Integrand size = 26, antiderivative size = 77 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^2} \, dx=-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}+\frac {b x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )} \]
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Time = 0.02 (sec) , antiderivative size = 77, 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} \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^2} \, dx=\frac {b x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )} \]
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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^2} \, 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^2}+b^2 x\right ) \, dx}{a b+b^2 x^3} \\ & = -\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}+\frac {b 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 = 38, normalized size of antiderivative = 0.49 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^2} \, dx=\frac {\left (-2 a+b x^3\right ) \sqrt {\left (a+b x^3\right )^2}}{2 x \left (a+b x^3\right )} \]
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Time = 2.39 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.47
method | result | size |
gosper | \(-\frac {\left (-b \,x^{3}+2 a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}{2 x \left (b \,x^{3}+a \right )}\) | \(36\) |
default | \(-\frac {\left (-b \,x^{3}+2 a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}{2 x \left (b \,x^{3}+a \right )}\) | \(36\) |
risch | \(-\frac {a \sqrt {\left (b \,x^{3}+a \right )^{2}}}{x \left (b \,x^{3}+a \right )}+\frac {b \,x^{2} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{2 b \,x^{3}+2 a}\) | \(54\) |
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Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^2} \, dx=\frac {b x^{3} - 2 \, a}{2 \, x} \]
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\[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^2} \, dx=\int \frac {\sqrt {\left (a + b x^{3}\right )^{2}}}{x^{2}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^2} \, dx=\frac {b x^{3} - 2 \, a}{2 \, x} \]
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Time = 0.31 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.38 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^2} \, dx=\frac {1}{2} \, b x^{2} \mathrm {sgn}\left (b x^{3} + a\right ) - \frac {a \mathrm {sgn}\left (b x^{3} + a\right )}{x} \]
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Timed out. \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^2} \, dx=\int \frac {\sqrt {{\left (b\,x^3+a\right )}^2}}{x^2} \,d x \]
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