Optimal. Leaf size=28 \[ -\frac {a B-A c x}{a c \sqrt {a+c x^2}} \]
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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} \[ -\frac {a B-A c x}{a c \sqrt {a+c x^2}} \]
Antiderivative was successfully verified.
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Rule 637
Rubi steps
\begin {align*} \int \frac {A+B x}{\left (a+c x^2\right )^{3/2}} \, dx &=-\frac {a B-A c x}{a c \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.96 \[ \frac {A c x-a B}{a c \sqrt {a+c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 35, normalized size = 1.25 \[ \frac {{\left (A c x - B a\right )} \sqrt {c x^{2} + a}}{a c^{2} x^{2} + a^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 23, normalized size = 0.82 \[ \frac {\frac {A x}{a} - \frac {B}{c}}{\sqrt {c x^{2} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 26, normalized size = 0.93 \[ \frac {A c x -B a}{\sqrt {c \,x^{2}+a}\, a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 31, normalized size = 1.11 \[ \frac {A x}{\sqrt {c x^{2} + a} a} - \frac {B}{\sqrt {c x^{2} + a} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 24, normalized size = 0.86 \[ -\frac {\frac {B}{c}-\frac {A\,x}{a}}{\sqrt {c\,x^2+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.23, size = 46, normalized size = 1.64 \[ \frac {A x}{a^{\frac {3}{2}} \sqrt {1 + \frac {c x^{2}}{a}}} + B \left (\begin {cases} - \frac {1}{c \sqrt {a + c x^{2}}} & \text {for}\: c \neq 0 \\\frac {x^{2}}{2 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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