Integrand size = 26, antiderivative size = 162 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^9} \, dx=-\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac {3 a^2 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}-\frac {3 a b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}+\frac {b^3 x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]
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Time = 0.03 (sec) , antiderivative size = 162, 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} \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^9} \, dx=-\frac {3 a^2 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}-\frac {3 a b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}+\frac {b^3 x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )} \]
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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^9} \, 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 (b^6+\frac {a^3 b^3}{x^9}+\frac {3 a^2 b^4}{x^6}+\frac {3 a b^5}{x^3}\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}}{8 x^8 \left (a+b x^3\right )}-\frac {3 a^2 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}-\frac {3 a b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}+\frac {b^3 x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \\ \end{align*}
Time = 1.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.38 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^9} \, dx=-\frac {\sqrt {\left (a+b x^3\right )^2} \left (5 a^3+24 a^2 b x^3+60 a b^2 x^6-40 b^3 x^9\right )}{40 x^8 \left (a+b x^3\right )} \]
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Time = 8.78 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.36
method | result | size |
gosper | \(-\frac {\left (-40 b^{3} x^{9}+60 b^{2} x^{6} a +24 a^{2} b \,x^{3}+5 a^{3}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {3}{2}}}{40 \left (b \,x^{3}+a \right )^{3} x^{8}}\) | \(58\) |
default | \(-\frac {\left (-40 b^{3} x^{9}+60 b^{2} x^{6} a +24 a^{2} b \,x^{3}+5 a^{3}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {3}{2}}}{40 \left (b \,x^{3}+a \right )^{3} x^{8}}\) | \(58\) |
risch | \(\frac {b^{3} x \sqrt {\left (b \,x^{3}+a \right )^{2}}}{b \,x^{3}+a}+\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (-\frac {3}{2} b^{2} x^{6} a -\frac {3}{5} a^{2} b \,x^{3}-\frac {1}{8} a^{3}\right )}{\left (b \,x^{3}+a \right ) x^{8}}\) | \(75\) |
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Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.23 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^9} \, dx=\frac {40 \, b^{3} x^{9} - 60 \, a b^{2} x^{6} - 24 \, a^{2} b x^{3} - 5 \, a^{3}}{40 \, x^{8}} \]
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\[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^9} \, dx=\int \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {3}{2}}}{x^{9}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.23 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^9} \, dx=\frac {40 \, b^{3} x^{9} - 60 \, a b^{2} x^{6} - 24 \, a^{2} b x^{3} - 5 \, a^{3}}{40 \, x^{8}} \]
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Time = 0.30 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.41 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^9} \, dx=b^{3} x \mathrm {sgn}\left (b x^{3} + a\right ) - \frac {60 \, a b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 24 \, a^{2} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 5 \, a^{3} \mathrm {sgn}\left (b x^{3} + a\right )}{40 \, x^{8}} \]
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Timed out. \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^9} \, dx=\int \frac {{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{3/2}}{x^9} \,d x \]
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