Optimal. Leaf size=28 \[ -\frac {a B-A b x}{a b \sqrt {a+b x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {637} \[ -\frac {a B-A b x}{a b \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 637
Rubi steps
\begin {align*} \int \frac {A+B x}{\left (a+b x^2\right )^{3/2}} \, dx &=-\frac {a B-A b x}{a b \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 27, normalized size = 0.96 \[ \frac {A b x-a B}{a b \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 35, normalized size = 1.25 \[ \frac {{\left (A b x - B a\right )} \sqrt {b x^{2} + a}}{a b^{2} x^{2} + a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 23, normalized size = 0.82 \[ \frac {\frac {A x}{a} - \frac {B}{b}}{\sqrt {b x^{2} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 26, normalized size = 0.93 \[ \frac {A b x -B a}{\sqrt {b \,x^{2}+a}\, a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 31, normalized size = 1.11 \[ \frac {A x}{\sqrt {b x^{2} + a} a} - \frac {B}{\sqrt {b x^{2} + a} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.91, size = 24, normalized size = 0.86 \[ -\frac {\frac {B}{b}-\frac {A\,x}{a}}{\sqrt {b\,x^2+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.22, size = 46, normalized size = 1.64 \[ \frac {A x}{a^{\frac {3}{2}} \sqrt {1 + \frac {b x^{2}}{a}}} + B \left (\begin {cases} - \frac {1}{b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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