Integrand size = 26, antiderivative size = 163 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^3} \, dx=-\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}+\frac {3 a^2 b x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {3 a b^2 x^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac {b^3 x^7 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )} \]
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Time = 0.03 (sec) , antiderivative size = 163, 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} \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^3} \, dx=\frac {3 a^2 b x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {3 a b^2 x^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac {b^3 x^7 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}-\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )} \]
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Rule 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^3} \, 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 (3 a^2 b^4+\frac {a^3 b^3}{x^3}+3 a b^5 x^3+b^6 x^6\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}}{2 x^2 \left (a+b x^3\right )}+\frac {3 a^2 b x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {3 a b^2 x^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac {b^3 x^7 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )} \\ \end{align*}
Time = 1.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.37 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^3} \, dx=\frac {\sqrt {\left (a+b x^3\right )^2} \left (-14 a^3+84 a^2 b x^3+21 a b^2 x^6+4 b^3 x^9\right )}{28 x^2 \left (a+b x^3\right )} \]
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Time = 3.13 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.36
method | result | size |
gosper | \(-\frac {\left (-4 b^{3} x^{9}-21 b^{2} x^{6} a -84 a^{2} b \,x^{3}+14 a^{3}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {3}{2}}}{28 x^{2} \left (b \,x^{3}+a \right )^{3}}\) | \(58\) |
default | \(-\frac {\left (-4 b^{3} x^{9}-21 b^{2} x^{6} a -84 a^{2} b \,x^{3}+14 a^{3}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {3}{2}}}{28 x^{2} \left (b \,x^{3}+a \right )^{3}}\) | \(58\) |
risch | \(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, b \left (\frac {1}{7} b^{2} x^{7}+\frac {3}{4} a b \,x^{4}+3 a^{2} x \right )}{b \,x^{3}+a}-\frac {a^{3} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{2 x^{2} \left (b \,x^{3}+a \right )}\) | \(74\) |
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Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.23 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^3} \, dx=\frac {4 \, b^{3} x^{9} + 21 \, a b^{2} x^{6} + 84 \, a^{2} b x^{3} - 14 \, a^{3}}{28 \, x^{2}} \]
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\[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^3} \, dx=\int \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {3}{2}}}{x^{3}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.23 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^3} \, dx=\frac {4 \, b^{3} x^{9} + 21 \, a b^{2} x^{6} + 84 \, a^{2} b x^{3} - 14 \, a^{3}}{28 \, x^{2}} \]
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Time = 0.31 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.40 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^3} \, dx=\frac {1}{7} \, b^{3} x^{7} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {3}{4} \, a b^{2} x^{4} \mathrm {sgn}\left (b x^{3} + a\right ) + 3 \, a^{2} b x \mathrm {sgn}\left (b x^{3} + a\right ) - \frac {a^{3} \mathrm {sgn}\left (b x^{3} + a\right )}{2 \, x^{2}} \]
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Timed out. \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^3} \, dx=\int \frac {{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{3/2}}{x^3} \,d x \]
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