Integrand size = 26, antiderivative size = 167 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^{15}} \, dx=-\frac {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \left (a+b x^2\right )}-\frac {a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 x^{12} \left (a+b x^2\right )}-\frac {3 a b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{10 x^{10} \left (a+b x^2\right )}-\frac {b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left (a+b x^2\right )} \]
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Time = 0.07 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1125, 660, 45} \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^{15}} \, dx=-\frac {a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 x^{12} \left (a+b x^2\right )}-\frac {3 a b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{10 x^{10} \left (a+b x^2\right )}-\frac {b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left (a+b x^2\right )}-\frac {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \left (a+b x^2\right )} \]
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Rule 45
Rule 660
Rule 1125
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^8} \, dx,x,x^2\right ) \\ & = \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \text {Subst}\left (\int \frac {\left (a b+b^2 x\right )^3}{x^8} \, dx,x,x^2\right )}{2 b^2 \left (a b+b^2 x^2\right )} \\ & = \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \text {Subst}\left (\int \left (\frac {a^3 b^3}{x^8}+\frac {3 a^2 b^4}{x^7}+\frac {3 a b^5}{x^6}+\frac {b^6}{x^5}\right ) \, dx,x,x^2\right )}{2 b^2 \left (a b+b^2 x^2\right )} \\ & = -\frac {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \left (a+b x^2\right )}-\frac {a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 x^{12} \left (a+b x^2\right )}-\frac {3 a b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{10 x^{10} \left (a+b x^2\right )}-\frac {b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left (a+b x^2\right )} \\ \end{align*}
Time = 1.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.37 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^{15}} \, dx=-\frac {\sqrt {\left (a+b x^2\right )^2} \left (20 a^3+70 a^2 b x^2+84 a b^2 x^4+35 b^3 x^6\right )}{280 x^{14} \left (a+b x^2\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.28
method | result | size |
pseudoelliptic | \(-\frac {\left (35 b^{3} x^{6}+84 b^{2} x^{4} a +70 a^{2} b \,x^{2}+20 a^{3}\right ) \operatorname {csgn}\left (b \,x^{2}+a \right )}{280 x^{14}}\) | \(46\) |
risch | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (-\frac {1}{14} a^{3}-\frac {1}{4} a^{2} b \,x^{2}-\frac {3}{10} b^{2} x^{4} a -\frac {1}{8} b^{3} x^{6}\right )}{\left (b \,x^{2}+a \right ) x^{14}}\) | \(57\) |
gosper | \(-\frac {\left (35 b^{3} x^{6}+84 b^{2} x^{4} a +70 a^{2} b \,x^{2}+20 a^{3}\right ) {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {3}{2}}}{280 x^{14} \left (b \,x^{2}+a \right )^{3}}\) | \(58\) |
default | \(-\frac {\left (35 b^{3} x^{6}+84 b^{2} x^{4} a +70 a^{2} b \,x^{2}+20 a^{3}\right ) {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {3}{2}}}{280 x^{14} \left (b \,x^{2}+a \right )^{3}}\) | \(58\) |
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none
Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.22 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^{15}} \, dx=-\frac {35 \, b^{3} x^{6} + 84 \, a b^{2} x^{4} + 70 \, a^{2} b x^{2} + 20 \, a^{3}}{280 \, x^{14}} \]
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\[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^{15}} \, dx=\int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}{x^{15}}\, dx \]
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none
Time = 0.21 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.21 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^{15}} \, dx=-\frac {b^{3}}{8 \, x^{8}} - \frac {3 \, a b^{2}}{10 \, x^{10}} - \frac {a^{2} b}{4 \, x^{12}} - \frac {a^{3}}{14 \, x^{14}} \]
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Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.41 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^{15}} \, dx=-\frac {35 \, b^{3} x^{6} \mathrm {sgn}\left (b x^{2} + a\right ) + 84 \, a b^{2} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 70 \, a^{2} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 20 \, a^{3} \mathrm {sgn}\left (b x^{2} + a\right )}{280 \, x^{14}} \]
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Time = 13.24 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^{15}} \, dx=-\frac {a^3\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{14\,x^{14}\,\left (b\,x^2+a\right )}-\frac {b^3\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{8\,x^8\,\left (b\,x^2+a\right )}-\frac {3\,a\,b^2\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{10\,x^{10}\,\left (b\,x^2+a\right )}-\frac {a^2\,b\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{4\,x^{12}\,\left (b\,x^2+a\right )} \]
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