3.2548 \(\int \frac{(5-x) (2+5 x+3 x^2)^3}{(3+2 x)^{5/2}} \, dx\)

Optimal. Leaf size=105 \[ -\frac{27 (2 x+3)^{11/2}}{1408}+\frac{63}{128} (2 x+3)^{9/2}-\frac{3519}{896} (2 x+3)^{7/2}+\frac{2095}{128} (2 x+3)^{5/2}-\frac{17201}{384} (2 x+3)^{3/2}+\frac{16005}{128} \sqrt{2 x+3}+\frac{7925}{128 \sqrt{2 x+3}}-\frac{1625}{384 (2 x+3)^{3/2}} \]

[Out]

-1625/(384*(3 + 2*x)^(3/2)) + 7925/(128*Sqrt[3 + 2*x]) + (16005*Sqrt[3 + 2*x])/128 - (17201*(3 + 2*x)^(3/2))/3
84 + (2095*(3 + 2*x)^(5/2))/128 - (3519*(3 + 2*x)^(7/2))/896 + (63*(3 + 2*x)^(9/2))/128 - (27*(3 + 2*x)^(11/2)
)/1408

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Rubi [A]  time = 0.0303977, antiderivative size = 105, 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} \[ -\frac{27 (2 x+3)^{11/2}}{1408}+\frac{63}{128} (2 x+3)^{9/2}-\frac{3519}{896} (2 x+3)^{7/2}+\frac{2095}{128} (2 x+3)^{5/2}-\frac{17201}{384} (2 x+3)^{3/2}+\frac{16005}{128} \sqrt{2 x+3}+\frac{7925}{128 \sqrt{2 x+3}}-\frac{1625}{384 (2 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1625/(384*(3 + 2*x)^(3/2)) + 7925/(128*Sqrt[3 + 2*x]) + (16005*Sqrt[3 + 2*x])/128 - (17201*(3 + 2*x)^(3/2))/3
84 + (2095*(3 + 2*x)^(5/2))/128 - (3519*(3 + 2*x)^(7/2))/896 + (63*(3 + 2*x)^(9/2))/128 - (27*(3 + 2*x)^(11/2)
)/1408

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}{(3+2 x)^{5/2}} \, dx &=\int \left (\frac{1625}{128 (3+2 x)^{5/2}}-\frac{7925}{128 (3+2 x)^{3/2}}+\frac{16005}{128 \sqrt{3+2 x}}-\frac{17201}{128} \sqrt{3+2 x}+\frac{10475}{128} (3+2 x)^{3/2}-\frac{3519}{128} (3+2 x)^{5/2}+\frac{567}{128} (3+2 x)^{7/2}-\frac{27}{128} (3+2 x)^{9/2}\right ) \, dx\\ &=-\frac{1625}{384 (3+2 x)^{3/2}}+\frac{7925}{128 \sqrt{3+2 x}}+\frac{16005}{128} \sqrt{3+2 x}-\frac{17201}{384} (3+2 x)^{3/2}+\frac{2095}{128} (3+2 x)^{5/2}-\frac{3519}{896} (3+2 x)^{7/2}+\frac{63}{128} (3+2 x)^{9/2}-\frac{27 (3+2 x)^{11/2}}{1408}\\ \end{align*}

Mathematica [A]  time = 0.0195251, size = 48, normalized size = 0.46 \[ -\frac{567 x^7-1323 x^6-9666 x^5-21360 x^4-17663 x^3-42003 x^2-184566 x-181486}{231 (2 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-181486 - 184566*x - 42003*x^2 - 17663*x^3 - 21360*x^4 - 9666*x^5 - 1323*x^6 + 567*x^7)/(231*(3 + 2*x)^(3/2)
)

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Maple [A]  time = 0.005, size = 45, normalized size = 0.4 \begin{align*} -{\frac{567\,{x}^{7}-1323\,{x}^{6}-9666\,{x}^{5}-21360\,{x}^{4}-17663\,{x}^{3}-42003\,{x}^{2}-184566\,x-181486}{231} \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/(3+2*x)^(5/2),x)

