Optimal. Leaf size=50 \[ -\frac {\sqrt {b} x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {d x^2}}-\frac {1}{a \sqrt {d x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {15, 325, 205} \[ -\frac {\sqrt {b} x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {d x^2}}-\frac {1}{a \sqrt {d x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 205
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {d x^2} \left (a+b x^2\right )} \, dx &=\frac {x \int \frac {1}{x^2 \left (a+b x^2\right )} \, dx}{\sqrt {d x^2}}\\ &=-\frac {1}{a \sqrt {d x^2}}-\frac {(b x) \int \frac {1}{a+b x^2} \, dx}{a \sqrt {d x^2}}\\ &=-\frac {1}{a \sqrt {d x^2}}-\frac {\sqrt {b} x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 46, normalized size = 0.92 \[ -\frac {d x^2 \left (\sqrt {b} x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )+\sqrt {a}\right )}{a^{3/2} \left (d x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 132, normalized size = 2.64 \[ \left [\frac {d x^{2} \sqrt {-\frac {b}{a d}} \log \left (\frac {b x^{2} - 2 \, \sqrt {d x^{2}} a \sqrt {-\frac {b}{a d}} - a}{b x^{2} + a}\right ) - 2 \, \sqrt {d x^{2}}}{2 \, a d x^{2}}, -\frac {d x^{2} \sqrt {\frac {b}{a d}} \arctan \left (\sqrt {d x^{2}} \sqrt {\frac {b}{a d}}\right ) + \sqrt {d x^{2}}}{a d x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 41, normalized size = 0.82 \[ -\frac {b \arctan \left (\frac {\sqrt {d x^{2}} b}{\sqrt {a b d}}\right )}{\sqrt {a b d} a} - \frac {1}{\sqrt {d x^{2}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 36, normalized size = 0.72 \[ -\frac {b x \arctan \left (\frac {b x}{\sqrt {a b}}\right )+\sqrt {a b}}{\sqrt {d \,x^{2}}\, \sqrt {a b}\, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.99, size = 35, normalized size = 0.70 \[ -\frac {b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a \sqrt {d}} - \frac {1}{a \sqrt {d} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.62, size = 38, normalized size = 0.76 \[ -\frac {1}{a\,\sqrt {d}\,\sqrt {x^2}}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x^2}}{\sqrt {a}}\right )}{a^{3/2}\,\sqrt {d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {d x^{2}} \left (a + b x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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