Integrand size = 26, antiderivative size = 74 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^3} \, dx=-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}+\frac {b x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]
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Time = 0.01 (sec) , antiderivative size = 74, 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} \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^3} \, dx=\frac {b x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )} \]
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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^3} \, dx}{a b+b^2 x^3} \\ & = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (b^2+\frac {a b}{x^3}\right ) \, dx}{a b+b^2 x^3} \\ & = -\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}+\frac {b 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 = 37, normalized size of antiderivative = 0.50 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^3} \, dx=-\frac {\left (a-2 b x^3\right ) \sqrt {\left (a+b x^3\right )^2}}{2 x^2 \left (a+b x^3\right )} \]
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Time = 3.16 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.46
method | result | size |
gosper | \(-\frac {\left (-2 b \,x^{3}+a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}{2 x^{2} \left (b \,x^{3}+a \right )}\) | \(34\) |
default | \(-\frac {\left (-2 b \,x^{3}+a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}{2 x^{2} \left (b \,x^{3}+a \right )}\) | \(34\) |
risch | \(-\frac {a \sqrt {\left (b \,x^{3}+a \right )^{2}}}{2 x^{2} \left (b \,x^{3}+a \right )}+\frac {b x \sqrt {\left (b \,x^{3}+a \right )^{2}}}{b \,x^{3}+a}\) | \(51\) |
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Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.20 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^3} \, dx=\frac {2 \, b x^{3} - a}{2 \, x^{2}} \]
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Timed out. \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^3} \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.20 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^3} \, dx=\frac {2 \, b x^{3} - a}{2 \, x^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.35 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^3} \, dx=b x \mathrm {sgn}\left (b x^{3} + a\right ) - \frac {a \mathrm {sgn}\left (b x^{3} + a\right )}{2 \, x^{2}} \]
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Timed out. \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^3} \, dx=\int \frac {\sqrt {{\left (b\,x^3+a\right )}^2}}{x^3} \,d x \]
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