3.2537 \(\int (5-x) \sqrt{3+2 x} (2+5 x+3 x^2)^2 \, dx\)

Optimal. Leaf size=79 \[ -\frac{9}{416} (2 x+3)^{13/2}+\frac{15}{32} (2 x+3)^{11/2}-\frac{359}{144} (2 x+3)^{9/2}+\frac{93}{16} (2 x+3)^{7/2}-\frac{213}{32} (2 x+3)^{5/2}+\frac{325}{96} (2 x+3)^{3/2} \]

[Out]

(325*(3 + 2*x)^(3/2))/96 - (213*(3 + 2*x)^(5/2))/32 + (93*(3 + 2*x)^(7/2))/16 - (359*(3 + 2*x)^(9/2))/144 + (1
5*(3 + 2*x)^(11/2))/32 - (9*(3 + 2*x)^(13/2))/416

________________________________________________________________________________________

Rubi [A]  time = 0.0225127, 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} \[ -\frac{9}{416} (2 x+3)^{13/2}+\frac{15}{32} (2 x+3)^{11/2}-\frac{359}{144} (2 x+3)^{9/2}+\frac{93}{16} (2 x+3)^{7/2}-\frac{213}{32} (2 x+3)^{5/2}+\frac{325}{96} (2 x+3)^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(325*(3 + 2*x)^(3/2))/96 - (213*(3 + 2*x)^(5/2))/32 + (93*(3 + 2*x)^(7/2))/16 - (359*(3 + 2*x)^(9/2))/144 + (1
5*(3 + 2*x)^(11/2))/32 - (9*(3 + 2*x)^(13/2))/416

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 (5-x) \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^2 \, dx &=\int \left (\frac{325}{32} \sqrt{3+2 x}-\frac{1065}{32} (3+2 x)^{3/2}+\frac{651}{16} (3+2 x)^{5/2}-\frac{359}{16} (3+2 x)^{7/2}+\frac{165}{32} (3+2 x)^{9/2}-\frac{9}{32} (3+2 x)^{11/2}\right ) \, dx\\ &=\frac{325}{96} (3+2 x)^{3/2}-\frac{213}{32} (3+2 x)^{5/2}+\frac{93}{16} (3+2 x)^{7/2}-\frac{359}{144} (3+2 x)^{9/2}+\frac{15}{32} (3+2 x)^{11/2}-\frac{9}{416} (3+2 x)^{13/2}\\ \end{align*}

Mathematica [A]  time = 0.0151189, size = 38, normalized size = 0.48 \[ -\frac{1}{117} (2 x+3)^{3/2} \left (81 x^5-270 x^4-1109 x^3-1332 x^2-648 x-132\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((3 + 2*x)^(3/2)*(-132 - 648*x - 1332*x^2 - 1109*x^3 - 270*x^4 + 81*x^5))/117

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 35, normalized size = 0.4 \begin{align*} -{\frac{81\,{x}^{5}-270\,{x}^{4}-1109\,{x}^{3}-1332\,{x}^{2}-648\,x-132}{117} \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)^2*(3+2*x)^(1/2),x)

[Out]

-1/117*(81*x^5-270*x^4-1109*x^3-1332*x^2-648*x-132)*(3+2*x)^(3/2)

________________________________________________________________________________________

Maxima [A]  time = 0.975527, size = 74, normalized size = 0.94 \begin{align*} -\frac{9}{416} \,{\left (2 \, x + 3\right )}^{\frac{13}{2}} + \frac{15}{32} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} - \frac{359}{144} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} + \frac{93}{16} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - \frac{213}{32} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{325}{96} \,{\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)^2*(3+2*x)^(1/2),x, algorithm="maxima")

[Out]

-9/416*(2*x + 3)^(13/2) + 15/32*(2*x + 3)^(11/2) - 359/144*(2*x + 3)^(9/2) + 93/16*(2*x + 3)^(7/2) - 213/32*(2
*x + 3)^(5/2) + 325/96*(2*x + 3)^(3/2)

________________________________________________________________________________________

Fricas [A]  time = 1.67844, size = 122, normalized size = 1.54 \begin{align*} -\frac{1}{117} \,{\left (162 \, x^{6} - 297 \, x^{5} - 3028 \, x^{4} - 5991 \, x^{3} - 5292 \, x^{2} - 2208 \, x - 396\right )} \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)^(1/2),x, algorithm="fricas")

[Out]

-1/117*(162*x^6 - 297*x^5 - 3028*x^4 - 5991*x^3 - 5292*x^2 - 2208*x - 396)*sqrt(2*x + 3)

________________________________________________________________________________________

Sympy [A]  time = 2.82986, size = 70, normalized size = 0.89 \begin{align*} - \frac{9 \left (2 x + 3\right )^{\frac{13}{2}}}{416} + \frac{15 \left (2 x + 3\right )^{\frac{11}{2}}}{32} - \frac{359 \left (2 x + 3\right )^{\frac{9}{2}}}{144} + \frac{93 \left (2 x + 3\right )^{\frac{7}{2}}}{16} - \frac{213 \left (2 x + 3\right )^{\frac{5}{2}}}{32} + \frac{325 \left (2 x + 3\right )^{\frac{3}{2}}}{96} \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)**(1/2),x)

[Out]

-9*(2*x + 3)**(13/2)/416 + 15*(2*x + 3)**(11/2)/32 - 359*(2*x + 3)**(9/2)/144 + 93*(2*x + 3)**(7/2)/16 - 213*(
2*x + 3)**(5/2)/32 + 325*(2*x + 3)**(3/2)/96

________________________________________________________________________________________

Giac [A]  time = 1.11675, size = 74, normalized size = 0.94 \begin{align*} -\frac{9}{416} \,{\left (2 \, x + 3\right )}^{\frac{13}{2}} + \frac{15}{32} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} - \frac{359}{144} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} + \frac{93}{16} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - \frac{213}{32} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{325}{96} \,{\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)^2*(3+2*x)^(1/2),x, algorithm="giac")

[Out]

-9/416*(2*x + 3)^(13/2) + 15/32*(2*x + 3)^(11/2) - 359/144*(2*x + 3)^(9/2) + 93/16*(2*x + 3)^(7/2) - 213/32*(2
*x + 3)^(5/2) + 325/96*(2*x + 3)^(3/2)