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 = 0.0165593, antiderivative size = 50, normalized size of antiderivative = 1., 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 325
Rule 205
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*}
Mathematica [A] time = 0.0163537, 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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Maple [A] time = 0.006, size = 36, normalized size = 0.7 \begin{align*} -{\frac{1}{a} \left ( b\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ) x+\sqrt{ab} \right ){\frac{1}{\sqrt{d{x}^{2}}}}{\frac{1}{\sqrt{ab}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28342, size = 273, normalized size = 5.46 \begin{align*} \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 ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{d x^{2}} \left (a + b x^{2}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09595, size = 65, normalized size = 1.3 \begin{align*} -d{\left (\frac{b \arctan \left (\frac{\sqrt{d x^{2}} b}{\sqrt{a b d}}\right )}{\sqrt{a b d} a d} + \frac{1}{\sqrt{d x^{2}} a d}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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