3.92 \(\int \sqrt{e^{a+b x}} x^2 \, dx\)

Optimal. Leaf size=53 \[ \frac{16 \sqrt{e^{a+b x}}}{b^3}-\frac{8 x \sqrt{e^{a+b x}}}{b^2}+\frac{2 x^2 \sqrt{e^{a+b x}}}{b} \]

[Out]

(16*Sqrt[E^(a + b*x)])/b^3 - (8*Sqrt[E^(a + b*x)]*x)/b^2 + (2*Sqrt[E^(a + b*x)]*
x^2)/b

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Rubi [A]  time = 0.0973366, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{16 \sqrt{e^{a+b x}}}{b^3}-\frac{8 x \sqrt{e^{a+b x}}}{b^2}+\frac{2 x^2 \sqrt{e^{a+b x}}}{b} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[E^(a + b*x)]*x^2,x]

[Out]

(16*Sqrt[E^(a + b*x)])/b^3 - (8*Sqrt[E^(a + b*x)]*x)/b^2 + (2*Sqrt[E^(a + b*x)]*
x^2)/b

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Rubi in Sympy [A]  time = 5.87901, size = 48, normalized size = 0.91 \[ \frac{2 x^{2} \sqrt{e^{a + b x}}}{b} - \frac{8 x \sqrt{e^{a + b x}}}{b^{2}} + \frac{16 \sqrt{e^{a + b x}}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*exp(b*x+a)**(1/2),x)

[Out]

2*x**2*sqrt(exp(a + b*x))/b - 8*x*sqrt(exp(a + b*x))/b**2 + 16*sqrt(exp(a + b*x)
)/b**3

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Mathematica [A]  time = 0.00913231, size = 29, normalized size = 0.55 \[ \frac{2 \left (b^2 x^2-4 b x+8\right ) \sqrt{e^{a+b x}}}{b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[E^(a + b*x)]*x^2,x]

[Out]

(2*Sqrt[E^(a + b*x)]*(8 - 4*b*x + b^2*x^2))/b^3

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Maple [A]  time = 0.006, size = 27, normalized size = 0.5 \[ 2\,{\frac{ \left ({x}^{2}{b}^{2}-4\,bx+8 \right ) \sqrt{{{\rm e}^{bx+a}}}}{{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*exp(b*x+a)^(1/2),x)

[Out]

2*(b^2*x^2-4*b*x+8)*exp(b*x+a)^(1/2)/b^3

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Maxima [A]  time = 0.777594, size = 49, normalized size = 0.92 \[ \frac{2 \,{\left (b^{2} x^{2} e^{\left (\frac{1}{2} \, a\right )} - 4 \, b x e^{\left (\frac{1}{2} \, a\right )} + 8 \, e^{\left (\frac{1}{2} \, a\right )}\right )} e^{\left (\frac{1}{2} \, b x\right )}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*e^(1/2*b*x + 1/2*a),x, algorithm="maxima")

[Out]

2*(b^2*x^2*e^(1/2*a) - 4*b*x*e^(1/2*a) + 8*e^(1/2*a))*e^(1/2*b*x)/b^3

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Fricas [A]  time = 0.248536, size = 36, normalized size = 0.68 \[ \frac{2 \,{\left (b^{2} x^{2} - 4 \, b x + 8\right )} e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*e^(1/2*b*x + 1/2*a),x, algorithm="fricas")

[Out]

2*(b^2*x^2 - 4*b*x + 8)*e^(1/2*b*x + 1/2*a)/b^3

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Sympy [A]  time = 0.200027, size = 34, normalized size = 0.64 \[ \begin{cases} \frac{\left (2 b^{2} x^{2} - 8 b x + 16\right ) \sqrt{e^{a + b x}}}{b^{3}} & \text{for}\: b^{3} \neq 0 \\\frac{x^{3}}{3} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*exp(b*x+a)**(1/2),x)

[Out]

Piecewise(((2*b**2*x**2 - 8*b*x + 16)*sqrt(exp(a + b*x))/b**3, Ne(b**3, 0)), (x*
*3/3, True))

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GIAC/XCAS [A]  time = 0.239023, size = 36, normalized size = 0.68 \[ \frac{2 \,{\left (b^{2} x^{2} - 4 \, b x + 8\right )} e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*e^(1/2*b*x + 1/2*a),x, algorithm="giac")

[Out]

2*(b^2*x^2 - 4*b*x + 8)*e^(1/2*b*x + 1/2*a)/b^3