Integrand size = 19, antiderivative size = 109 \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{10}} \, dx=-\frac {c \sqrt {b x^2+c x^4}}{8 x^5}-\frac {c^2 \sqrt {b x^2+c x^4}}{16 b x^3}-\frac {\left (b x^2+c x^4\right )^{3/2}}{6 x^9}+\frac {c^3 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{16 b^{3/2}} \]
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Time = 0.09 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2045, 2050, 2033, 212} \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{10}} \, dx=\frac {c^3 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{16 b^{3/2}}-\frac {c^2 \sqrt {b x^2+c x^4}}{16 b x^3}-\frac {\left (b x^2+c x^4\right )^{3/2}}{6 x^9}-\frac {c \sqrt {b x^2+c x^4}}{8 x^5} \]
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Rule 212
Rule 2033
Rule 2045
Rule 2050
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (b x^2+c x^4\right )^{3/2}}{6 x^9}+\frac {1}{2} c \int \frac {\sqrt {b x^2+c x^4}}{x^6} \, dx \\ & = -\frac {c \sqrt {b x^2+c x^4}}{8 x^5}-\frac {\left (b x^2+c x^4\right )^{3/2}}{6 x^9}+\frac {1}{8} c^2 \int \frac {1}{x^2 \sqrt {b x^2+c x^4}} \, dx \\ & = -\frac {c \sqrt {b x^2+c x^4}}{8 x^5}-\frac {c^2 \sqrt {b x^2+c x^4}}{16 b x^3}-\frac {\left (b x^2+c x^4\right )^{3/2}}{6 x^9}-\frac {c^3 \int \frac {1}{\sqrt {b x^2+c x^4}} \, dx}{16 b} \\ & = -\frac {c \sqrt {b x^2+c x^4}}{8 x^5}-\frac {c^2 \sqrt {b x^2+c x^4}}{16 b x^3}-\frac {\left (b x^2+c x^4\right )^{3/2}}{6 x^9}+\frac {c^3 \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {b x^2+c x^4}}\right )}{16 b} \\ & = -\frac {c \sqrt {b x^2+c x^4}}{8 x^5}-\frac {c^2 \sqrt {b x^2+c x^4}}{16 b x^3}-\frac {\left (b x^2+c x^4\right )^{3/2}}{6 x^9}+\frac {c^3 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{16 b^{3/2}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.95 \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{10}} \, dx=\frac {\sqrt {x^2 \left (b+c x^2\right )} \left (-\sqrt {b} \sqrt {b+c x^2} \left (8 b^2+14 b c x^2+3 c^2 x^4\right )+3 c^3 x^6 \text {arctanh}\left (\frac {\sqrt {b+c x^2}}{\sqrt {b}}\right )\right )}{48 b^{3/2} x^7 \sqrt {b+c x^2}} \]
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Time = 0.80 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.92
method | result | size |
risch | \(-\frac {\left (3 c^{2} x^{4}+14 b c \,x^{2}+8 b^{2}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{48 x^{7} b}+\frac {c^{3} \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {c \,x^{2}+b}}{x}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{16 b^{\frac {3}{2}} x \sqrt {c \,x^{2}+b}}\) | \(100\) |
default | \(\frac {\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (3 b^{\frac {3}{2}} \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {c \,x^{2}+b}}{x}\right ) c^{3} x^{6}-\left (c \,x^{2}+b \right )^{\frac {3}{2}} c^{3} x^{6}+\left (c \,x^{2}+b \right )^{\frac {5}{2}} c^{2} x^{4}-3 \sqrt {c \,x^{2}+b}\, b \,c^{3} x^{6}+2 \left (c \,x^{2}+b \right )^{\frac {5}{2}} b c \,x^{2}-8 \left (c \,x^{2}+b \right )^{\frac {5}{2}} b^{2}\right )}{48 x^{9} \left (c \,x^{2}+b \right )^{\frac {3}{2}} b^{3}}\) | \(145\) |
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Time = 0.27 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.70 \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{10}} \, dx=\left [\frac {3 \, \sqrt {b} c^{3} x^{7} \log \left (-\frac {c x^{3} + 2 \, b x + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {b}}{x^{3}}\right ) - 2 \, {\left (3 \, b c^{2} x^{4} + 14 \, b^{2} c x^{2} + 8 \, b^{3}\right )} \sqrt {c x^{4} + b x^{2}}}{96 \, b^{2} x^{7}}, -\frac {3 \, \sqrt {-b} c^{3} x^{7} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-b}}{c x^{3} + b x}\right ) + {\left (3 \, b c^{2} x^{4} + 14 \, b^{2} c x^{2} + 8 \, b^{3}\right )} \sqrt {c x^{4} + b x^{2}}}{48 \, b^{2} x^{7}}\right ] \]
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\[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{10}} \, dx=\int \frac {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}{x^{10}}\, dx \]
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\[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{10}} \, dx=\int { \frac {{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}{x^{10}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.92 \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{10}} \, dx=-\frac {\frac {3 \, c^{4} \arctan \left (\frac {\sqrt {c x^{2} + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-b} b} + \frac {3 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} c^{4} \mathrm {sgn}\left (x\right ) + 8 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} b c^{4} \mathrm {sgn}\left (x\right ) - 3 \, \sqrt {c x^{2} + b} b^{2} c^{4} \mathrm {sgn}\left (x\right )}{b c^{3} x^{6}}}{48 \, c} \]
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Timed out. \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{10}} \, dx=\int \frac {{\left (c\,x^4+b\,x^2\right )}^{3/2}}{x^{10}} \,d x \]
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