3.90 \(\int \sqrt{e^{a+b x}} x^4 \, dx\)

Optimal. Leaf size=91 \[ \frac{768 \sqrt{e^{a+b x}}}{b^5}-\frac{384 x \sqrt{e^{a+b x}}}{b^4}+\frac{96 x^2 \sqrt{e^{a+b x}}}{b^3}-\frac{16 x^3 \sqrt{e^{a+b x}}}{b^2}+\frac{2 x^4 \sqrt{e^{a+b x}}}{b} \]

[Out]

(768*Sqrt[E^(a + b*x)])/b^5 - (384*Sqrt[E^(a + b*x)]*x)/b^4 + (96*Sqrt[E^(a + b*
x)]*x^2)/b^3 - (16*Sqrt[E^(a + b*x)]*x^3)/b^2 + (2*Sqrt[E^(a + b*x)]*x^4)/b

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Rubi [A]  time = 0.213147, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{768 \sqrt{e^{a+b x}}}{b^5}-\frac{384 x \sqrt{e^{a+b x}}}{b^4}+\frac{96 x^2 \sqrt{e^{a+b x}}}{b^3}-\frac{16 x^3 \sqrt{e^{a+b x}}}{b^2}+\frac{2 x^4 \sqrt{e^{a+b x}}}{b} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[E^(a + b*x)]*x^4,x]

[Out]

(768*Sqrt[E^(a + b*x)])/b^5 - (384*Sqrt[E^(a + b*x)]*x)/b^4 + (96*Sqrt[E^(a + b*
x)]*x^2)/b^3 - (16*Sqrt[E^(a + b*x)]*x^3)/b^2 + (2*Sqrt[E^(a + b*x)]*x^4)/b

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Rubi in Sympy [A]  time = 12.8547, size = 85, normalized size = 0.93 \[ \frac{2 x^{4} \sqrt{e^{a + b x}}}{b} - \frac{16 x^{3} \sqrt{e^{a + b x}}}{b^{2}} + \frac{96 x^{2} \sqrt{e^{a + b x}}}{b^{3}} - \frac{384 x \sqrt{e^{a + b x}}}{b^{4}} + \frac{768 \sqrt{e^{a + b x}}}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*exp(b*x+a)**(1/2),x)

[Out]

2*x**4*sqrt(exp(a + b*x))/b - 16*x**3*sqrt(exp(a + b*x))/b**2 + 96*x**2*sqrt(exp
(a + b*x))/b**3 - 384*x*sqrt(exp(a + b*x))/b**4 + 768*sqrt(exp(a + b*x))/b**5

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Mathematica [A]  time = 0.0143247, size = 45, normalized size = 0.49 \[ \frac{2 \left (b^4 x^4-8 b^3 x^3+48 b^2 x^2-192 b x+384\right ) \sqrt{e^{a+b x}}}{b^5} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[E^(a + b*x)]*x^4,x]

[Out]

(2*Sqrt[E^(a + b*x)]*(384 - 192*b*x + 48*b^2*x^2 - 8*b^3*x^3 + b^4*x^4))/b^5

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Maple [A]  time = 0.006, size = 43, normalized size = 0.5 \[ 2\,{\frac{ \left ({x}^{4}{b}^{4}-8\,{x}^{3}{b}^{3}+48\,{x}^{2}{b}^{2}-192\,bx+384 \right ) \sqrt{{{\rm e}^{bx+a}}}}{{b}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*exp(b*x+a)^(1/2),x)

[Out]

2*(b^4*x^4-8*b^3*x^3+48*b^2*x^2-192*b*x+384)*exp(b*x+a)^(1/2)/b^5

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Maxima [A]  time = 0.750999, size = 81, normalized size = 0.89 \[ \frac{2 \,{\left (b^{4} x^{4} e^{\left (\frac{1}{2} \, a\right )} - 8 \, b^{3} x^{3} e^{\left (\frac{1}{2} \, a\right )} + 48 \, b^{2} x^{2} e^{\left (\frac{1}{2} \, a\right )} - 192 \, b x e^{\left (\frac{1}{2} \, a\right )} + 384 \, e^{\left (\frac{1}{2} \, a\right )}\right )} e^{\left (\frac{1}{2} \, b x\right )}}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4*e^(1/2*b*x + 1/2*a),x, algorithm="maxima")

[Out]

2*(b^4*x^4*e^(1/2*a) - 8*b^3*x^3*e^(1/2*a) + 48*b^2*x^2*e^(1/2*a) - 192*b*x*e^(1
/2*a) + 384*e^(1/2*a))*e^(1/2*b*x)/b^5

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Fricas [A]  time = 0.247205, size = 58, normalized size = 0.64 \[ \frac{2 \,{\left (b^{4} x^{4} - 8 \, b^{3} x^{3} + 48 \, b^{2} x^{2} - 192 \, b x + 384\right )} e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4*e^(1/2*b*x + 1/2*a),x, algorithm="fricas")

[Out]

2*(b^4*x^4 - 8*b^3*x^3 + 48*b^2*x^2 - 192*b*x + 384)*e^(1/2*b*x + 1/2*a)/b^5

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Sympy [A]  time = 0.229877, size = 51, normalized size = 0.56 \[ \begin{cases} \frac{\left (2 b^{4} x^{4} - 16 b^{3} x^{3} + 96 b^{2} x^{2} - 384 b x + 768\right ) \sqrt{e^{a + b x}}}{b^{5}} & \text{for}\: b^{5} \neq 0 \\\frac{x^{5}}{5} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*exp(b*x+a)**(1/2),x)

[Out]

Piecewise(((2*b**4*x**4 - 16*b**3*x**3 + 96*b**2*x**2 - 384*b*x + 768)*sqrt(exp(
a + b*x))/b**5, Ne(b**5, 0)), (x**5/5, True))

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GIAC/XCAS [A]  time = 0.236409, size = 58, normalized size = 0.64 \[ \frac{2 \,{\left (b^{4} x^{4} - 8 \, b^{3} x^{3} + 48 \, b^{2} x^{2} - 192 \, b x + 384\right )} e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4*e^(1/2*b*x + 1/2*a),x, algorithm="giac")

[Out]

2*(b^4*x^4 - 8*b^3*x^3 + 48*b^2*x^2 - 192*b*x + 384)*e^(1/2*b*x + 1/2*a)/b^5