} a \,b^{4} x^{15}+\frac {1}{18} b^{5} x^{18}\) | \(58\) |
norman | \(\frac {1}{3} a^{5} x^{3}+\frac {5}{6} a^{4} b \,x^{6}+\frac {10}{9} a^{3} b^{2} x^{9}+\frac {5}{6} a^{2} b^{3} x^{12}+\frac {1}{3} a \,b^{4} x^{15}+\frac {1}{18} b^{5} x^{18}\) | \(58\) |
parallelrisch | \(\frac {1}{3} a^{5} x^{3}+\frac {5}{6} a^{4} b \,x^{6}+\frac {10}{9} a^{3} b^{2} x^{9}+\frac {5}{6} a^{2} b^{3} x^{12}+\frac {1}{3} a \,b^{4} x^{15}+\frac {1}{18} b^{5} x^{18}\) | \(58\) |
risch | \(\frac {b^{5} x^{18}}{18}+\frac {a \,b^{4} x^{15}}{3}+\frac {5 a^{2} b^{3} x^{12}}{6}+\frac {10 a^{3} b^{2} x^{9}}{9}+\frac {5 a^{4} b \,x^{6}}{6}+\frac {a^{5} x^{3}}{3}+\frac {a^{6}}{18 b}\) | \(66\) |
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Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (14) = 28\).
Time = 0.26 (sec) , antiderivative size = 57, normalized size of antiderivative = 3.56 \[ \int x^2 \left (a+b x^3\right )^5 \, dx=\frac {1}{18} \, b^{5} x^{18} + \frac {1}{3} \, a b^{4} x^{15} + \frac {5}{6} \, a^{2} b^{3} x^{12} + \frac {10}{9} \, a^{3} b^{2} x^{9} + \frac {5}{6} \, a^{4} b x^{6} + \frac {1}{3} \, a^{5} x^{3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (10) = 20\).
Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 4.06 \[ \int x^2 \left (a+b x^3\right )^5 \, dx=\frac {a^{5} x^{3}}{3} + \frac {5 a^{4} b x^{6}}{6} + \frac {10 a^{3} b^{2} x^{9}}{9} + \frac {5 a^{2} b^{3} x^{12}}{6} + \frac {a b^{4} x^{15}}{3} + \frac {b^{5} x^{18}}{18} \]
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none
Time = 0.21 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int x^2 \left (a+b x^3\right )^5 \, dx=\frac {{\left (b x^{3} + a\right )}^{6}}{18 \, b} \]
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none
Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int x^2 \left (a+b x^3\right )^5 \, dx=\frac {{\left (b x^{3} + a\right )}^{6}}{18 \, b} \]
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Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 3.56 \[ \int x^2 \left (a+b x^3\right )^5 \, dx=\frac {a^5\,x^3}{3}+\frac {5\,a^4\,b\,x^6}{6}+\frac {10\,a^3\,b^2\,x^9}{9}+\frac {5\,a^2\,b^3\,x^{12}}{6}+\frac {a\,b^4\,x^{15}}{3}+\frac {b^5\,x^{18}}{18} \]
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