Optimal. Leaf size=34 \[ -\frac {4 \sqrt {e^{a+b x}}}{b^2}+\frac {2 \sqrt {e^{a+b x}} x}{b} \]
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Rubi [A]
time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2207, 2225}
\begin {gather*} \frac {2 x \sqrt {e^{a+b x}}}{b}-\frac {4 \sqrt {e^{a+b x}}}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2207
Rule 2225
Rubi steps
\begin {align*} \int \sqrt {e^{a+b x}} x \, dx &=\frac {2 \sqrt {e^{a+b x}} x}{b}-\frac {2 \int \sqrt {e^{a+b x}} \, dx}{b}\\ &=-\frac {4 \sqrt {e^{a+b x}}}{b^2}+\frac {2 \sqrt {e^{a+b x}} x}{b}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 21, normalized size = 0.62 \begin {gather*} \frac {2 \sqrt {e^{a+b x}} (-2+b x)}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 19, normalized size = 0.56
method | result | size |
gosper | \(\frac {2 \left (b x -2\right ) \sqrt {{\mathrm e}^{b x +a}}}{b^{2}}\) | \(19\) |
risch | \(\frac {2 \left (b x -2\right ) \sqrt {{\mathrm e}^{b x +a}}}{b^{2}}\) | \(19\) |
meijerg | \(\frac {4 \sqrt {{\mathrm e}^{b x +a}}\, {\mathrm e}^{-a -\frac {b x \,{\mathrm e}^{\frac {a}{2}}}{2}} \left (1-\frac {\left (-b x \,{\mathrm e}^{\frac {a}{2}}+2\right ) {\mathrm e}^{\frac {b x \,{\mathrm e}^{\frac {a}{2}}}{2}}}{2}\right )}{b^{2}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 24, normalized size = 0.71 \begin {gather*} \frac {2 \, {\left (b x e^{\left (\frac {1}{2} \, a\right )} - 2 \, e^{\left (\frac {1}{2} \, a\right )}\right )} e^{\left (\frac {1}{2} \, b x\right )}}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 19, normalized size = 0.56 \begin {gather*} \frac {2 \, {\left (b x - 2\right )} e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )}}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 26, normalized size = 0.76 \begin {gather*} \begin {cases} \frac {\left (2 b x - 4\right ) \sqrt {e^{a + b x}}}{b^{2}} & \text {for}\: b^{2} \neq 0 \\\frac {x^{2}}{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.99, size = 19, normalized size = 0.56 \begin {gather*} \frac {2 \, {\left (b x - 2\right )} e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )}}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 18, normalized size = 0.53 \begin {gather*} \frac {2\,\sqrt {{\mathrm {e}}^{a+b\,x}}\,\left (b\,x-2\right )}{b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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