3.2532 \(\int \frac{(5-x) (2+5 x+3 x^2)}{(3+2 x)^{5/2}} \, dx\)

Optimal. Leaf size=53 \[ -\frac{1}{8} (2 x+3)^{3/2}+\frac{47}{8} \sqrt{2 x+3}+\frac{109}{8 \sqrt{2 x+3}}-\frac{65}{24 (2 x+3)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0162295, 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} \[ -\frac{1}{8} (2 x+3)^{3/2}+\frac{47}{8} \sqrt{2 x+3}+\frac{109}{8 \sqrt{2 x+3}}-\frac{65}{24 (2 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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)^{5/2}} \, dx &=\int \left (\frac{65}{8 (3+2 x)^{5/2}}-\frac{109}{8 (3+2 x)^{3/2}}+\frac{47}{8 \sqrt{3+2 x}}-\frac{3}{8} \sqrt{3+2 x}\right ) \, dx\\ &=-\frac{65}{24 (3+2 x)^{3/2}}+\frac{109}{8 \sqrt{3+2 x}}+\frac{47}{8} \sqrt{3+2 x}-\frac{1}{8} (3+2 x)^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0122776, size = 28, normalized size = 0.53 \[ -\frac{3 x^3-57 x^2-273 x-263}{3 (2 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-263 - 273*x - 57*x^2 + 3*x^3)/(3*(3 + 2*x)^(3/2))

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Maple [A]  time = 0.003, size = 25, normalized size = 0.5 \begin{align*} -{\frac{3\,{x}^{3}-57\,{x}^{2}-273\,x-263}{3} \left ( 3+2\,x \right ) ^{-{\frac{3}{2}}}} \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)^(5/2),x)

[Out]

-1/3*(3*x^3-57*x^2-273*x-263)/(3+2*x)^(3/2)

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Maxima [A]  time = 0.983456, size = 45, normalized size = 0.85 \begin{align*} -\frac{1}{8} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} + \frac{47}{8} \, \sqrt{2 \, x + 3} + \frac{327 \, x + 458}{12 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}}} \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)^(5/2),x, algorithm="maxima")

[Out]

-1/8*(2*x + 3)^(3/2) + 47/8*sqrt(2*x + 3) + 1/12*(327*x + 458)/(2*x + 3)^(3/2)

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Fricas [A]  time = 1.74869, size = 95, normalized size = 1.79 \begin{align*} -\frac{{\left (3 \, x^{3} - 57 \, x^{2} - 273 \, x - 263\right )} \sqrt{2 \, x + 3}}{3 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} \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)^(5/2),x, algorithm="fricas")

[Out]

-1/3*(3*x^3 - 57*x^2 - 273*x - 263)*sqrt(2*x + 3)/(4*x^2 + 12*x + 9)

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Sympy [B]  time = 0.687744, size = 102, normalized size = 1.92 \begin{align*} - \frac{3 x^{3}}{6 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}} + \frac{57 x^{2}}{6 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}} + \frac{273 x}{6 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}} + \frac{263}{6 x \sqrt{2 x + 3} + 9 \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)**(5/2),x)

[Out]

-3*x**3/(6*x*sqrt(2*x + 3) + 9*sqrt(2*x + 3)) + 57*x**2/(6*x*sqrt(2*x + 3) + 9*sqrt(2*x + 3)) + 273*x/(6*x*sqr
t(2*x + 3) + 9*sqrt(2*x + 3)) + 263/(6*x*sqrt(2*x + 3) + 9*sqrt(2*x + 3))

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Giac [A]  time = 1.11615, size = 45, normalized size = 0.85 \begin{align*} -\frac{1}{8} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} + \frac{47}{8} \, \sqrt{2 \, x + 3} + \frac{327 \, x + 458}{12 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}}} \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)^(5/2),x, algorithm="giac")

[Out]

-1/8*(2*x + 3)^(3/2) + 47/8*sqrt(2*x + 3) + 1/12*(327*x + 458)/(2*x + 3)^(3/2)