Integrand size = 21, antiderivative size = 71 \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^4} \, dx=-\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{3 x^3}-\frac {b \left (c x^2\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{\sqrt {a}}\right )}{3 \sqrt {a} x^3} \]
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Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {375, 272, 43, 65, 214} \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^4} \, dx=-\frac {b \left (c x^2\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{\sqrt {a}}\right )}{3 \sqrt {a} x^3}-\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{3 x^3} \]
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Rule 43
Rule 65
Rule 214
Rule 272
Rule 375
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c x^2\right )^{3/2} \text {Subst}\left (\int \frac {\sqrt {a+b x^3}}{x^4} \, dx,x,\sqrt {c x^2}\right )}{x^3} \\ & = \frac {\left (c x^2\right )^{3/2} \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2} \, dx,x,\left (c x^2\right )^{3/2}\right )}{3 x^3} \\ & = -\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{3 x^3}+\frac {\left (b \left (c x^2\right )^{3/2}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\left (c x^2\right )^{3/2}\right )}{6 x^3} \\ & = -\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{3 x^3}+\frac {\left (c x^2\right )^{3/2} \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \left (c x^2\right )^{3/2}}\right )}{3 x^3} \\ & = -\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{3 x^3}-\frac {b \left (c x^2\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{\sqrt {a}}\right )}{3 \sqrt {a} x^3} \\ \end{align*}
Time = 1.57 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^4} \, dx=\frac {-a-b \left (c x^2\right )^{3/2}-b \left (c x^2\right )^{3/2} \sqrt {1+\frac {b \left (c x^2\right )^{3/2}}{a}} \text {arctanh}\left (\sqrt {1+\frac {b \left (c x^2\right )^{3/2}}{a}}\right )}{3 x^3 \sqrt {a+b \left (c x^2\right )^{3/2}}} \]
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\[\int \frac {\sqrt {a +b \left (c \,x^{2}\right )^{\frac {3}{2}}}}{x^{4}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (53) = 106\).
Time = 0.28 (sec) , antiderivative size = 206, normalized size of antiderivative = 2.90 \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^4} \, dx=\left [\frac {b c x^{3} \sqrt {\frac {c}{a}} \log \left (\frac {b c^{2} x^{4} - 2 \, \sqrt {\sqrt {c x^{2}} b c x^{2} + a} a x \sqrt {\frac {c}{a}} + 2 \, \sqrt {c x^{2}} a}{x^{4}}\right ) - 2 \, \sqrt {\sqrt {c x^{2}} b c x^{2} + a}}{6 \, x^{3}}, -\frac {b c x^{3} \sqrt {-\frac {c}{a}} \arctan \left (-\frac {{\left (a b c^{2} x^{4} \sqrt {-\frac {c}{a}} - \sqrt {c x^{2}} a^{2} \sqrt {-\frac {c}{a}}\right )} \sqrt {\sqrt {c x^{2}} b c x^{2} + a}}{b^{2} c^{4} x^{7} - a^{2} c x}\right ) + \sqrt {\sqrt {c x^{2}} b c x^{2} + a}}{3 \, x^{3}}\right ] \]
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\[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^4} \, dx=\int \frac {\sqrt {a + b \left (c x^{2}\right )^{\frac {3}{2}}}}{x^{4}}\, dx \]
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\[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^4} \, dx=\int { \frac {\sqrt {\left (c x^{2}\right )^{\frac {3}{2}} b + a}}{x^{4}} \,d x } \]
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none
Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^4} \, dx=\frac {\frac {b^{2} c^{3} \arctan \left (\frac {\sqrt {b c^{\frac {3}{2}} x^{3} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {\sqrt {b c^{\frac {3}{2}} x^{3} + a} b c^{\frac {3}{2}}}{x^{3}}}{3 \, b c^{\frac {3}{2}}} \]
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Timed out. \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^4} \, dx=\int \frac {\sqrt {a+b\,{\left (c\,x^2\right )}^{3/2}}}{x^4} \,d x \]
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