Integrand size = 26, antiderivative size = 36 \[ \int x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {\left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{18 b} \]
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Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1366, 623} \[ \int x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {\left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{18 b} \]
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Rule 623
Rule 1366
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx,x,x^3\right ) \\ & = \frac {\left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{18 b} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(82\) vs. \(2(36)=72\).
Time = 1.02 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.28 \[ \int x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {x^3 \sqrt {\left (a+b x^3\right )^2} \left (6 a^5+15 a^4 b x^3+20 a^3 b^2 x^6+15 a^2 b^3 x^9+6 a b^4 x^{12}+b^5 x^{15}\right )}{18 \left (a+b x^3\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.10 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.64
method | result | size |
pseudoelliptic | \(\frac {\left (b \,x^{3}+a \right )^{6} \operatorname {csgn}\left (b \,x^{3}+a \right )}{18 b}\) | \(23\) |
default | \(\frac {\left (b \,x^{3}+a \right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {5}{2}}}{18 b}\) | \(24\) |
risch | \(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (b \,x^{3}+a \right )^{5}}{18 b}\) | \(26\) |
gosper | \(\frac {x^{3} \left (b^{5} x^{15}+6 a \,b^{4} x^{12}+15 a^{2} b^{3} x^{9}+20 a^{3} b^{2} x^{6}+15 a^{4} b \,x^{3}+6 a^{5}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {5}{2}}}{18 \left (b \,x^{3}+a \right )^{5}}\) | \(79\) |
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none
Time = 0.29 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.58 \[ \int x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {1}{18} \, b^{5} x^{18} + \frac {1}{3} \, a b^{4} x^{15} + \frac {5}{6} \, a^{2} b^{3} x^{12} + \frac {10}{9} \, a^{3} b^{2} x^{9} + \frac {5}{6} \, a^{4} b x^{6} + \frac {1}{3} \, a^{5} x^{3} \]
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\[ \int x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\int x^{2} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}\, dx \]
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none
Time = 0.22 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.44 \[ \int x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {1}{18} \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} x^{3} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} a}{18 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (32) = 64\).
Time = 0.32 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.83 \[ \int x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {1}{18} \, {\left (3 \, {\left (b x^{6} + 2 \, a x^{3}\right )} a^{4} + 3 \, {\left (b x^{6} + 2 \, a x^{3}\right )}^{2} a^{2} b + {\left (b x^{6} + 2 \, a x^{3}\right )}^{3} b^{2}\right )} \mathrm {sgn}\left (b x^{3} + a\right ) \]
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Time = 8.21 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {\left (b^2\,x^3+a\,b\right )\,{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2}}{18\,b^2} \]
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