3.2540 \(\int \frac{(5-x) (2+5 x+3 x^2)^2}{(3+2 x)^{5/2}} \, dx\)

Optimal. Leaf size=79 \[ -\frac{9}{224} (2 x+3)^{7/2}+\frac{33}{32} (2 x+3)^{5/2}-\frac{359}{48} (2 x+3)^{3/2}+\frac{651}{16} \sqrt{2 x+3}+\frac{1065}{32 \sqrt{2 x+3}}-\frac{325}{96 (2 x+3)^{3/2}} \]

[Out]

-325/(96*(3 + 2*x)^(3/2)) + 1065/(32*Sqrt[3 + 2*x]) + (651*Sqrt[3 + 2*x])/16 - (359*(3 + 2*x)^(3/2))/48 + (33*
(3 + 2*x)^(5/2))/32 - (9*(3 + 2*x)^(7/2))/224

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Rubi [A]  time = 0.026224, 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} \[ -\frac{9}{224} (2 x+3)^{7/2}+\frac{33}{32} (2 x+3)^{5/2}-\frac{359}{48} (2 x+3)^{3/2}+\frac{651}{16} \sqrt{2 x+3}+\frac{1065}{32 \sqrt{2 x+3}}-\frac{325}{96 (2 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-325/(96*(3 + 2*x)^(3/2)) + 1065/(32*Sqrt[3 + 2*x]) + (651*Sqrt[3 + 2*x])/16 - (359*(3 + 2*x)^(3/2))/48 + (33*
(3 + 2*x)^(5/2))/32 - (9*(3 + 2*x)^(7/2))/224

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)^{5/2}} \, dx &=\int \left (\frac{325}{32 (3+2 x)^{5/2}}-\frac{1065}{32 (3+2 x)^{3/2}}+\frac{651}{16 \sqrt{3+2 x}}-\frac{359}{16} \sqrt{3+2 x}+\frac{165}{32} (3+2 x)^{3/2}-\frac{9}{32} (3+2 x)^{5/2}\right ) \, dx\\ &=-\frac{325}{96 (3+2 x)^{3/2}}+\frac{1065}{32 \sqrt{3+2 x}}+\frac{651}{16} \sqrt{3+2 x}-\frac{359}{48} (3+2 x)^{3/2}+\frac{33}{32} (3+2 x)^{5/2}-\frac{9}{224} (3+2 x)^{7/2}\\ \end{align*}

Mathematica [A]  time = 0.0162293, size = 38, normalized size = 0.48 \[ -\frac{27 x^5-144 x^4-215 x^3-1530 x^2-7164 x-7024}{21 (2 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-7024 - 7164*x - 1530*x^2 - 215*x^3 - 144*x^4 + 27*x^5)/(21*(3 + 2*x)^(3/2))

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Maple [A]  time = 0.003, size = 35, normalized size = 0.4 \begin{align*} -{\frac{27\,{x}^{5}-144\,{x}^{4}-215\,{x}^{3}-1530\,{x}^{2}-7164\,x-7024}{21} \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)^(5/2),x)

[Out]

-1/21*(27*x^5-144*x^4-215*x^3-1530*x^2-7164*x-7024)/(3+2*x)^(3/2)

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Maxima [A]  time = 0.99555, size = 69, normalized size = 0.87 \begin{align*} -\frac{9}{224} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} + \frac{33}{32} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} - \frac{359}{48} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} + \frac{651}{16} \, \sqrt{2 \, x + 3} + \frac{5 \,{\left (639 \, x + 926\right )}}{48 \,{\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)^(5/2),x, algorithm="maxima")

[Out]

-9/224*(2*x + 3)^(7/2) + 33/32*(2*x + 3)^(5/2) - 359/48*(2*x + 3)^(3/2) + 651/16*sqrt(2*x + 3) + 5/48*(639*x +
 926)/(2*x + 3)^(3/2)

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Fricas [A]  time = 1.83846, size = 130, normalized size = 1.65 \begin{align*} -\frac{{\left (27 \, x^{5} - 144 \, x^{4} - 215 \, x^{3} - 1530 \, x^{2} - 7164 \, x - 7024\right )} \sqrt{2 \, x + 3}}{21 \,{\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)^2/(3+2*x)^(5/2),x, algorithm="fricas")

[Out]

-1/21*(27*x^5 - 144*x^4 - 215*x^3 - 1530*x^2 - 7164*x - 7024)*sqrt(2*x + 3)/(4*x^2 + 12*x + 9)

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Sympy [A]  time = 33.5901, size = 70, normalized size = 0.89 \begin{align*} - \frac{9 \left (2 x + 3\right )^{\frac{7}{2}}}{224} + \frac{33 \left (2 x + 3\right )^{\frac{5}{2}}}{32} - \frac{359 \left (2 x + 3\right )^{\frac{3}{2}}}{48} + \frac{651 \sqrt{2 x + 3}}{16} + \frac{1065}{32 \sqrt{2 x + 3}} - \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)**(5/2),x)

[Out]

-9*(2*x + 3)**(7/2)/224 + 33*(2*x + 3)**(5/2)/32 - 359*(2*x + 3)**(3/2)/48 + 651*sqrt(2*x + 3)/16 + 1065/(32*s
qrt(2*x + 3)) - 325/(96*(2*x + 3)**(3/2))

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Giac [A]  time = 1.08219, size = 69, normalized size = 0.87 \begin{align*} -\frac{9}{224} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} + \frac{33}{32} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} - \frac{359}{48} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} + \frac{651}{16} \, \sqrt{2 \, x + 3} + \frac{5 \,{\left (639 \, x + 926\right )}}{48 \,{\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)^(5/2),x, algorithm="giac")

[Out]

-9/224*(2*x + 3)^(7/2) + 33/32*(2*x + 3)^(5/2) - 359/48*(2*x + 3)^(3/2) + 651/16*sqrt(2*x + 3) + 5/48*(639*x +
 926)/(2*x + 3)^(3/2)