Optimal. Leaf size=47 \[ -\frac{x (2 A b-a B)}{a^2 \sqrt{a+b x^2}}-\frac{A}{a x \sqrt{a+b x^2}} \]
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Rubi [A] time = 0.0188691, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {453, 191} \[ -\frac{x (2 A b-a B)}{a^2 \sqrt{a+b x^2}}-\frac{A}{a x \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 453
Rule 191
Rubi steps
\begin{align*} \int \frac{A+B x^2}{x^2 \left (a+b x^2\right )^{3/2}} \, dx &=-\frac{A}{a x \sqrt{a+b x^2}}-\frac{(2 A b-a B) \int \frac{1}{\left (a+b x^2\right )^{3/2}} \, dx}{a}\\ &=-\frac{A}{a x \sqrt{a+b x^2}}-\frac{(2 A b-a B) x}{a^2 \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [A] time = 0.0116595, size = 36, normalized size = 0.77 \[ \frac{-a A+a B x^2-2 A b x^2}{a^2 x \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 36, normalized size = 0.8 \begin{align*} -{\frac{2\,Ab{x}^{2}-Ba{x}^{2}+Aa}{x{a}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \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 = 1.49501, size = 85, normalized size = 1.81 \begin{align*} \frac{{\left ({\left (B a - 2 \, A b\right )} x^{2} - A a\right )} \sqrt{b x^{2} + a}}{a^{2} b x^{3} + a^{3} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.80439, size = 68, normalized size = 1.45 \begin{align*} A \left (- \frac{1}{a \sqrt{b} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{2 \sqrt{b}}{a^{2} \sqrt{\frac{a}{b x^{2}} + 1}}\right ) + \frac{B x}{a^{\frac{3}{2}} \sqrt{1 + \frac{b x^{2}}{a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12052, size = 77, normalized size = 1.64 \begin{align*} \frac{2 \, A \sqrt{b}}{{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )} a} + \frac{{\left (B a - A b\right )} x}{\sqrt{b x^{2} + a} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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