3.91 \(\int \sqrt{e^{a+b x}} x^3 \, dx\)

Optimal. Leaf size=72 \[ -\frac{96 \sqrt{e^{a+b x}}}{b^4}+\frac{48 x \sqrt{e^{a+b x}}}{b^3}-\frac{12 x^2 \sqrt{e^{a+b x}}}{b^2}+\frac{2 x^3 \sqrt{e^{a+b x}}}{b} \]

[Out]

(-96*Sqrt[E^(a + b*x)])/b^4 + (48*Sqrt[E^(a + b*x)]*x)/b^3 - (12*Sqrt[E^(a + b*x
)]*x^2)/b^2 + (2*Sqrt[E^(a + b*x)]*x^3)/b

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Rubi [A]  time = 0.152313, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{96 \sqrt{e^{a+b x}}}{b^4}+\frac{48 x \sqrt{e^{a+b x}}}{b^3}-\frac{12 x^2 \sqrt{e^{a+b x}}}{b^2}+\frac{2 x^3 \sqrt{e^{a+b x}}}{b} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[E^(a + b*x)]*x^3,x]

[Out]

(-96*Sqrt[E^(a + b*x)])/b^4 + (48*Sqrt[E^(a + b*x)]*x)/b^3 - (12*Sqrt[E^(a + b*x
)]*x^2)/b^2 + (2*Sqrt[E^(a + b*x)]*x^3)/b

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Rubi in Sympy [A]  time = 9.15162, size = 66, normalized size = 0.92 \[ \frac{2 x^{3} \sqrt{e^{a + b x}}}{b} - \frac{12 x^{2} \sqrt{e^{a + b x}}}{b^{2}} + \frac{48 x \sqrt{e^{a + b x}}}{b^{3}} - \frac{96 \sqrt{e^{a + b x}}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*exp(b*x+a)**(1/2),x)

[Out]

2*x**3*sqrt(exp(a + b*x))/b - 12*x**2*sqrt(exp(a + b*x))/b**2 + 48*x*sqrt(exp(a
+ b*x))/b**3 - 96*sqrt(exp(a + b*x))/b**4

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Mathematica [A]  time = 0.0106193, size = 37, normalized size = 0.51 \[ \frac{2 \left (b^3 x^3-6 b^2 x^2+24 b x-48\right ) \sqrt{e^{a+b x}}}{b^4} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[E^(a + b*x)]*x^3,x]

[Out]

(2*Sqrt[E^(a + b*x)]*(-48 + 24*b*x - 6*b^2*x^2 + b^3*x^3))/b^4

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Maple [A]  time = 0.006, size = 35, normalized size = 0.5 \[ 2\,{\frac{ \left ({x}^{3}{b}^{3}-6\,{x}^{2}{b}^{2}+24\,bx-48 \right ) \sqrt{{{\rm e}^{bx+a}}}}{{b}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*exp(b*x+a)^(1/2),x)

[Out]

2*(b^3*x^3-6*b^2*x^2+24*b*x-48)*exp(b*x+a)^(1/2)/b^4

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Maxima [A]  time = 0.835351, size = 65, normalized size = 0.9 \[ \frac{2 \,{\left (b^{3} x^{3} e^{\left (\frac{1}{2} \, a\right )} - 6 \, b^{2} x^{2} e^{\left (\frac{1}{2} \, a\right )} + 24 \, b x e^{\left (\frac{1}{2} \, a\right )} - 48 \, e^{\left (\frac{1}{2} \, a\right )}\right )} e^{\left (\frac{1}{2} \, b x\right )}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*e^(1/2*b*x + 1/2*a),x, algorithm="maxima")

[Out]

2*(b^3*x^3*e^(1/2*a) - 6*b^2*x^2*e^(1/2*a) + 24*b*x*e^(1/2*a) - 48*e^(1/2*a))*e^
(1/2*b*x)/b^4

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Fricas [A]  time = 0.260096, size = 47, normalized size = 0.65 \[ \frac{2 \,{\left (b^{3} x^{3} - 6 \, b^{2} x^{2} + 24 \, b x - 48\right )} e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*e^(1/2*b*x + 1/2*a),x, algorithm="fricas")

[Out]

2*(b^3*x^3 - 6*b^2*x^2 + 24*b*x - 48)*e^(1/2*b*x + 1/2*a)/b^4

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Sympy [A]  time = 0.218076, size = 42, normalized size = 0.58 \[ \begin{cases} \frac{\left (2 b^{3} x^{3} - 12 b^{2} x^{2} + 48 b x - 96\right ) \sqrt{e^{a + b x}}}{b^{4}} & \text{for}\: b^{4} \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*exp(b*x+a)**(1/2),x)

[Out]

Piecewise(((2*b**3*x**3 - 12*b**2*x**2 + 48*b*x - 96)*sqrt(exp(a + b*x))/b**4, N
e(b**4, 0)), (x**4/4, True))

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GIAC/XCAS [A]  time = 0.25041, size = 47, normalized size = 0.65 \[ \frac{2 \,{\left (b^{3} x^{3} - 6 \, b^{2} x^{2} + 24 \, b x - 48\right )} e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*e^(1/2*b*x + 1/2*a),x, algorithm="giac")

[Out]

2*(b^3*x^3 - 6*b^2*x^2 + 24*b*x - 48)*e^(1/2*b*x + 1/2*a)/b^4