Optimal. Leaf size=47 \[ \frac{2 A x}{3 a^2 \sqrt{a+b x^2}}+\frac{x \left (A+B x^2\right )}{3 a \left (a+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0098935, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {378, 191} \[ \frac{2 A x}{3 a^2 \sqrt{a+b x^2}}+\frac{x \left (A+B x^2\right )}{3 a \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 378
Rule 191
Rubi steps
\begin{align*} \int \frac{A+B x^2}{\left (a+b x^2\right )^{5/2}} \, dx &=\frac{x \left (A+B x^2\right )}{3 a \left (a+b x^2\right )^{3/2}}+\frac{(2 A) \int \frac{1}{\left (a+b x^2\right )^{3/2}} \, dx}{3 a}\\ &=\frac{2 A x}{3 a^2 \sqrt{a+b x^2}}+\frac{x \left (A+B x^2\right )}{3 a \left (a+b x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.013434, size = 37, normalized size = 0.79 \[ \frac{x \left (3 a A+a B x^2+2 A b x^2\right )}{3 a^2 \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 34, normalized size = 0.7 \begin{align*}{\frac{x \left ( 2\,Ab{x}^{2}+Ba{x}^{2}+3\,aA \right ) }{3\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.24358, size = 92, normalized size = 1.96 \begin{align*} \frac{2 \, A x}{3 \, \sqrt{b x^{2} + a} a^{2}} + \frac{A x}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a} - \frac{B x}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b} + \frac{B x}{3 \, \sqrt{b x^{2} + a} a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54768, size = 115, normalized size = 2.45 \begin{align*} \frac{{\left ({\left (B a + 2 \, A b\right )} x^{3} + 3 \, A a x\right )} \sqrt{b x^{2} + a}}{3 \,{\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 10.2009, size = 144, normalized size = 3.06 \begin{align*} A \left (\frac{3 a x}{3 a^{\frac{7}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 3 a^{\frac{5}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{2 b x^{3}}{3 a^{\frac{7}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 3 a^{\frac{5}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}}}\right ) + \frac{B x^{3}}{3 a^{\frac{5}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 3 a^{\frac{3}{2}} b x^{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 = 1.14196, size = 54, normalized size = 1.15 \begin{align*} \frac{x{\left (\frac{3 \, A}{a} + \frac{{\left (B a b + 2 \, A b^{2}\right )} x^{2}}{a^{2} b}\right )}}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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