Integrand size = 26, antiderivative size = 165 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^8} \, dx=-\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac {3 a^2 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}-\frac {3 a b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}+\frac {b^3 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )} \]
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Time = 0.03 (sec) , antiderivative size = 165, 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} \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^8} \, dx=-\frac {3 a b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}-\frac {3 a^2 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}+\frac {b^3 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}-\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )} \]
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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^8} \, 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^8}+\frac {3 a^2 b^4}{x^5}+\frac {3 a b^5}{x^2}+b^6 x\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}}{7 x^7 \left (a+b x^3\right )}-\frac {3 a^2 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}-\frac {3 a b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}+\frac {b^3 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )} \\ \end{align*}
Time = 1.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.37 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^8} \, dx=-\frac {\sqrt {\left (a+b x^3\right )^2} \left (4 a^3+21 a^2 b x^3+84 a b^2 x^6-14 b^3 x^9\right )}{28 x^7 \left (a+b x^3\right )} \]
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Time = 7.54 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.35
method | result | size |
gosper | \(-\frac {\left (-14 b^{3} x^{9}+84 b^{2} x^{6} a +21 a^{2} b \,x^{3}+4 a^{3}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {3}{2}}}{28 \left (b \,x^{3}+a \right )^{3} x^{7}}\) | \(58\) |
default | \(-\frac {\left (-14 b^{3} x^{9}+84 b^{2} x^{6} a +21 a^{2} b \,x^{3}+4 a^{3}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {3}{2}}}{28 \left (b \,x^{3}+a \right )^{3} x^{7}}\) | \(58\) |
risch | \(\frac {b^{3} x^{2} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{2 b \,x^{3}+2 a}+\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (-3 b^{2} x^{6} a -\frac {3}{4} a^{2} b \,x^{3}-\frac {1}{7} a^{3}\right )}{\left (b \,x^{3}+a \right ) x^{7}}\) | \(78\) |
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Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.22 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^8} \, dx=\frac {14 \, b^{3} x^{9} - 84 \, a b^{2} x^{6} - 21 \, a^{2} b x^{3} - 4 \, a^{3}}{28 \, x^{7}} \]
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\[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^8} \, dx=\int \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {3}{2}}}{x^{8}}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.22 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^8} \, dx=\frac {14 \, b^{3} x^{9} - 84 \, a b^{2} x^{6} - 21 \, a^{2} b x^{3} - 4 \, a^{3}}{28 \, x^{7}} \]
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Time = 0.34 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.42 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^8} \, dx=\frac {1}{2} \, b^{3} x^{2} \mathrm {sgn}\left (b x^{3} + a\right ) - \frac {84 \, a b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 21 \, a^{2} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 4 \, a^{3} \mathrm {sgn}\left (b x^{3} + a\right )}{28 \, x^{7}} \]
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Timed out. \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^8} \, dx=\int \frac {{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{3/2}}{x^8} \,d x \]
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