Optimal. Leaf size=40 \[ \frac {2 \sqrt {a+b x} (A b-a B)}{b^2}+\frac {2 B (a+b x)^{3/2}}{3 b^2} \]
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Rubi [A] time = 0.01, antiderivative size = 40, 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} \[ \frac {2 \sqrt {a+b x} (A b-a B)}{b^2}+\frac {2 B (a+b x)^{3/2}}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {A+B x}{\sqrt {a+b x}} \, dx &=\int \left (\frac {A b-a B}{b \sqrt {a+b x}}+\frac {B \sqrt {a+b x}}{b}\right ) \, dx\\ &=\frac {2 (A b-a B) \sqrt {a+b x}}{b^2}+\frac {2 B (a+b x)^{3/2}}{3 b^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 29, normalized size = 0.72 \[ \frac {2 \sqrt {a+b x} (-2 a B+3 A b+b B x)}{3 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 25, normalized size = 0.62 \[ \frac {2 \, {\left (B b x - 2 \, B a + 3 \, A b\right )} \sqrt {b x + a}}{3 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.23, size = 39, normalized size = 0.98 \[ \frac {2 \, {\left (3 \, \sqrt {b x + a} A + \frac {{\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} B}{b}\right )}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 26, normalized size = 0.65 \[ \frac {2 \sqrt {b x +a}\, \left (B b x +3 A b -2 B a \right )}{3 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.86, size = 39, normalized size = 0.98 \[ \frac {2 \, {\left (3 \, \sqrt {b x + a} A + \frac {{\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} B}{b}\right )}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 28, normalized size = 0.70 \[ \frac {2\,\sqrt {a+b\,x}\,\left (3\,A\,b-3\,B\,a+B\,\left (a+b\,x\right )\right )}{3\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.62, size = 121, normalized size = 3.02 \[ \begin {cases} \frac {- \frac {2 A a}{\sqrt {a + b x}} - 2 A \left (- \frac {a}{\sqrt {a + b x}} - \sqrt {a + b x}\right ) - \frac {2 B a \left (- \frac {a}{\sqrt {a + b x}} - \sqrt {a + b x}\right )}{b} - \frac {2 B \left (\frac {a^{2}}{\sqrt {a + b x}} + 2 a \sqrt {a + b x} - \frac {\left (a + b x\right )^{\frac {3}{2}}}{3}\right )}{b}}{b} & \text {for}\: b \neq 0 \\\frac {A x + \frac {B x^{2}}{2}}{\sqrt {a}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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