Integrand size = 26, antiderivative size = 160 \[ \int x^{11} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=-\frac {a^3 \left (a+b x^3\right )^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{18 b^4}+\frac {a^2 \left (a+b x^3\right )^6 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 b^4}-\frac {a \left (a+b x^3\right )^7 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 b^4}+\frac {\left (a+b x^3\right )^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{27 b^4} \]
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Time = 0.08 (sec) , antiderivative size = 160, 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} \[ \int x^{11} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^8}{27 b^4}-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^7}{8 b^4}+\frac {a^2 \sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^6}{7 b^4}-\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^5}{18 b^4} \]
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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^{11} \left (a b+b^2 x^3\right )^5 \, dx}{b^4 \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^3 \left (a b+b^2 x\right )^5 \, dx,x,x^3\right )}{3 b^4 \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^3 \left (a b+b^2 x\right )^5}{b^3}+\frac {3 a^2 \left (a b+b^2 x\right )^6}{b^4}-\frac {3 a \left (a b+b^2 x\right )^7}{b^5}+\frac {\left (a b+b^2 x\right )^8}{b^6}\right ) \, dx,x,x^3\right )}{3 b^4 \left (a b+b^2 x^3\right )} \\ & = -\frac {a^3 \left (a+b x^3\right )^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{18 b^4}+\frac {a^2 \left (a+b x^3\right )^6 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 b^4}-\frac {a \left (a+b x^3\right )^7 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 b^4}+\frac {\left (a+b x^3\right )^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{27 b^4} \\ \end{align*}
Time = 0.81 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.84 \[ \int x^{11} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {x^{12} \left (126 a^5+504 a^4 b x^3+840 a^3 b^2 x^6+720 a^2 b^3 x^9+315 a b^4 x^{12}+56 b^5 x^{15}\right ) \left (\sqrt {a^2} b x^3+a \left (\sqrt {a^2}-\sqrt {\left (a+b x^3\right )^2}\right )\right )}{1512 \left (-a^2-a b x^3+\sqrt {a^2} \sqrt {\left (a+b x^3\right )^2}\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.28 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.33
method | result | size |
pseudoelliptic | \(-\frac {\operatorname {csgn}\left (b \,x^{3}+a \right ) \left (b \,x^{3}+a \right )^{6} \left (-56 b^{3} x^{9}+21 b^{2} x^{6} a -6 a^{2} b \,x^{3}+a^{3}\right )}{1512 b^{4}}\) | \(53\) |
gosper | \(\frac {x^{12} \left (56 b^{5} x^{15}+315 a \,b^{4} x^{12}+720 a^{2} b^{3} x^{9}+840 a^{3} b^{2} x^{6}+504 a^{4} b \,x^{3}+126 a^{5}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {5}{2}}}{1512 \left (b \,x^{3}+a \right )^{5}}\) | \(80\) |
default | \(\frac {x^{12} \left (56 b^{5} x^{15}+315 a \,b^{4} x^{12}+720 a^{2} b^{3} x^{9}+840 a^{3} b^{2} x^{6}+504 a^{4} b \,x^{3}+126 a^{5}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {5}{2}}}{1512 \left (b \,x^{3}+a \right )^{5}}\) | \(80\) |
risch | \(\frac {5 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, b^{4} a \,x^{24}}{24 \left (b \,x^{3}+a \right )}+\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, b^{5} x^{27}}{27 b \,x^{3}+27 a}+\frac {5 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, a^{3} b^{2} x^{18}}{9 \left (b \,x^{3}+a \right )}+\frac {10 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, a^{2} b^{3} x^{21}}{21 \left (b \,x^{3}+a \right )}+\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, a^{5} x^{12}}{12 b \,x^{3}+12 a}+\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, b \,a^{4} x^{15}}{3 b \,x^{3}+3 a}\) | \(178\) |
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Time = 0.26 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.36 \[ \int x^{11} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {1}{27} \, b^{5} x^{27} + \frac {5}{24} \, a b^{4} x^{24} + \frac {10}{21} \, a^{2} b^{3} x^{21} + \frac {5}{9} \, a^{3} b^{2} x^{18} + \frac {1}{3} \, a^{4} b x^{15} + \frac {1}{12} \, a^{5} x^{12} \]
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\[ \int x^{11} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\int x^{11} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.91 \[ \int x^{11} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} x^{6}}{27 \, b^{2}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} a^{3} x^{3}}{18 \, b^{3}} - \frac {11 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} a x^{3}}{216 \, b^{3}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} a^{4}}{18 \, b^{4}} + \frac {83 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} a^{2}}{1512 \, b^{4}} \]
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Time = 0.30 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.66 \[ \int x^{11} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {1}{27} \, b^{5} x^{27} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{24} \, a b^{4} x^{24} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {10}{21} \, a^{2} b^{3} x^{21} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{9} \, a^{3} b^{2} x^{18} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {1}{3} \, a^{4} b x^{15} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {1}{12} \, a^{5} x^{12} \mathrm {sgn}\left (b x^{3} + a\right ) \]
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Timed out. \[ \int x^{11} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\int x^{11}\,{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2} \,d x \]
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