Optimal. Leaf size=34 \[ \frac{2 x \sqrt{e^{a+b x}}}{b}-\frac{4 \sqrt{e^{a+b x}}}{b^2} \]
[Out]
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Rubi [A] time = 0.0434838, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 x \sqrt{e^{a+b x}}}{b}-\frac{4 \sqrt{e^{a+b x}}}{b^2} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[E^(a + b*x)]*x,x]
[Out]
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Rubi in Sympy [A] time = 3.05383, size = 29, normalized size = 0.85 \[ \frac{2 x \sqrt{e^{a + b x}}}{b} - \frac{4 \sqrt{e^{a + b x}}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*exp(b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.00642014, size = 21, normalized size = 0.62 \[ \frac{2 (b x-2) \sqrt{e^{a+b x}}}{b^2} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[E^(a + b*x)]*x,x]
[Out]
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Maple [A] time = 0.003, size = 19, normalized size = 0.6 \[ 2\,{\frac{ \left ( bx-2 \right ) \sqrt{{{\rm e}^{bx+a}}}}{{b}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*exp(b*x+a)^(1/2),x)
[Out]
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Maxima [A] time = 0.771949, size = 32, normalized size = 0.94 \[ \frac{2 \,{\left (b x e^{\left (\frac{1}{2} \, a\right )} - 2 \, e^{\left (\frac{1}{2} \, a\right )}\right )} e^{\left (\frac{1}{2} \, b x\right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*e^(1/2*b*x + 1/2*a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.245078, size = 26, normalized size = 0.76 \[ \frac{2 \,{\left (b x - 2\right )} e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*e^(1/2*b*x + 1/2*a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.17606, size = 26, normalized size = 0.76 \[ \begin{cases} \frac{\left (2 b x - 4\right ) \sqrt{e^{a + b x}}}{b^{2}} & \text{for}\: b^{2} \neq 0 \\\frac{x^{2}}{2} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*exp(b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.251503, size = 26, normalized size = 0.76 \[ \frac{2 \,{\left (b x - 2\right )} e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*e^(1/2*b*x + 1/2*a),x, algorithm="giac")
[Out]