Integrand size = 26, antiderivative size = 78 \[ \int x^5 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=-\frac {a \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{12 b^2}+\frac {\left (a+b x^3\right )^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{15 b^2} \]
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Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1369, 272, 45} \[ \int x^5 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\frac {\left (a+b x^3\right )^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{15 b^2}-\frac {a \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{12 b^2} \]
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Rule 45
Rule 272
Rule 1369
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int x^5 \left (a b+b^2 x^3\right )^3 \, dx}{b^2 \left (a b+b^2 x^3\right )} \\ & = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \text {Subst}\left (\int x \left (a b+b^2 x\right )^3 \, dx,x,x^3\right )}{3 b^2 \left (a b+b^2 x^3\right )} \\ & = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \text {Subst}\left (\int \left (-\frac {a \left (a b+b^2 x\right )^3}{b}+\frac {\left (a b+b^2 x\right )^4}{b^2}\right ) \, dx,x,x^3\right )}{3 b^2 \left (a b+b^2 x^3\right )} \\ & = -\frac {a \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{12 b^2}+\frac {\left (a+b x^3\right )^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{15 b^2} \\ \end{align*}
Time = 0.47 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.45 \[ \int x^5 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\frac {x^6 \left (10 a^3+20 a^2 b x^3+15 a b^2 x^6+4 b^3 x^9\right ) \left (\sqrt {a^2} b x^3+a \left (\sqrt {a^2}-\sqrt {\left (a+b x^3\right )^2}\right )\right )}{60 \left (-a^2-a b x^3+\sqrt {a^2} \sqrt {\left (a+b x^3\right )^2}\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.40
method | result | size |
pseudoelliptic | \(-\frac {\operatorname {csgn}\left (b \,x^{3}+a \right ) \left (b \,x^{3}+a \right )^{4} \left (-4 b \,x^{3}+a \right )}{60 b^{2}}\) | \(31\) |
gosper | \(\frac {x^{6} \left (4 b^{3} x^{9}+15 b^{2} x^{6} a +20 a^{2} b \,x^{3}+10 a^{3}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {3}{2}}}{60 \left (b \,x^{3}+a \right )^{3}}\) | \(58\) |
default | \(\frac {x^{6} \left (4 b^{3} x^{9}+15 b^{2} x^{6} a +20 a^{2} b \,x^{3}+10 a^{3}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {3}{2}}}{60 \left (b \,x^{3}+a \right )^{3}}\) | \(58\) |
risch | \(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, a^{3} x^{6}}{6 b \,x^{3}+6 a}+\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, a^{2} b \,x^{9}}{3 b \,x^{3}+3 a}+\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, b^{2} a \,x^{12}}{4 b \,x^{3}+4 a}+\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, b^{3} x^{15}}{15 b \,x^{3}+15 a}\) | \(116\) |
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Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.45 \[ \int x^5 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\frac {1}{15} \, b^{3} x^{15} + \frac {1}{4} \, a b^{2} x^{12} + \frac {1}{3} \, a^{2} b x^{9} + \frac {1}{6} \, a^{3} x^{6} \]
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\[ \int x^5 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\int x^{5} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac {3}{2}}\, dx \]
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Time = 0.25 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.06 \[ \int x^5 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=-\frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}} a x^{3}}{12 \, b} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}} a^{2}}{12 \, b^{2}} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}}}{15 \, b^{2}} \]
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Time = 0.31 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.58 \[ \int x^5 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\frac {1}{60} \, {\left (4 \, b^{3} x^{15} + 15 \, a b^{2} x^{12} + 20 \, a^{2} b x^{9} + 10 \, a^{3} x^{6}\right )} \mathrm {sgn}\left (b x^{3} + a\right ) \]
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Time = 8.32 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.59 \[ \int x^5 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\frac {{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{3/2}\,\left (-a^2+3\,a\,b\,x^3+4\,b^2\,x^6\right )}{60\,b^2} \]
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