3.2538 \(\int \frac{(5-x) (2+5 x+3 x^2)^2}{\sqrt{3+2 x}} \, dx\)

Optimal. Leaf size=79 \[ -\frac{9}{352} (2 x+3)^{11/2}+\frac{55}{96} (2 x+3)^{9/2}-\frac{359}{112} (2 x+3)^{7/2}+\frac{651}{80} (2 x+3)^{5/2}-\frac{355}{32} (2 x+3)^{3/2}+\frac{325}{32} \sqrt{2 x+3} \]

[Out]

(325*Sqrt[3 + 2*x])/32 - (355*(3 + 2*x)^(3/2))/32 + (651*(3 + 2*x)^(5/2))/80 - (359*(3 + 2*x)^(7/2))/112 + (55
*(3 + 2*x)^(9/2))/96 - (9*(3 + 2*x)^(11/2))/352

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Rubi [A]  time = 0.0234889, 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} \[ -\frac{9}{352} (2 x+3)^{11/2}+\frac{55}{96} (2 x+3)^{9/2}-\frac{359}{112} (2 x+3)^{7/2}+\frac{651}{80} (2 x+3)^{5/2}-\frac{355}{32} (2 x+3)^{3/2}+\frac{325}{32} \sqrt{2 x+3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(325*Sqrt[3 + 2*x])/32 - (355*(3 + 2*x)^(3/2))/32 + (651*(3 + 2*x)^(5/2))/80 - (359*(3 + 2*x)^(7/2))/112 + (55
*(3 + 2*x)^(9/2))/96 - (9*(3 + 2*x)^(11/2))/352

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}{\sqrt{3+2 x}} \, dx &=\int \left (\frac{325}{32 \sqrt{3+2 x}}-\frac{1065}{32} \sqrt{3+2 x}+\frac{651}{16} (3+2 x)^{3/2}-\frac{359}{16} (3+2 x)^{5/2}+\frac{165}{32} (3+2 x)^{7/2}-\frac{9}{32} (3+2 x)^{9/2}\right ) \, dx\\ &=\frac{325}{32} \sqrt{3+2 x}-\frac{355}{32} (3+2 x)^{3/2}+\frac{651}{80} (3+2 x)^{5/2}-\frac{359}{112} (3+2 x)^{7/2}+\frac{55}{96} (3+2 x)^{9/2}-\frac{9}{352} (3+2 x)^{11/2}\\ \end{align*}

Mathematica [A]  time = 0.01655, size = 38, normalized size = 0.48 \[ -\frac{\sqrt{2 x+3} \left (945 x^5-3500 x^4-12645 x^3-15354 x^2-6252 x-4344\right )}{1155} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[3 + 2*x]*(-4344 - 6252*x - 15354*x^2 - 12645*x^3 - 3500*x^4 + 945*x^5))/1155

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Maple [A]  time = 0.004, size = 35, normalized size = 0.4 \begin{align*} -{\frac{945\,{x}^{5}-3500\,{x}^{4}-12645\,{x}^{3}-15354\,{x}^{2}-6252\,x-4344}{1155}\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)^(1/2),x)

[Out]

-1/1155*(945*x^5-3500*x^4-12645*x^3-15354*x^2-6252*x-4344)*(3+2*x)^(1/2)

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Maxima [A]  time = 0.958316, size = 74, normalized size = 0.94 \begin{align*} -\frac{9}{352} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} + \frac{55}{96} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} - \frac{359}{112} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} + \frac{651}{80} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} - \frac{355}{32} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} + \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)^(1/2),x, algorithm="maxima")

[Out]

-9/352*(2*x + 3)^(11/2) + 55/96*(2*x + 3)^(9/2) - 359/112*(2*x + 3)^(7/2) + 651/80*(2*x + 3)^(5/2) - 355/32*(2
*x + 3)^(3/2) + 325/32*sqrt(2*x + 3)

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Fricas [A]  time = 1.7826, size = 113, normalized size = 1.43 \begin{align*} -\frac{1}{1155} \,{\left (945 \, x^{5} - 3500 \, x^{4} - 12645 \, x^{3} - 15354 \, x^{2} - 6252 \, x - 4344\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]

-1/1155*(945*x^5 - 3500*x^4 - 12645*x^3 - 15354*x^2 - 6252*x - 4344)*sqrt(2*x + 3)

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Sympy [A]  time = 72.3021, size = 70, normalized size = 0.89 \begin{align*} - \frac{9 \left (2 x + 3\right )^{\frac{11}{2}}}{352} + \frac{55 \left (2 x + 3\right )^{\frac{9}{2}}}{96} - \frac{359 \left (2 x + 3\right )^{\frac{7}{2}}}{112} + \frac{651 \left (2 x + 3\right )^{\frac{5}{2}}}{80} - \frac{355 \left (2 x + 3\right )^{\frac{3}{2}}}{32} + \frac{325 \sqrt{2 x + 3}}{32} \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)**(11/2)/352 + 55*(2*x + 3)**(9/2)/96 - 359*(2*x + 3)**(7/2)/112 + 651*(2*x + 3)**(5/2)/80 - 355*(
2*x + 3)**(3/2)/32 + 325*sqrt(2*x + 3)/32

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Giac [A]  time = 1.08663, size = 74, normalized size = 0.94 \begin{align*} -\frac{9}{352} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} + \frac{55}{96} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} - \frac{359}{112} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} + \frac{651}{80} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} - \frac{355}{32} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} + \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)^(1/2),x, algorithm="giac")

[Out]

-9/352*(2*x + 3)^(11/2) + 55/96*(2*x + 3)^(9/2) - 359/112*(2*x + 3)^(7/2) + 651/80*(2*x + 3)^(5/2) - 355/32*(2
*x + 3)^(3/2) + 325/32*sqrt(2*x + 3)