3.2539 \(\int \frac{(5-x) (2+5 x+3 x^2)^2}{(3+2 x)^{3/2}} \, dx\)

Optimal. Leaf size=79 \[ -\frac{1}{32} (2 x+3)^{9/2}+\frac{165}{224} (2 x+3)^{7/2}-\frac{359}{80} (2 x+3)^{5/2}+\frac{217}{16} (2 x+3)^{3/2}-\frac{1065}{32} \sqrt{2 x+3}-\frac{325}{32 \sqrt{2 x+3}} \]

[Out]

-325/(32*Sqrt[3 + 2*x]) - (1065*Sqrt[3 + 2*x])/32 + (217*(3 + 2*x)^(3/2))/16 - (359*(3 + 2*x)^(5/2))/80 + (165
*(3 + 2*x)^(7/2))/224 - (3 + 2*x)^(9/2)/32

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Rubi [A]  time = 0.0328533, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037, Rules used = {771} \[ -\frac{1}{32} (2 x+3)^{9/2}+\frac{165}{224} (2 x+3)^{7/2}-\frac{359}{80} (2 x+3)^{5/2}+\frac{217}{16} (2 x+3)^{3/2}-\frac{1065}{32} \sqrt{2 x+3}-\frac{325}{32 \sqrt{2 x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-325/(32*Sqrt[3 + 2*x]) - (1065*Sqrt[3 + 2*x])/32 + (217*(3 + 2*x)^(3/2))/16 - (359*(3 + 2*x)^(5/2))/80 + (165
*(3 + 2*x)^(7/2))/224 - (3 + 2*x)^(9/2)/32

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 )^2}{(3+2 x)^{3/2}} \, dx &=\int \left (\frac{325}{32 (3+2 x)^{3/2}}-\frac{1065}{32 \sqrt{3+2 x}}+\frac{651}{16} \sqrt{3+2 x}-\frac{359}{16} (3+2 x)^{3/2}+\frac{165}{32} (3+2 x)^{5/2}-\frac{9}{32} (3+2 x)^{7/2}\right ) \, dx\\ &=-\frac{325}{32 \sqrt{3+2 x}}-\frac{1065}{32} \sqrt{3+2 x}+\frac{217}{16} (3+2 x)^{3/2}-\frac{359}{80} (3+2 x)^{5/2}+\frac{165}{224} (3+2 x)^{7/2}-\frac{1}{32} (3+2 x)^{9/2}\\ \end{align*}

Mathematica [A]  time = 0.0162905, size = 38, normalized size = 0.48 \[ -\frac{35 x^5-150 x^4-431 x^3-632 x^2+432 x+1996}{35 \sqrt{2 x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1996 + 432*x - 632*x^2 - 431*x^3 - 150*x^4 + 35*x^5)/(35*Sqrt[3 + 2*x])

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Maple [A]  time = 0.004, size = 35, normalized size = 0.4 \begin{align*} -{\frac{35\,{x}^{5}-150\,{x}^{4}-431\,{x}^{3}-632\,{x}^{2}+432\,x+1996}{35}{\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)^2/(3+2*x)^(3/2),x)

[Out]

-1/35*(35*x^5-150*x^4-431*x^3-632*x^2+432*x+1996)/(3+2*x)^(1/2)

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Maxima [A]  time = 0.991004, size = 74, normalized size = 0.94 \begin{align*} -\frac{1}{32} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} + \frac{165}{224} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - \frac{359}{80} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{217}{16} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - \frac{1065}{32} \, \sqrt{2 \, x + 3} - \frac{325}{32 \, \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)^2/(3+2*x)^(3/2),x, algorithm="maxima")

[Out]

-1/32*(2*x + 3)^(9/2) + 165/224*(2*x + 3)^(7/2) - 359/80*(2*x + 3)^(5/2) + 217/16*(2*x + 3)^(3/2) - 1065/32*sq
rt(2*x + 3) - 325/32/sqrt(2*x + 3)

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Fricas [A]  time = 1.7479, size = 101, normalized size = 1.28 \begin{align*} -\frac{35 \, x^{5} - 150 \, x^{4} - 431 \, x^{3} - 632 \, x^{2} + 432 \, x + 1996}{35 \, \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)^2/(3+2*x)^(3/2),x, algorithm="fricas")

[Out]

-1/35*(35*x^5 - 150*x^4 - 431*x^3 - 632*x^2 + 432*x + 1996)/sqrt(2*x + 3)

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Sympy [A]  time = 27.979, size = 68, normalized size = 0.86 \begin{align*} - \frac{\left (2 x + 3\right )^{\frac{9}{2}}}{32} + \frac{165 \left (2 x + 3\right )^{\frac{7}{2}}}{224} - \frac{359 \left (2 x + 3\right )^{\frac{5}{2}}}{80} + \frac{217 \left (2 x + 3\right )^{\frac{3}{2}}}{16} - \frac{1065 \sqrt{2 x + 3}}{32} - \frac{325}{32 \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)**2/(3+2*x)**(3/2),x)

[Out]

-(2*x + 3)**(9/2)/32 + 165*(2*x + 3)**(7/2)/224 - 359*(2*x + 3)**(5/2)/80 + 217*(2*x + 3)**(3/2)/16 - 1065*sqr
t(2*x + 3)/32 - 325/(32*sqrt(2*x + 3))

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Giac [A]  time = 1.14232, size = 74, normalized size = 0.94 \begin{align*} -\frac{1}{32} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} + \frac{165}{224} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - \frac{359}{80} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{217}{16} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - \frac{1065}{32} \, \sqrt{2 \, x + 3} - \frac{325}{32 \, \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)^2/(3+2*x)^(3/2),x, algorithm="giac")

[Out]

-1/32*(2*x + 3)^(9/2) + 165/224*(2*x + 3)^(7/2) - 359/80*(2*x + 3)^(5/2) + 217/16*(2*x + 3)^(3/2) - 1065/32*sq
rt(2*x + 3) - 325/32/sqrt(2*x + 3)