Integrand size = 26, antiderivative size = 167 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^{15}} \, dx=-\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{14 x^{14} \left (a+b x^3\right )}-\frac {3 a^2 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac {3 a b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac {b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )} \]
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Time = 0.03 (sec) , antiderivative size = 167, 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} \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^{15}} \, dx=-\frac {3 a^2 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac {3 a b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac {b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}-\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{14 x^{14} \left (a+b x^3\right )} \]
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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 \frac {\left (a b+b^2 x^3\right )^3}{x^{15}} \, dx}{b^2 \left (a b+b^2 x^3\right )} \\ & = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (\frac {a^3 b^3}{x^{15}}+\frac {3 a^2 b^4}{x^{12}}+\frac {3 a b^5}{x^9}+\frac {b^6}{x^6}\right ) \, dx}{b^2 \left (a b+b^2 x^3\right )} \\ & = -\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{14 x^{14} \left (a+b x^3\right )}-\frac {3 a^2 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac {3 a b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac {b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )} \\ \end{align*}
Time = 1.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.37 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^{15}} \, dx=-\frac {\sqrt {\left (a+b x^3\right )^2} \left (220 a^3+840 a^2 b x^3+1155 a b^2 x^6+616 b^3 x^9\right )}{3080 x^{14} \left (a+b x^3\right )} \]
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Time = 19.19 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.34
method | result | size |
risch | \(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (-\frac {1}{14} a^{3}-\frac {3}{11} a^{2} b \,x^{3}-\frac {3}{8} b^{2} x^{6} a -\frac {1}{5} b^{3} x^{9}\right )}{\left (b \,x^{3}+a \right ) x^{14}}\) | \(57\) |
gosper | \(-\frac {\left (616 b^{3} x^{9}+1155 b^{2} x^{6} a +840 a^{2} b \,x^{3}+220 a^{3}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {3}{2}}}{3080 x^{14} \left (b \,x^{3}+a \right )^{3}}\) | \(58\) |
default | \(-\frac {\left (616 b^{3} x^{9}+1155 b^{2} x^{6} a +840 a^{2} b \,x^{3}+220 a^{3}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {3}{2}}}{3080 x^{14} \left (b \,x^{3}+a \right )^{3}}\) | \(58\) |
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Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.22 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^{15}} \, dx=-\frac {616 \, b^{3} x^{9} + 1155 \, a b^{2} x^{6} + 840 \, a^{2} b x^{3} + 220 \, a^{3}}{3080 \, x^{14}} \]
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\[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^{15}} \, dx=\int \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {3}{2}}}{x^{15}}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.22 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^{15}} \, dx=-\frac {616 \, b^{3} x^{9} + 1155 \, a b^{2} x^{6} + 840 \, a^{2} b x^{3} + 220 \, a^{3}}{3080 \, x^{14}} \]
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Time = 0.33 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.41 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^{15}} \, dx=-\frac {616 \, b^{3} x^{9} \mathrm {sgn}\left (b x^{3} + a\right ) + 1155 \, a b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 840 \, a^{2} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 220 \, a^{3} \mathrm {sgn}\left (b x^{3} + a\right )}{3080 \, x^{14}} \]
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Time = 8.12 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^{15}} \, dx=-\frac {a^3\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{14\,x^{14}\,\left (b\,x^3+a\right )}-\frac {b^3\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{5\,x^5\,\left (b\,x^3+a\right )}-\frac {3\,a\,b^2\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{8\,x^8\,\left (b\,x^3+a\right )}-\frac {3\,a^2\,b\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{11\,x^{11}\,\left (b\,x^3+a\right )} \]
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