Optimal. Leaf size=34 \[ \frac{x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \sqrt{d x^2}} \]
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Rubi [A] time = 0.0082262, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 205} \[ \frac{x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \sqrt{d x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 205
Rubi steps
\begin{align*} \int \frac{x}{\sqrt{d x^2} \left (a+b x^2\right )} \, dx &=\frac{x \int \frac{1}{a+b x^2} \, dx}{\sqrt{d x^2}}\\ &=\frac{x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \sqrt{d x^2}}\\ \end{align*}
Mathematica [A] time = 0.0072771, size = 34, normalized size = 1. \[ \frac{x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \sqrt{d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 24, normalized size = 0.7 \begin{align*}{x\arctan \left ({bx{\frac{1}{\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 = 1.35109, size = 205, normalized size = 6.03 \begin{align*} \left [-\frac{\sqrt{-a b d} \log \left (\frac{b d x^{2} - a d - 2 \, \sqrt{-a b d} \sqrt{d x^{2}}}{b x^{2} + a}\right )}{2 \, a b d}, \frac{\sqrt{a b d} \arctan \left (\frac{\sqrt{a b d} \sqrt{d x^{2}}}{a d}\right )}{a b d}\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{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 = 1.12092, size = 31, normalized size = 0.91 \begin{align*} \frac{\arctan \left (\frac{\sqrt{d x^{2}} b}{\sqrt{a b d}}\right )}{\sqrt{a b d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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