Integrand size = 26, antiderivative size = 79 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^6} \, dx=-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}-\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )} \]
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Time = 0.02 (sec) , antiderivative size = 79, 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, 14} \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^6} \, dx=-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}-\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )} \]
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Rule 14
Rule 1369
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {a b+b^2 x^3}{x^6} \, dx}{a b+b^2 x^3} \\ & = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (\frac {a b}{x^6}+\frac {b^2}{x^3}\right ) \, dx}{a b+b^2 x^3} \\ & = -\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}-\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )} \\ \end{align*}
Time = 1.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.49 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^6} \, dx=-\frac {\sqrt {\left (a+b x^3\right )^2} \left (2 a+5 b x^3\right )}{10 x^5 \left (a+b x^3\right )} \]
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Time = 5.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.44
method | result | size |
risch | \(\frac {\left (-\frac {b \,x^{3}}{2}-\frac {a}{5}\right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}{x^{5} \left (b \,x^{3}+a \right )}\) | \(35\) |
gosper | \(-\frac {\left (5 b \,x^{3}+2 a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}{10 \left (b \,x^{3}+a \right ) x^{5}}\) | \(36\) |
default | \(-\frac {\left (5 b \,x^{3}+2 a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}{10 \left (b \,x^{3}+a \right ) x^{5}}\) | \(36\) |
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none
Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.19 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^6} \, dx=-\frac {5 \, b x^{3} + 2 \, a}{10 \, x^{5}} \]
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Timed out. \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^6} \, dx=\text {Timed out} \]
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none
Time = 0.22 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.19 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^6} \, dx=-\frac {5 \, b x^{3} + 2 \, a}{10 \, x^{5}} \]
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none
Time = 0.32 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.39 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^6} \, dx=-\frac {5 \, b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 2 \, a \mathrm {sgn}\left (b x^{3} + a\right )}{10 \, x^{5}} \]
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Time = 8.17 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.44 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^6} \, dx=-\frac {\left (5\,b\,x^3+2\,a\right )\,\sqrt {{\left (b\,x^3+a\right )}^2}}{10\,x^5\,\left (b\,x^3+a\right )} \]
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