Optimal. Leaf size=47 \[ \frac{B x}{3 a b \sqrt{a+b x^2}}-\frac{A+B x}{3 b \left (a+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0144857, antiderivative size = 47, 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 = {778, 191} \[ \frac{B x}{3 a b \sqrt{a+b x^2}}-\frac{A+B x}{3 b \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 778
Rule 191
Rubi steps
\begin{align*} \int \frac{x (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx &=-\frac{A+B x}{3 b \left (a+b x^2\right )^{3/2}}+\frac{B \int \frac{1}{\left (a+b x^2\right )^{3/2}} \, dx}{3 b}\\ &=-\frac{A+B x}{3 b \left (a+b x^2\right )^{3/2}}+\frac{B x}{3 a b \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [A] time = 0.0171072, size = 32, normalized size = 0.68 \[ \frac{b B x^3-a A}{3 a b \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 29, normalized size = 0.6 \begin{align*} -{\frac{-bB{x}^{3}+Aa}{3\,ab} \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 = 0.991677, size = 69, normalized size = 1.47 \begin{align*} -\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} - \frac{A}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54832, size = 99, normalized size = 2.11 \begin{align*} \frac{{\left (B b x^{3} - A a\right )} \sqrt{b x^{2} + a}}{3 \,{\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.03744, size = 95, normalized size = 2.02 \begin{align*} A \left (\begin{cases} - \frac{1}{3 a b \sqrt{a + b x^{2}} + 3 b^{2} x^{2} \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{\frac{5}{2}}} & \text{otherwise} \end{cases}\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 = 1.16797, size = 35, normalized size = 0.74 \begin{align*} \frac{\frac{B x^{3}}{a} - \frac{A}{b}}{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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