Integrand size = 21, antiderivative size = 97 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^3} \, dx=-\frac {\sqrt {a+b \sqrt {c x^2}}}{2 x^2}-\frac {b c \sqrt {a+b \sqrt {c x^2}}}{4 a \sqrt {c x^2}}+\frac {b^2 c \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{4 a^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 97, 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, 43, 44, 65, 214} \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^3} \, dx=\frac {b^2 c \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{4 a^{3/2}}-\frac {b c \sqrt {a+b \sqrt {c x^2}}}{4 a \sqrt {c x^2}}-\frac {\sqrt {a+b \sqrt {c x^2}}}{2 x^2} \]
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Rule 43
Rule 44
Rule 65
Rule 214
Rule 375
Rubi steps \begin{align*} \text {integral}& = c \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x^3} \, dx,x,\sqrt {c x^2}\right ) \\ & = -\frac {\sqrt {a+b \sqrt {c x^2}}}{2 x^2}+\frac {1}{4} (b c) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,\sqrt {c x^2}\right ) \\ & = -\frac {\sqrt {a+b \sqrt {c x^2}}}{2 x^2}-\frac {b c \sqrt {a+b \sqrt {c x^2}}}{4 a \sqrt {c x^2}}-\frac {\left (b^2 c\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sqrt {c x^2}\right )}{8 a} \\ & = -\frac {\sqrt {a+b \sqrt {c x^2}}}{2 x^2}-\frac {b c \sqrt {a+b \sqrt {c x^2}}}{4 a \sqrt {c x^2}}-\frac {(b c) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sqrt {c x^2}}\right )}{4 a} \\ & = -\frac {\sqrt {a+b \sqrt {c x^2}}}{2 x^2}-\frac {b c \sqrt {a+b \sqrt {c x^2}}}{4 a \sqrt {c x^2}}+\frac {b^2 c \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{4 a^{3/2}} \\ \end{align*}
Time = 0.94 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^3} \, dx=-\frac {\sqrt {a+b \sqrt {c x^2}} \left (2 a+b \sqrt {c x^2}\right )}{4 a x^2}+\frac {b^2 c \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{4 a^{3/2}} \]
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Time = 3.96 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.74
method | result | size |
default | \(-\frac {-\operatorname {arctanh}\left (\frac {\sqrt {a +b \sqrt {c \,x^{2}}}}{\sqrt {a}}\right ) a \,b^{2} c \,x^{2}+a^{\frac {3}{2}} \left (a +b \sqrt {c \,x^{2}}\right )^{\frac {3}{2}}+a^{\frac {5}{2}} \sqrt {a +b \sqrt {c \,x^{2}}}}{4 a^{\frac {5}{2}} x^{2}}\) | \(72\) |
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Time = 0.25 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.79 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^3} \, dx=\left [\frac {\sqrt {a} b^{2} c x^{2} \log \left (\frac {b c x^{2} + 2 \, \sqrt {c x^{2}} \sqrt {\sqrt {c x^{2}} b + a} \sqrt {a} + 2 \, \sqrt {c x^{2}} a}{x^{2}}\right ) - 2 \, {\left (\sqrt {c x^{2}} a b + 2 \, a^{2}\right )} \sqrt {\sqrt {c x^{2}} b + a}}{8 \, a^{2} x^{2}}, -\frac {\sqrt {-a} b^{2} c x^{2} \arctan \left (\frac {\sqrt {\sqrt {c x^{2}} b + a} \sqrt {-a}}{a}\right ) + {\left (\sqrt {c x^{2}} a b + 2 \, a^{2}\right )} \sqrt {\sqrt {c x^{2}} b + a}}{4 \, a^{2} x^{2}}\right ] \]
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\[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^3} \, dx=\int \frac {\sqrt {a + b \sqrt {c x^{2}}}}{x^{3}}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.30 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^3} \, dx=-\frac {1}{8} \, {\left (\frac {b^{2} \log \left (\frac {\sqrt {\sqrt {c x^{2}} b + a} - \sqrt {a}}{\sqrt {\sqrt {c x^{2}} b + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2 \, {\left ({\left (\sqrt {c x^{2}} b + a\right )}^{\frac {3}{2}} b^{2} + \sqrt {\sqrt {c x^{2}} b + a} a b^{2}\right )}}{{\left (\sqrt {c x^{2}} b + a\right )}^{2} a - 2 \, {\left (\sqrt {c x^{2}} b + a\right )} a^{2} + a^{3}}\right )} c \]
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Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^3} \, dx=-\frac {\frac {b^{3} c^{\frac {3}{2}} \arctan \left (\frac {\sqrt {b \sqrt {c} x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {{\left (b \sqrt {c} x + a\right )}^{\frac {3}{2}} b^{3} c^{\frac {3}{2}} + \sqrt {b \sqrt {c} x + a} a b^{3} c^{\frac {3}{2}}}{a b^{2} c x^{2}}}{4 \, b \sqrt {c}} \]
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Timed out. \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^3} \, dx=\int \frac {\sqrt {a+b\,\sqrt {c\,x^2}}}{x^3} \,d x \]
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