Optimal. Leaf size=39 \[ \frac{(A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}+\frac{B x}{b} \]
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Rubi [A] time = 0.0168032, antiderivative size = 39, 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} \[ \frac{(A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}+\frac{B x}{b} \]
Antiderivative was successfully verified.
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Rule 388
Rule 205
Rubi steps
\begin{align*} \int \frac{A+B x^2}{a+b x^2} \, dx &=\frac{B x}{b}-\frac{(-A b+a B) \int \frac{1}{a+b x^2} \, dx}{b}\\ &=\frac{B x}{b}+\frac{(A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0250127, size = 40, normalized size = 1.03 \[ \frac{B x}{b}-\frac{(a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 45, normalized size = 1.2 \begin{align*}{\frac{Bx}{b}}+{A\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{Ba}{b}\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 = 1.20962, size = 223, normalized size = 5.72 \begin{align*} \left [\frac{2 \, B a b x +{\left (B a - A b\right )} \sqrt{-a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right )}{2 \, a b^{2}}, \frac{B a b x -{\left (B a - A b\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{a b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.400456, size = 82, normalized size = 2.1 \begin{align*} \frac{B x}{b} + \frac{\sqrt{- \frac{1}{a b^{3}}} \left (- A b + B a\right ) \log{\left (- a b \sqrt{- \frac{1}{a b^{3}}} + x \right )}}{2} - \frac{\sqrt{- \frac{1}{a b^{3}}} \left (- A b + B a\right ) \log{\left (a b \sqrt{- \frac{1}{a b^{3}}} + x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19269, size = 46, normalized size = 1.18 \begin{align*} \frac{B x}{b} - \frac{{\left (B a - A b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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