Optimal. Leaf size=51 \[ \frac{2 A x}{3 a^2 \sqrt{a+b x^2}}-\frac{a B-A b x}{3 a b \left (a+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0104064, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {639, 191} \[ \frac{2 A x}{3 a^2 \sqrt{a+b x^2}}-\frac{a B-A b x}{3 a b \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 639
Rule 191
Rubi steps
\begin{align*} \int \frac{A+B x}{\left (a+b x^2\right )^{5/2}} \, dx &=-\frac{a B-A b x}{3 a b \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{a B-A b x}{3 a b \left (a+b x^2\right )^{3/2}}+\frac{2 A x}{3 a^2 \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [A] time = 0.0199786, size = 43, normalized size = 0.84 \[ \frac{-a^2 B+3 a A b x+2 A b^2 x^3}{3 a^2 b \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 40, normalized size = 0.8 \begin{align*}{\frac{2\,A{b}^{2}{x}^{3}+3\,Axab-B{a}^{2}}{3\,{a}^{2}b} \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 = 1.0094, size = 65, normalized size = 1.27 \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}{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 = 1.5894, size = 126, normalized size = 2.47 \begin{align*} \frac{{\left (2 \, A b^{2} x^{3} + 3 \, A a b x - B a^{2}\right )} \sqrt{b x^{2} + a}}{3 \,{\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 8.9493, size = 146, normalized size = 2.86 \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 ) + B \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22283, size = 50, normalized size = 0.98 \begin{align*} \frac{{\left (\frac{2 \, A b x^{2}}{a^{2}} + \frac{3 \, A}{a}\right )} x - \frac{B}{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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