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

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

[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 = 0.143393, 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, Rules used = {2176, 2194} \[ -\frac{16 x^3 \sqrt{e^{a+b x}}}{b^2}+\frac{96 x^2 \sqrt{e^{a+b x}}}{b^3}-\frac{384 x \sqrt{e^{a+b x}}}{b^4}+\frac{768 \sqrt{e^{a+b x}}}{b^5}+\frac{2 x^4 \sqrt{e^{a+b x}}}{b} \]

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

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^4 \, dx &=\frac{2 \sqrt{e^{a+b x}} x^4}{b}-\frac{8 \int \sqrt{e^{a+b x}} x^3 \, dx}{b}\\ &=-\frac{16 \sqrt{e^{a+b x}} x^3}{b^2}+\frac{2 \sqrt{e^{a+b x}} x^4}{b}+\frac{48 \int \sqrt{e^{a+b x}} x^2 \, dx}{b^2}\\ &=\frac{96 \sqrt{e^{a+b x}} x^2}{b^3}-\frac{16 \sqrt{e^{a+b x}} x^3}{b^2}+\frac{2 \sqrt{e^{a+b x}} x^4}{b}-\frac{192 \int \sqrt{e^{a+b x}} x \, dx}{b^3}\\ &=-\frac{384 \sqrt{e^{a+b x}} x}{b^4}+\frac{96 \sqrt{e^{a+b x}} x^2}{b^3}-\frac{16 \sqrt{e^{a+b x}} x^3}{b^2}+\frac{2 \sqrt{e^{a+b x}} x^4}{b}+\frac{384 \int \sqrt{e^{a+b x}} \, dx}{b^4}\\ &=\frac{768 \sqrt{e^{a+b x}}}{b^5}-\frac{384 \sqrt{e^{a+b x}} x}{b^4}+\frac{96 \sqrt{e^{a+b x}} x^2}{b^3}-\frac{16 \sqrt{e^{a+b x}} x^3}{b^2}+\frac{2 \sqrt{e^{a+b x}} x^4}{b}\\ \end{align*}

Mathematica [A]  time = 0.0206717, 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 = 0.003, size = 43, normalized size = 0.5 \begin{align*} 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}}} \end{align*}

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 = 1.027, size = 81, normalized size = 0.89 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*exp(b*x+a)^(1/2),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 = 1.44212, size = 105, normalized size = 1.15 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*exp(b*x+a)^(1/2),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 = 0.114484, size = 51, normalized size = 0.56 \begin{align*} \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} \end{align*}

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 [A]  time = 1.31474, size = 58, normalized size = 0.64 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*exp(b*x+a)^(1/2),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