Optimal. Leaf size=74 \[ -\frac {\sqrt {a+\frac {b}{x^2}}}{4 x^3}-\frac {a \sqrt {a+\frac {b}{x^2}}}{8 b x}+\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^2}} x}\right )}{8 b^{3/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {342, 285, 327,
223, 212} \begin {gather*} \frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {b}}{x \sqrt {a+\frac {b}{x^2}}}\right )}{8 b^{3/2}}-\frac {a \sqrt {a+\frac {b}{x^2}}}{8 b x}-\frac {\sqrt {a+\frac {b}{x^2}}}{4 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 285
Rule 327
Rule 342
Rubi steps
\begin {align*} \int \frac {\sqrt {a+\frac {b}{x^2}}}{x^4} \, dx &=-\text {Subst}\left (\int x^2 \sqrt {a+b x^2} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\sqrt {a+\frac {b}{x^2}}}{4 x^3}-\frac {1}{4} a \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\sqrt {a+\frac {b}{x^2}}}{4 x^3}-\frac {a \sqrt {a+\frac {b}{x^2}}}{8 b x}+\frac {a^2 \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{x}\right )}{8 b}\\ &=-\frac {\sqrt {a+\frac {b}{x^2}}}{4 x^3}-\frac {a \sqrt {a+\frac {b}{x^2}}}{8 b x}+\frac {a^2 \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x^2}} x}\right )}{8 b}\\ &=-\frac {\sqrt {a+\frac {b}{x^2}}}{4 x^3}-\frac {a \sqrt {a+\frac {b}{x^2}}}{8 b x}+\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^2}} x}\right )}{8 b^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 76, normalized size = 1.03 \begin {gather*} \frac {\sqrt {a+\frac {b}{x^2}} \left (-\sqrt {b} \left (2 b+a x^2\right )+\frac {a^2 x^4 \tanh ^{-1}\left (\frac {\sqrt {b+a x^2}}{\sqrt {b}}\right )}{\sqrt {b+a x^2}}\right )}{8 b^{3/2} x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 106, normalized size = 1.43
method | result | size |
risch | \(-\frac {\left (a \,x^{2}+2 b \right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}{8 x^{3} b}+\frac {a^{2} \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {a \,x^{2}+b}}{x}\right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x}{8 b^{\frac {3}{2}} \sqrt {a \,x^{2}+b}}\) | \(86\) |
default | \(\frac {\sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, \left (\sqrt {b}\, \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {a \,x^{2}+b}}{x}\right ) a^{2} x^{4}-\sqrt {a \,x^{2}+b}\, a^{2} x^{4}+\left (a \,x^{2}+b \right )^{\frac {3}{2}} a \,x^{2}-2 \left (a \,x^{2}+b \right )^{\frac {3}{2}} b \right )}{8 x^{3} \sqrt {a \,x^{2}+b}\, b^{2}}\) | \(106\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 114, normalized size = 1.54 \begin {gather*} -\frac {a^{2} \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} x - \sqrt {b}}{\sqrt {a + \frac {b}{x^{2}}} x + \sqrt {b}}\right )}{16 \, b^{\frac {3}{2}}} - \frac {{\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} a^{2} x^{3} + \sqrt {a + \frac {b}{x^{2}}} a^{2} b x}{8 \, {\left ({\left (a + \frac {b}{x^{2}}\right )}^{2} b x^{4} - 2 \, {\left (a + \frac {b}{x^{2}}\right )} b^{2} x^{2} + b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 158, normalized size = 2.14 \begin {gather*} \left [\frac {a^{2} \sqrt {b} x^{3} \log \left (-\frac {a x^{2} + 2 \, \sqrt {b} x \sqrt {\frac {a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ) - 2 \, {\left (a b x^{2} + 2 \, b^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{16 \, b^{2} x^{3}}, -\frac {a^{2} \sqrt {-b} x^{3} \arctan \left (\frac {\sqrt {-b} x \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) + {\left (a b x^{2} + 2 \, b^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{8 \, b^{2} x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.08, size = 92, normalized size = 1.24 \begin {gather*} - \frac {a^{\frac {3}{2}}}{8 b x \sqrt {1 + \frac {b}{a x^{2}}}} - \frac {3 \sqrt {a}}{8 x^{3} \sqrt {1 + \frac {b}{a x^{2}}}} + \frac {a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} x} \right )}}{8 b^{\frac {3}{2}}} - \frac {b}{4 \sqrt {a} x^{5} \sqrt {1 + \frac {b}{a x^{2}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.46, size = 78, normalized size = 1.05 \begin {gather*} -\frac {\frac {a^{3} \arctan \left (\frac {\sqrt {a x^{2} + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-b} b} + \frac {{\left (a x^{2} + b\right )}^{\frac {3}{2}} a^{3} \mathrm {sgn}\left (x\right ) + \sqrt {a x^{2} + b} a^{3} b \mathrm {sgn}\left (x\right )}{a^{2} b x^{4}}}{8 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {a+\frac {b}{x^2}}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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