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

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

[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 = 0.101391, 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, Rules used = {2176, 2194} \[ -\frac{12 x^2 \sqrt{e^{a+b x}}}{b^2}+\frac{48 x \sqrt{e^{a+b x}}}{b^3}-\frac{96 \sqrt{e^{a+b x}}}{b^4}+\frac{2 x^3 \sqrt{e^{a+b x}}}{b} \]

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

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^3 \, dx &=\frac{2 \sqrt{e^{a+b x}} x^3}{b}-\frac{6 \int \sqrt{e^{a+b x}} x^2 \, dx}{b}\\ &=-\frac{12 \sqrt{e^{a+b x}} x^2}{b^2}+\frac{2 \sqrt{e^{a+b x}} x^3}{b}+\frac{24 \int \sqrt{e^{a+b x}} x \, dx}{b^2}\\ &=\frac{48 \sqrt{e^{a+b x}} x}{b^3}-\frac{12 \sqrt{e^{a+b x}} x^2}{b^2}+\frac{2 \sqrt{e^{a+b x}} x^3}{b}-\frac{48 \int \sqrt{e^{a+b x}} \, dx}{b^3}\\ &=-\frac{96 \sqrt{e^{a+b x}}}{b^4}+\frac{48 \sqrt{e^{a+b x}} x}{b^3}-\frac{12 \sqrt{e^{a+b x}} x^2}{b^2}+\frac{2 \sqrt{e^{a+b x}} x^3}{b}\\ \end{align*}

Mathematica [A]  time = 0.0134152, 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 = 0.003, size = 35, normalized size = 0.5 \begin{align*} 2\,{\frac{ \left ({x}^{3}{b}^{3}-6\,{x}^{2}{b}^{2}+24\,bx-48 \right ) \sqrt{{{\rm e}^{bx+a}}}}{{b}^{4}}} \end{align*}

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 = 1.09332, size = 65, normalized size = 0.9 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*exp(b*x+a)^(1/2),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 = 1.45923, size = 85, normalized size = 1.18 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*exp(b*x+a)^(1/2),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 = 0.108289, size = 42, normalized size = 0.58 \begin{align*} \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} \end{align*}

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, Ne(b**4, 0)), (x**4/4, True))

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Giac [A]  time = 1.25374, size = 47, normalized size = 0.65 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*exp(b*x+a)^(1/2),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