Optimal. Leaf size=41 \[ -\frac {1}{a \sqrt {a+\frac {b}{x^2}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{a^{3/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {272, 53, 65,
214} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a \sqrt {a+\frac {b}{x^2}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 65
Rule 214
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x} \, dx &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\frac {1}{a \sqrt {a+\frac {b}{x^2}}}-\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^2}\right )}{2 a}\\ &=-\frac {1}{a \sqrt {a+\frac {b}{x^2}}}-\frac {\text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^2}}\right )}{a b}\\ &=-\frac {1}{a \sqrt {a+\frac {b}{x^2}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{a^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 63, normalized size = 1.54 \begin {gather*} \frac {-\sqrt {a} x-\sqrt {b+a x^2} \log \left (-\sqrt {a} x+\sqrt {b+a x^2}\right )}{a^{3/2} \sqrt {a+\frac {b}{x^2}} x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 63, normalized size = 1.54
method | result | size |
default | \(-\frac {\left (a \,x^{2}+b \right ) \left (x \,a^{\frac {3}{2}}-\ln \left (x \sqrt {a}+\sqrt {a \,x^{2}+b}\right ) a \sqrt {a \,x^{2}+b}\right )}{\left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}} x^{3} a^{\frac {5}{2}}}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 52, normalized size = 1.27 \begin {gather*} -\frac {\log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{2}}} + \sqrt {a}}\right )}{2 \, a^{\frac {3}{2}}} - \frac {1}{\sqrt {a + \frac {b}{x^{2}}} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 81 vs.
\(2 (33) = 66\).
time = 0.38, size = 163, normalized size = 3.98 \begin {gather*} \left [-\frac {2 \, a x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} - {\left (a x^{2} + b\right )} \sqrt {a} \log \left (-2 \, a x^{2} - 2 \, \sqrt {a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} - b\right )}{2 \, {\left (a^{3} x^{2} + a^{2} b\right )}}, -\frac {a x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} + {\left (a x^{2} + b\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right )}{a^{3} x^{2} + a^{2} b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 187 vs.
\(2 (34) = 68\).
time = 0.89, size = 187, normalized size = 4.56 \begin {gather*} - \frac {2 a^{3} x^{2} \sqrt {1 + \frac {b}{a x^{2}}}}{2 a^{\frac {9}{2}} x^{2} + 2 a^{\frac {7}{2}} b} - \frac {a^{3} x^{2} \log {\left (\frac {b}{a x^{2}} \right )}}{2 a^{\frac {9}{2}} x^{2} + 2 a^{\frac {7}{2}} b} + \frac {2 a^{3} x^{2} \log {\left (\sqrt {1 + \frac {b}{a x^{2}}} + 1 \right )}}{2 a^{\frac {9}{2}} x^{2} + 2 a^{\frac {7}{2}} b} - \frac {a^{2} b \log {\left (\frac {b}{a x^{2}} \right )}}{2 a^{\frac {9}{2}} x^{2} + 2 a^{\frac {7}{2}} b} + \frac {2 a^{2} b \log {\left (\sqrt {1 + \frac {b}{a x^{2}}} + 1 \right )}}{2 a^{\frac {9}{2}} x^{2} + 2 a^{\frac {7}{2}} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.38, size = 57, normalized size = 1.39 \begin {gather*} \frac {\log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (x\right )}{2 \, a^{\frac {3}{2}}} - \frac {x}{\sqrt {a x^{2} + b} a \mathrm {sgn}\left (x\right )} - \frac {\log \left ({\left | -\sqrt {a} x + \sqrt {a x^{2} + b} \right |}\right )}{a^{\frac {3}{2}} \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.32, size = 33, normalized size = 0.80 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a\,\sqrt {a+\frac {b}{x^2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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