3.2531 \(\int \frac{(5-x) (2+5 x+3 x^2)}{(3+2 x)^{3/2}} \, dx\)

Optimal. Leaf size=53 \[ -\frac{3}{40} (2 x+3)^{5/2}+\frac{47}{24} (2 x+3)^{3/2}-\frac{109}{8} \sqrt{2 x+3}-\frac{65}{8 \sqrt{2 x+3}} \]

[Out]

-65/(8*Sqrt[3 + 2*x]) - (109*Sqrt[3 + 2*x])/8 + (47*(3 + 2*x)^(3/2))/24 - (3*(3 + 2*x)^(5/2))/40

________________________________________________________________________________________

Rubi [A]  time = 0.015937, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {771} \[ -\frac{3}{40} (2 x+3)^{5/2}+\frac{47}{24} (2 x+3)^{3/2}-\frac{109}{8} \sqrt{2 x+3}-\frac{65}{8 \sqrt{2 x+3}} \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2))/(3 + 2*x)^(3/2),x]

[Out]

-65/(8*Sqrt[3 + 2*x]) - (109*Sqrt[3 + 2*x])/8 + (47*(3 + 2*x)^(3/2))/24 - (3*(3 + 2*x)^(5/2))/40

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{(5-x) \left (2+5 x+3 x^2\right )}{(3+2 x)^{3/2}} \, dx &=\int \left (\frac{65}{8 (3+2 x)^{3/2}}-\frac{109}{8 \sqrt{3+2 x}}+\frac{47}{8} \sqrt{3+2 x}-\frac{3}{8} (3+2 x)^{3/2}\right ) \, dx\\ &=-\frac{65}{8 \sqrt{3+2 x}}-\frac{109}{8} \sqrt{3+2 x}+\frac{47}{24} (3+2 x)^{3/2}-\frac{3}{40} (3+2 x)^{5/2}\\ \end{align*}

Mathematica [A]  time = 0.0120587, size = 28, normalized size = 0.53 \[ -\frac{9 x^3-77 x^2+117 x+501}{15 \sqrt{2 x+3}} \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2))/(3 + 2*x)^(3/2),x]

[Out]

-(501 + 117*x - 77*x^2 + 9*x^3)/(15*Sqrt[3 + 2*x])

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 25, normalized size = 0.5 \begin{align*} -{\frac{9\,{x}^{3}-77\,{x}^{2}+117\,x+501}{15}{\frac{1}{\sqrt{3+2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+5*x+2)/(3+2*x)^(3/2),x)

[Out]

-1/15*(9*x^3-77*x^2+117*x+501)/(3+2*x)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.02658, size = 50, normalized size = 0.94 \begin{align*} -\frac{3}{40} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{47}{24} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - \frac{109}{8} \, \sqrt{2 \, x + 3} - \frac{65}{8 \, \sqrt{2 \, x + 3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)/(3+2*x)^(3/2),x, algorithm="maxima")

[Out]

-3/40*(2*x + 3)^(5/2) + 47/24*(2*x + 3)^(3/2) - 109/8*sqrt(2*x + 3) - 65/8/sqrt(2*x + 3)

________________________________________________________________________________________

Fricas [A]  time = 1.75775, size = 70, normalized size = 1.32 \begin{align*} -\frac{9 \, x^{3} - 77 \, x^{2} + 117 \, x + 501}{15 \, \sqrt{2 \, x + 3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)/(3+2*x)^(3/2),x, algorithm="fricas")

[Out]

-1/15*(9*x^3 - 77*x^2 + 117*x + 501)/sqrt(2*x + 3)

________________________________________________________________________________________

Sympy [A]  time = 15.0716, size = 46, normalized size = 0.87 \begin{align*} - \frac{3 \left (2 x + 3\right )^{\frac{5}{2}}}{40} + \frac{47 \left (2 x + 3\right )^{\frac{3}{2}}}{24} - \frac{109 \sqrt{2 x + 3}}{8} - \frac{65}{8 \sqrt{2 x + 3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x**2+5*x+2)/(3+2*x)**(3/2),x)

[Out]

-3*(2*x + 3)**(5/2)/40 + 47*(2*x + 3)**(3/2)/24 - 109*sqrt(2*x + 3)/8 - 65/(8*sqrt(2*x + 3))

________________________________________________________________________________________

Giac [A]  time = 1.09219, size = 50, normalized size = 0.94 \begin{align*} -\frac{3}{40} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{47}{24} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - \frac{109}{8} \, \sqrt{2 \, x + 3} - \frac{65}{8 \, \sqrt{2 \, x + 3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)/(3+2*x)^(3/2),x, algorithm="giac")

[Out]

-3/40*(2*x + 3)^(5/2) + 47/24*(2*x + 3)^(3/2) - 109/8*sqrt(2*x + 3) - 65/8/sqrt(2*x + 3)