Optimal. Leaf size=40 \[ \frac{b x}{d}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} d^{3/2}} \]
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Rubi [A] time = 0.0198071, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {388, 205} \[ \frac{b x}{d}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} d^{3/2}} \]
Antiderivative was successfully verified.
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Rule 388
Rule 205
Rubi steps
\begin{align*} \int \frac{a+b x^2}{c+d x^2} \, dx &=\frac{b x}{d}-\frac{(b c-a d) \int \frac{1}{c+d x^2} \, dx}{d}\\ &=\frac{b x}{d}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} d^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0225555, size = 40, normalized size = 1. \[ \frac{b x}{d}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} d^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 45, normalized size = 1.1 \begin{align*}{\frac{bx}{d}}+{a\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{bc}{d}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \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 = 1.77492, size = 223, normalized size = 5.58 \begin{align*} \left [\frac{2 \, b c d x +{\left (b c - a d\right )} \sqrt{-c d} \log \left (\frac{d x^{2} - 2 \, \sqrt{-c d} x - c}{d x^{2} + c}\right )}{2 \, c d^{2}}, \frac{b c d x -{\left (b c - a d\right )} \sqrt{c d} \arctan \left (\frac{\sqrt{c d} x}{c}\right )}{c d^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.387133, size = 82, normalized size = 2.05 \begin{align*} \frac{b x}{d} - \frac{\sqrt{- \frac{1}{c d^{3}}} \left (a d - b c\right ) \log{\left (- c d \sqrt{- \frac{1}{c d^{3}}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{c d^{3}}} \left (a d - b c\right ) \log{\left (c d \sqrt{- \frac{1}{c d^{3}}} + x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06938, size = 46, normalized size = 1.15 \begin{align*} \frac{b x}{d} - \frac{{\left (b c - a d\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{\sqrt{c d} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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