Integrand size = 26, antiderivative size = 255 \[ \int x^{13} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {a^5 x^{14} \sqrt {a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}+\frac {5 a^4 b x^{17} \sqrt {a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}+\frac {a^3 b^2 x^{20} \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac {10 a^2 b^3 x^{23} \sqrt {a^2+2 a b x^3+b^2 x^6}}{23 \left (a+b x^3\right )}+\frac {5 a b^4 x^{26} \sqrt {a^2+2 a b x^3+b^2 x^6}}{26 \left (a+b x^3\right )}+\frac {b^5 x^{29} \sqrt {a^2+2 a b x^3+b^2 x^6}}{29 \left (a+b x^3\right )} \]
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Time = 0.04 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1369, 276} \[ \int x^{13} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {b^5 x^{29} \sqrt {a^2+2 a b x^3+b^2 x^6}}{29 \left (a+b x^3\right )}+\frac {5 a b^4 x^{26} \sqrt {a^2+2 a b x^3+b^2 x^6}}{26 \left (a+b x^3\right )}+\frac {10 a^2 b^3 x^{23} \sqrt {a^2+2 a b x^3+b^2 x^6}}{23 \left (a+b x^3\right )}+\frac {a^5 x^{14} \sqrt {a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}+\frac {5 a^4 b x^{17} \sqrt {a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}+\frac {a^3 b^2 x^{20} \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )} \]
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Rule 276
Rule 1369
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int x^{13} \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} \int \left (a^5 b^5 x^{13}+5 a^4 b^6 x^{16}+10 a^3 b^7 x^{19}+10 a^2 b^8 x^{22}+5 a b^9 x^{25}+b^{10} x^{28}\right ) \, dx}{b^4 \left (a b+b^2 x^3\right )} \\ & = \frac {a^5 x^{14} \sqrt {a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}+\frac {5 a^4 b x^{17} \sqrt {a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}+\frac {a^3 b^2 x^{20} \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac {10 a^2 b^3 x^{23} \sqrt {a^2+2 a b x^3+b^2 x^6}}{23 \left (a+b x^3\right )}+\frac {5 a b^4 x^{26} \sqrt {a^2+2 a b x^3+b^2 x^6}}{26 \left (a+b x^3\right )}+\frac {b^5 x^{29} \sqrt {a^2+2 a b x^3+b^2 x^6}}{29 \left (a+b x^3\right )} \\ \end{align*}
Time = 1.02 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.33 \[ \int x^{13} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {x^{14} \sqrt {\left (a+b x^3\right )^2} \left (147407 a^5+606970 a^4 b x^3+1031849 a^3 b^2 x^6+897260 a^2 b^3 x^9+396865 a b^4 x^{12}+71162 b^5 x^{15}\right )}{2063698 \left (a+b x^3\right )} \]
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Time = 14.54 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.31
method | result | size |
gosper | \(\frac {x^{14} \left (71162 b^{5} x^{15}+396865 a \,b^{4} x^{12}+897260 a^{2} b^{3} x^{9}+1031849 a^{3} b^{2} x^{6}+606970 a^{4} b \,x^{3}+147407 a^{5}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {5}{2}}}{2063698 \left (b \,x^{3}+a \right )^{5}}\) | \(80\) |
default | \(\frac {x^{14} \left (71162 b^{5} x^{15}+396865 a \,b^{4} x^{12}+897260 a^{2} b^{3} x^{9}+1031849 a^{3} b^{2} x^{6}+606970 a^{4} b \,x^{3}+147407 a^{5}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {5}{2}}}{2063698 \left (b \,x^{3}+a \right )^{5}}\) | \(80\) |
risch | \(\frac {a^{5} x^{14} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{14 b \,x^{3}+14 a}+\frac {5 a^{4} b \,x^{17} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{17 \left (b \,x^{3}+a \right )}+\frac {a^{3} b^{2} x^{20} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{2 b \,x^{3}+2 a}+\frac {10 a^{2} b^{3} x^{23} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{23 \left (b \,x^{3}+a \right )}+\frac {5 a \,b^{4} x^{26} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{26 \left (b \,x^{3}+a \right )}+\frac {b^{5} x^{29} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{29 b \,x^{3}+29 a}\) | \(178\) |
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Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.22 \[ \int x^{13} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {1}{29} \, b^{5} x^{29} + \frac {5}{26} \, a b^{4} x^{26} + \frac {10}{23} \, a^{2} b^{3} x^{23} + \frac {1}{2} \, a^{3} b^{2} x^{20} + \frac {5}{17} \, a^{4} b x^{17} + \frac {1}{14} \, a^{5} x^{14} \]
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\[ \int x^{13} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\int x^{13} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.22 \[ \int x^{13} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {1}{29} \, b^{5} x^{29} + \frac {5}{26} \, a b^{4} x^{26} + \frac {10}{23} \, a^{2} b^{3} x^{23} + \frac {1}{2} \, a^{3} b^{2} x^{20} + \frac {5}{17} \, a^{4} b x^{17} + \frac {1}{14} \, a^{5} x^{14} \]
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Time = 0.29 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.41 \[ \int x^{13} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {1}{29} \, b^{5} x^{29} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{26} \, a b^{4} x^{26} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {10}{23} \, a^{2} b^{3} x^{23} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {1}{2} \, a^{3} b^{2} x^{20} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{17} \, a^{4} b x^{17} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {1}{14} \, a^{5} x^{14} \mathrm {sgn}\left (b x^{3} + a\right ) \]
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Timed out. \[ \int x^{13} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\int x^{13}\,{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2} \,d x \]
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