Integrand size = 19, antiderivative size = 73 \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^4} \, dx=\frac {b \sqrt {b x^2+c x^4}}{x}+\frac {\left (b x^2+c x^4\right )^{3/2}}{3 x^3}-b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2046, 2033, 212} \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^4} \, dx=b^{3/2} \left (-\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )\right )+\frac {b \sqrt {b x^2+c x^4}}{x}+\frac {\left (b x^2+c x^4\right )^{3/2}}{3 x^3} \]
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Rule 212
Rule 2033
Rule 2046
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b x^2+c x^4\right )^{3/2}}{3 x^3}+b \int \frac {\sqrt {b x^2+c x^4}}{x^2} \, dx \\ & = \frac {b \sqrt {b x^2+c x^4}}{x}+\frac {\left (b x^2+c x^4\right )^{3/2}}{3 x^3}+b^2 \int \frac {1}{\sqrt {b x^2+c x^4}} \, dx \\ & = \frac {b \sqrt {b x^2+c x^4}}{x}+\frac {\left (b x^2+c x^4\right )^{3/2}}{3 x^3}-b^2 \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {b x^2+c x^4}}\right ) \\ & = \frac {b \sqrt {b x^2+c x^4}}{x}+\frac {\left (b x^2+c x^4\right )^{3/2}}{3 x^3}-b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.04 \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^4} \, dx=\frac {x \left (4 b^2+5 b c x^2+c^2 x^4-3 b^{3/2} \sqrt {b+c x^2} \text {arctanh}\left (\frac {\sqrt {b+c x^2}}{\sqrt {b}}\right )\right )}{3 \sqrt {x^2 \left (b+c x^2\right )}} \]
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Time = 0.30 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.07
method | result | size |
default | \(-\frac {\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (3 b^{\frac {3}{2}} \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {c \,x^{2}+b}}{x}\right )-\left (c \,x^{2}+b \right )^{\frac {3}{2}}-3 \sqrt {c \,x^{2}+b}\, b \right )}{3 x^{3} \left (c \,x^{2}+b \right )^{\frac {3}{2}}}\) | \(78\) |
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Time = 0.27 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.92 \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^4} \, dx=\left [\frac {3 \, b^{\frac {3}{2}} x \log \left (-\frac {c x^{3} + 2 \, b x - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {b}}{x^{3}}\right ) + 2 \, \sqrt {c x^{4} + b x^{2}} {\left (c x^{2} + 4 \, b\right )}}{6 \, x}, \frac {3 \, \sqrt {-b} b x \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-b}}{c x^{3} + b x}\right ) + \sqrt {c x^{4} + b x^{2}} {\left (c x^{2} + 4 \, b\right )}}{3 \, x}\right ] \]
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\[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^4} \, dx=\int \frac {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}{x^{4}}\, dx \]
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\[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^4} \, dx=\int { \frac {{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}{x^{4}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.22 \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^4} \, dx=\frac {b^{2} \arctan \left (\frac {\sqrt {c x^{2} + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-b}} + \frac {1}{3} \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} \mathrm {sgn}\left (x\right ) + \sqrt {c x^{2} + b} b \mathrm {sgn}\left (x\right ) - \frac {{\left (3 \, b^{2} \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + 4 \, \sqrt {-b} b^{\frac {3}{2}}\right )} \mathrm {sgn}\left (x\right )}{3 \, \sqrt {-b}} \]
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Timed out. \[ \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^4} \, dx=\int \frac {{\left (c\,x^4+b\,x^2\right )}^{3/2}}{x^4} \,d x \]
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