3.93 \(\int \sqrt{e^{a+b x}} x \, dx\)

Optimal. Leaf size=34 \[ \frac{2 x \sqrt{e^{a+b x}}}{b}-\frac{4 \sqrt{e^{a+b x}}}{b^2} \]

[Out]

(-4*Sqrt[E^(a + b*x)])/b^2 + (2*Sqrt[E^(a + b*x)]*x)/b

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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]

(-4*Sqrt[E^(a + b*x)])/b^2 + (2*Sqrt[E^(a + b*x)]*x)/b

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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]

2*x*sqrt(exp(a + b*x))/b - 4*sqrt(exp(a + b*x))/b**2

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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]

(2*Sqrt[E^(a + b*x)]*(-2 + b*x))/b^2

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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]

2*(b*x-2)*exp(b*x+a)^(1/2)/b^2

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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]

2*(b*x*e^(1/2*a) - 2*e^(1/2*a))*e^(1/2*b*x)/b^2

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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]

2*(b*x - 2)*e^(1/2*b*x + 1/2*a)/b^2

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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]

Piecewise(((2*b*x - 4)*sqrt(exp(a + b*x))/b**2, Ne(b**2, 0)), (x**2/2, True))

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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]

2*(b*x - 2)*e^(1/2*b*x + 1/2*a)/b^2