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 = 0.0285417, 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, Rules used = {2176, 2194} \[ \frac{2 x \sqrt{e^{a+b x}}}{b}-\frac{4 \sqrt{e^{a+b x}}}{b^2} \]

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

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

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*}

Mathematica [A]  time = 0.0101217, 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 = 0.001, size = 19, normalized size = 0.6 \begin{align*} 2\,{\frac{ \left ( bx-2 \right ) \sqrt{{{\rm e}^{bx+a}}}}{{b}^{2}}} \end{align*}

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 = 1.0455, size = 32, normalized size = 0.94 \begin{align*} \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*exp(b*x+a)^(1/2),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 = 1.5013, size = 50, normalized size = 1.47 \begin{align*} \frac{2 \,{\left (b x - 2\right )} e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*exp(b*x+a)^(1/2),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 = 0.094429, size = 26, normalized size = 0.76 \begin{align*} \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{align*}

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 [A]  time = 1.2432, size = 26, normalized size = 0.76 \begin{align*} \frac{2 \,{\left (b x - 2\right )} e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*exp(b*x+a)^(1/2),x, algorithm="giac")

[Out]

2*(b*x - 2)*e^(1/2*b*x + 1/2*a)/b^2