[Out]

-1/231*(567*x^7-1323*x^6-9666*x^5-21360*x^4-17663*x^3-42003*x^2-184566*x-181486)/(3+2*x)^(3/2)

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Maxima [A]  time = 1.01697, size = 93, normalized size = 0.89 \begin{align*} -\frac{27}{1408} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} + \frac{63}{128} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} - \frac{3519}{896} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} + \frac{2095}{128} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} - \frac{17201}{384} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} + \frac{16005}{128} \, \sqrt{2 \, x + 3} + \frac{25 \,{\left (951 \, x + 1394\right )}}{192 \,{\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/(3+2*x)^(5/2),x, algorithm="maxima")

[Out]

-27/1408*(2*x + 3)^(11/2) + 63/128*(2*x + 3)^(9/2) - 3519/896*(2*x + 3)^(7/2) + 2095/128*(2*x + 3)^(5/2) - 172
01/384*(2*x + 3)^(3/2) + 16005/128*sqrt(2*x + 3) + 25/192*(951*x + 1394)/(2*x + 3)^(3/2)

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Fricas [A]  time = 1.75836, size = 174, normalized size = 1.66 \begin{align*} -\frac{{\left (567 \, x^{7} - 1323 \, x^{6} - 9666 \, x^{5} - 21360 \, x^{4} - 17663 \, x^{3} - 42003 \, x^{2} - 184566 \, x - 181486\right )} \sqrt{2 \, x + 3}}{231 \,{\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/(3+2*x)^(5/2),x, algorithm="fricas")

[Out]

-1/231*(567*x^7 - 1323*x^6 - 9666*x^5 - 21360*x^4 - 17663*x^3 - 42003*x^2 - 184566*x - 181486)*sqrt(2*x + 3)/(
4*x^2 + 12*x + 9)

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Sympy [A]  time = 43.7615, size = 94, normalized size = 0.9 \begin{align*} - \frac{27 \left (2 x + 3\right )^{\frac{11}{2}}}{1408} + \frac{63 \left (2 x + 3\right )^{\frac{9}{2}}}{128} - \frac{3519 \left (2 x + 3\right )^{\frac{7}{2}}}{896} + \frac{2095 \left (2 x + 3\right )^{\frac{5}{2}}}{128} - \frac{17201 \left (2 x + 3\right )^{\frac{3}{2}}}{384} + \frac{16005 \sqrt{2 x + 3}}{128} + \frac{7925}{128 \sqrt{2 x + 3}} - \frac{1625}{384 \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/(3+2*x)**(5/2),x)

[Out]

-27*(2*x + 3)**(11/2)/1408 + 63*(2*x + 3)**(9/2)/128 - 3519*(2*x + 3)**(7/2)/896 + 2095*(2*x + 3)**(5/2)/128 -
17201*(2*x + 3)**(3/2)/384 + 16005*sqrt(2*x + 3)/128 + 7925/(128*sqrt(2*x + 3)) - 1625/(384*(2*x + 3)**(3/2))

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Giac [A]  time = 1.10684, size = 93, normalized size = 0.89 \begin{align*} -\frac{27}{1408} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} + \frac{63}{128} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} - \frac{3519}{896} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} + \frac{2095}{128} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} - \frac{17201}{384} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} + \frac{16005}{128} \, \sqrt{2 \, x + 3} + \frac{25 \,{\left (951 \, x + 1394\right )}}{192 \,{\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/(3+2*x)^(5/2),x, algorithm="giac")

[Out]

-27/1408*(2*x + 3)^(11/2) + 63/128*(2*x + 3)^(9/2) - 3519/896*(2*x + 3)^(7/2) + 2095/128*(2*x + 3)^(5/2) - 172
01/384*(2*x + 3)^(3/2) + 16005/128*sqrt(2*x + 3) + 25/192*(951*x + 1394)/(2*x + 3)^(3/2)