Optimal. Leaf size=53 \[ \frac{\sqrt{a+b x^2} (2 A b-3 a B)}{3 a^2 x}-\frac{A \sqrt{a+b x^2}}{3 a x^3} \]
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Rubi [A] time = 0.0197902, antiderivative size = 53, 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, 264} \[ \frac{\sqrt{a+b x^2} (2 A b-3 a B)}{3 a^2 x}-\frac{A \sqrt{a+b x^2}}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 453
Rule 264
Rubi steps
\begin{align*} \int \frac{A+B x^2}{x^4 \sqrt{a+b x^2}} \, dx &=-\frac{A \sqrt{a+b x^2}}{3 a x^3}-\frac{(2 A b-3 a B) \int \frac{1}{x^2 \sqrt{a+b x^2}} \, dx}{3 a}\\ &=-\frac{A \sqrt{a+b x^2}}{3 a x^3}+\frac{(2 A b-3 a B) \sqrt{a+b x^2}}{3 a^2 x}\\ \end{align*}
Mathematica [A] time = 0.0126149, size = 39, normalized size = 0.74 \[ -\frac{\sqrt{a+b x^2} \left (a \left (A+3 B x^2\right )-2 A b x^2\right )}{3 a^2 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 36, normalized size = 0.7 \begin{align*} -{\frac{-2\,Ab{x}^{2}+3\,Ba{x}^{2}+Aa}{3\,{x}^{3}{a}^{2}}\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 = 1.61191, size = 81, normalized size = 1.53 \begin{align*} -\frac{{\left ({\left (3 \, B a - 2 \, A b\right )} x^{2} + A a\right )} \sqrt{b x^{2} + a}}{3 \, a^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.68327, size = 70, normalized size = 1.32 \begin{align*} - \frac{A \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a x^{2}} + \frac{2 A b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{2}} - \frac{B \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13835, size = 162, normalized size = 3.06 \begin{align*} \frac{2 \,{\left (3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B \sqrt{b} - 6 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a \sqrt{b} + 6 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A b^{\frac{3}{2}} + 3 \, B a^{2} \sqrt{b} - 2 \, A a b^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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