Integrand size = 24, antiderivative size = 151 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^9} \, dx=-\frac {a^3 \sqrt {a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)}-\frac {3 a^2 b \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac {a b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^6 (a+b x)}-\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)} \]
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Time = 0.02 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {660, 45} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^9} \, dx=-\frac {3 a^2 b \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac {a b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^6 (a+b x)}-\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac {a^3 \sqrt {a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)} \]
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Rule 45
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^9} \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a^3 b^3}{x^9}+\frac {3 a^2 b^4}{x^8}+\frac {3 a b^5}{x^7}+\frac {b^6}{x^6}\right ) \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = -\frac {a^3 \sqrt {a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)}-\frac {3 a^2 b \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac {a b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^6 (a+b x)}-\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)} \\ \end{align*}
Time = 0.43 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.36 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^9} \, dx=-\frac {\sqrt {(a+b x)^2} \left (35 a^3+120 a^2 b x+140 a b^2 x^2+56 b^3 x^3\right )}{280 x^8 (a+b x)} \]
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Time = 2.50 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.34
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {1}{5} b^{3} x^{3}-\frac {1}{2} a \,b^{2} x^{2}-\frac {3}{7} a^{2} b x -\frac {1}{8} a^{3}\right )}{\left (b x +a \right ) x^{8}}\) | \(51\) |
gosper | \(-\frac {\left (56 b^{3} x^{3}+140 a \,b^{2} x^{2}+120 a^{2} b x +35 a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{280 x^{8} \left (b x +a \right )^{3}}\) | \(52\) |
default | \(-\frac {\left (56 b^{3} x^{3}+140 a \,b^{2} x^{2}+120 a^{2} b x +35 a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{280 x^{8} \left (b x +a \right )^{3}}\) | \(52\) |
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Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.23 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^9} \, dx=-\frac {56 \, b^{3} x^{3} + 140 \, a b^{2} x^{2} + 120 \, a^{2} b x + 35 \, a^{3}}{280 \, x^{8}} \]
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\[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^9} \, dx=\int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{x^{9}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (99) = 198\).
Time = 0.19 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.68 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^9} \, dx=\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{8}}{4 \, a^{8}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{7}}{4 \, a^{7} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{6}}{4 \, a^{8} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{5}}{4 \, a^{7} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{4}}{4 \, a^{6} x^{4}} + \frac {69 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{3}}{280 \, a^{5} x^{5}} - \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{2}}{56 \, a^{4} x^{6}} + \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b}{56 \, a^{3} x^{7}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}}{8 \, a^{2} x^{8}} \]
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Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.49 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^9} \, dx=-\frac {b^{8} \mathrm {sgn}\left (b x + a\right )}{280 \, a^{5}} - \frac {56 \, b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 140 \, a b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 120 \, a^{2} b x \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{3} \mathrm {sgn}\left (b x + a\right )}{280 \, x^{8}} \]
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Time = 9.09 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^9} \, dx=-\frac {a^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{8\,x^8\,\left (a+b\,x\right )}-\frac {b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{5\,x^5\,\left (a+b\,x\right )}-\frac {a\,b^2\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,x^6\,\left (a+b\,x\right )}-\frac {3\,a^2\,b\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{7\,x^7\,\left (a+b\,x\right )} \]
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