Optimal. Leaf size=39 \[ \frac{(a B+A b) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}-\frac{B x}{b} \]
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Rubi [A] time = 0.018227, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {388, 208} \[ \frac{(a B+A b) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}-\frac{B x}{b} \]
Antiderivative was successfully verified.
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Rule 388
Rule 208
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) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0214197, size = 39, normalized size = 1. \[ \frac{(a B+A b) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}-\frac{B x}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 37, normalized size = 1. \begin{align*} -{\frac{Bx}{b}}-{\frac{-Ab-Ba}{b}{\it Artanh} \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 = 1.20187, size = 225, normalized size = 5.77 \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 = 0.408331, size = 75, normalized size = 1.92 \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 = 1.17008, size = 49, normalized size = 1.26 \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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