Optimal. Leaf size=43 \[ -\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}-\frac{c}{a x} \]
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Rubi [A] time = 0.0203093, antiderivative size = 43, 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 = {453, 205} \[ -\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}-\frac{c}{a x} \]
Antiderivative was successfully verified.
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Rule 453
Rule 205
Rubi steps
\begin{align*} \int \frac{c+d x^2}{x^2 \left (a+b x^2\right )} \, dx &=-\frac{c}{a x}-\frac{(b c-a d) \int \frac{1}{a+b x^2} \, dx}{a}\\ &=-\frac{c}{a x}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0253388, size = 42, normalized size = 0.98 \[ \frac{(a d-b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}-\frac{c}{a x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 48, normalized size = 1.1 \begin{align*} -{\frac{c}{ax}}+{d\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{bc}{a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\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 = 1.50154, size = 230, normalized size = 5.35 \begin{align*} \left [\frac{\sqrt{-a b}{\left (b c - a d\right )} x \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right ) - 2 \, a b c}{2 \, a^{2} b x}, -\frac{\sqrt{a b}{\left (b c - a d\right )} x \arctan \left (\frac{\sqrt{a b} x}{a}\right ) + a b c}{a^{2} b x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.459156, size = 82, normalized size = 1.91 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{3} b}} \left (a d - b c\right ) \log{\left (- a^{2} \sqrt{- \frac{1}{a^{3} b}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{a^{3} b}} \left (a d - b c\right ) \log{\left (a^{2} \sqrt{- \frac{1}{a^{3} b}} + x \right )}}{2} - \frac{c}{a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15237, size = 50, normalized size = 1.16 \begin{align*} -\frac{{\left (b c - a d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a} - \frac{c}{a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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