Optimal. Leaf size=40 \[ \frac {2 b (c+d x)^{3/2}}{3 d^2}-\frac {2 \sqrt {c+d x} (b c-a d)}{d^2} \]
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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} \[ \frac {2 b (c+d x)^{3/2}}{3 d^2}-\frac {2 \sqrt {c+d x} (b c-a d)}{d^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {a+b x}{\sqrt {c+d x}} \, dx &=\int \left (\frac {-b c+a d}{d \sqrt {c+d x}}+\frac {b \sqrt {c+d x}}{d}\right ) \, dx\\ &=-\frac {2 (b c-a d) \sqrt {c+d x}}{d^2}+\frac {2 b (c+d x)^{3/2}}{3 d^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 29, normalized size = 0.72 \[ \frac {2 \sqrt {c+d x} (3 a d-2 b c+b d x)}{3 d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 25, normalized size = 0.62 \[ \frac {2 \, {\left (b d x - 2 \, b c + 3 \, a d\right )} \sqrt {d x + c}}{3 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.88, size = 39, normalized size = 0.98 \[ \frac {2 \, {\left (3 \, \sqrt {d x + c} a + \frac {{\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} b}{d}\right )}}{3 \, d} \]
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 \[ \frac {2 \sqrt {d x +c}\, \left (b d x +3 a d -2 b c \right )}{3 d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 39, normalized size = 0.98 \[ \frac {2 \, {\left (3 \, \sqrt {d x + c} a + \frac {{\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} b}{d}\right )}}{3 \, d} \]
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 \[ \frac {2\,\sqrt {c+d\,x}\,\left (3\,a\,d-3\,b\,c+b\,\left (c+d\,x\right )\right )}{3\,d^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.78, size = 121, normalized size = 3.02 \[ \begin {cases} \frac {- \frac {2 a c}{\sqrt {c + d x}} - 2 a \left (- \frac {c}{\sqrt {c + d x}} - \sqrt {c + d x}\right ) - \frac {2 b c \left (- \frac {c}{\sqrt {c + d x}} - \sqrt {c + d x}\right )}{d} - \frac {2 b \left (\frac {c^{2}}{\sqrt {c + d x}} + 2 c \sqrt {c + d x} - \frac {\left (c + d x\right )^{\frac {3}{2}}}{3}\right )}{d}}{d} & \text {for}\: d \neq 0 \\\frac {a x + \frac {b x^{2}}{2}}{\sqrt {c}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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