Integrand size = 24, antiderivative size = 76 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^6} \, dx=-\frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{5 a x^5}+\frac {b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{20 a^2 x^4} \]
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Time = 0.02 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {660, 47, 37} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^6} \, dx=\frac {b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{20 a^2 x^4}-\frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{5 a x^5} \]
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Rule 37
Rule 47
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^3}{x^6} \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = -\frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{5 a x^5}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^3}{x^5} \, dx}{5 a b \left (a b+b^2 x\right )} \\ & = -\frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{5 a x^5}+\frac {b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{20 a^2 x^4} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.72 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^6} \, dx=-\frac {\sqrt {(a+b x)^2} \left (4 a^3+15 a^2 b x+20 a b^2 x^2+10 b^3 x^3\right )}{20 x^5 (a+b x)} \]
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Time = 2.58 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.67
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {1}{2} b^{3} x^{3}-a \,b^{2} x^{2}-\frac {3}{4} a^{2} b x -\frac {1}{5} a^{3}\right )}{\left (b x +a \right ) x^{5}}\) | \(51\) |
gosper | \(-\frac {\left (10 b^{3} x^{3}+20 a \,b^{2} x^{2}+15 a^{2} b x +4 a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{20 x^{5} \left (b x +a \right )^{3}}\) | \(52\) |
default | \(-\frac {\left (10 b^{3} x^{3}+20 a \,b^{2} x^{2}+15 a^{2} b x +4 a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{20 x^{5} \left (b x +a \right )^{3}}\) | \(52\) |
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Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.46 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^6} \, dx=-\frac {10 \, b^{3} x^{3} + 20 \, a b^{2} x^{2} + 15 \, a^{2} b x + 4 \, a^{3}}{20 \, x^{5}} \]
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\[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^6} \, dx=\int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{x^{6}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (50) = 100\).
Time = 0.19 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.20 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^6} \, dx=-\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{5}}{4 \, a^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{4}}{4 \, a^{4} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{3}}{4 \, a^{5} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{2}}{4 \, a^{4} x^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b}{4 \, a^{3} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}}{5 \, a^{2} x^{5}} \]
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Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^6} \, dx=\frac {b^{5} \mathrm {sgn}\left (b x + a\right )}{20 \, a^{2}} - \frac {10 \, b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 20 \, a b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b x \mathrm {sgn}\left (b x + a\right ) + 4 \, a^{3} \mathrm {sgn}\left (b x + a\right )}{20 \, x^{5}} \]
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Time = 9.06 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.78 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^6} \, dx=-\frac {a^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{5\,x^5\,\left (a+b\,x\right )}-\frac {b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,x^2\,\left (a+b\,x\right )}-\frac {a\,b^2\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^3\,\left (a+b\,x\right )}-\frac {3\,a^2\,b\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,x^4\,\left (a+b\,x\right )} \]
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