Integrand size = 19, antiderivative size = 52 \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{11}} \, dx=-\frac {\left (b x^2+c x^4\right )^{5/2}}{7 b x^{12}}+\frac {2 c \left (b x^2+c x^4\right )^{5/2}}{35 b^2 x^{10}} \]
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Time = 0.05 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2041, 2039} \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{11}} \, dx=\frac {2 c \left (b x^2+c x^4\right )^{5/2}}{35 b^2 x^{10}}-\frac {\left (b x^2+c x^4\right )^{5/2}}{7 b x^{12}} \]
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Rule 2039
Rule 2041
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (b x^2+c x^4\right )^{5/2}}{7 b x^{12}}-\frac {(2 c) \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^9} \, dx}{7 b} \\ & = -\frac {\left (b x^2+c x^4\right )^{5/2}}{7 b x^{12}}+\frac {2 c \left (b x^2+c x^4\right )^{5/2}}{35 b^2 x^{10}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.67 \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{11}} \, dx=\frac {\left (x^2 \left (b+c x^2\right )\right )^{5/2} \left (-5 b+2 c x^2\right )}{35 b^2 x^{12}} \]
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Time = 0.12 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.75
method | result | size |
gosper | \(-\frac {\left (c \,x^{2}+b \right ) \left (-2 c \,x^{2}+5 b \right ) \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}{35 x^{10} b^{2}}\) | \(39\) |
default | \(-\frac {\left (c \,x^{2}+b \right ) \left (-2 c \,x^{2}+5 b \right ) \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}{35 x^{10} b^{2}}\) | \(39\) |
pseudoelliptic | \(-\frac {\sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \left (-\frac {2 c \,x^{2}}{5}+b \right ) \left (c \,x^{2}+b \right )^{2}}{7 x^{8} b^{2}}\) | \(39\) |
trager | \(-\frac {\left (-2 c^{3} x^{6}+b \,c^{2} x^{4}+8 b^{2} c \,x^{2}+5 b^{3}\right ) \sqrt {c \,x^{4}+b \,x^{2}}}{35 b^{2} x^{8}}\) | \(53\) |
risch | \(-\frac {\sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \left (-2 c^{3} x^{6}+b \,c^{2} x^{4}+8 b^{2} c \,x^{2}+5 b^{3}\right )}{35 x^{8} b^{2}}\) | \(53\) |
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Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.02 \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{11}} \, dx=\frac {{\left (2 \, c^{3} x^{6} - b c^{2} x^{4} - 8 \, b^{2} c x^{2} - 5 \, b^{3}\right )} \sqrt {c x^{4} + b x^{2}}}{35 \, b^{2} x^{8}} \]
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\[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{11}} \, dx=\int \frac {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}{x^{11}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (44) = 88\).
Time = 0.20 (sec) , antiderivative size = 105, normalized size of antiderivative = 2.02 \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{11}} \, dx=\frac {2 \, \sqrt {c x^{4} + b x^{2}} c^{3}}{35 \, b^{2} x^{2}} - \frac {\sqrt {c x^{4} + b x^{2}} c^{2}}{35 \, b x^{4}} + \frac {3 \, \sqrt {c x^{4} + b x^{2}} c}{140 \, x^{6}} + \frac {3 \, \sqrt {c x^{4} + b x^{2}} b}{28 \, x^{8}} - \frac {{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}{4 \, x^{10}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (44) = 88\).
Time = 0.32 (sec) , antiderivative size = 178, normalized size of antiderivative = 3.42 \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{11}} \, dx=\frac {4 \, {\left (35 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{10} c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) + 35 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} b c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) + 70 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} b^{2} c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) + 14 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} b^{3} c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) + 7 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} b^{4} c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) - b^{5} c^{\frac {7}{2}} \mathrm {sgn}\left (x\right )\right )}}{35 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )}^{7}} \]
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Time = 13.71 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.67 \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{11}} \, dx=\frac {2\,c^3\,\sqrt {c\,x^4+b\,x^2}}{35\,b^2\,x^2}-\frac {8\,c\,\sqrt {c\,x^4+b\,x^2}}{35\,x^6}-\frac {c^2\,\sqrt {c\,x^4+b\,x^2}}{35\,b\,x^4}-\frac {b\,\sqrt {c\,x^4+b\,x^2}}{7\,x^8} \]
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