Optimal. Leaf size=38 \[ \frac {2 B \sqrt {a+b x}}{b^2}-\frac {2 (A b-a B)}{b^2 \sqrt {a+b x}} \]
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Rubi [A] time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {43} \[ \frac {2 B \sqrt {a+b x}}{b^2}-\frac {2 (A b-a B)}{b^2 \sqrt {a+b x}} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {A+B x}{(a+b x)^{3/2}} \, dx &=\int \left (\frac {A b-a B}{b (a+b x)^{3/2}}+\frac {B}{b \sqrt {a+b x}}\right ) \, dx\\ &=-\frac {2 (A b-a B)}{b^2 \sqrt {a+b x}}+\frac {2 B \sqrt {a+b x}}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 27, normalized size = 0.71 \[ \frac {2 (2 a B-A b+b B x)}{b^2 \sqrt {a+b x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 35, normalized size = 0.92 \[ \frac {2 \, {\left (B b x + 2 \, B a - A b\right )} \sqrt {b x + a}}{b^{3} x + a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.26, size = 34, normalized size = 0.89 \[ \frac {2 \, \sqrt {b x + a} B}{b^{2}} + \frac {2 \, {\left (B a - A b\right )}}{\sqrt {b x + a} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 26, normalized size = 0.68 \[ -\frac {2 \left (-B b x +A b -2 B a \right )}{\sqrt {b x +a}\, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 37, normalized size = 0.97 \[ \frac {2 \, {\left (\frac {\sqrt {b x + a} B}{b} + \frac {B a - A b}{\sqrt {b x + a} b}\right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.37, size = 25, normalized size = 0.66 \[ \frac {4\,B\,a-2\,A\,b+2\,B\,b\,x}{b^2\,\sqrt {a+b\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.62, size = 60, normalized size = 1.58 \[ \begin {cases} - \frac {2 A}{b \sqrt {a + b x}} + \frac {4 B a}{b^{2} \sqrt {a + b x}} + \frac {2 B x}{b \sqrt {a + b x}} & \text {for}\: b \neq 0 \\\frac {A x + \frac {B x^{2}}{2}}{a^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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