3.2547 \(\int \frac{(5-x) (2+5 x+3 x^2)^3}{(3+2 x)^{3/2}} \, dx\)

Optimal. Leaf size=105 \[ -\frac{27 (2 x+3)^{13/2}}{1664}+\frac{567 (2 x+3)^{11/2}}{1408}-\frac{391}{128} (2 x+3)^{9/2}+\frac{10475}{896} (2 x+3)^{7/2}-\frac{17201}{640} (2 x+3)^{5/2}+\frac{5335}{128} (2 x+3)^{3/2}-\frac{7925}{128} \sqrt{2 x+3}-\frac{1625}{128 \sqrt{2 x+3}} \]

[Out]

-1625/(128*Sqrt[3 + 2*x]) - (7925*Sqrt[3 + 2*x])/128 + (5335*(3 + 2*x)^(3/2))/128 - (17201*(3 + 2*x)^(5/2))/64
0 + (10475*(3 + 2*x)^(7/2))/896 - (391*(3 + 2*x)^(9/2))/128 + (567*(3 + 2*x)^(11/2))/1408 - (27*(3 + 2*x)^(13/
2))/1664

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Rubi [A]  time = 0.0311547, 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} \[ -\frac{27 (2 x+3)^{13/2}}{1664}+\frac{567 (2 x+3)^{11/2}}{1408}-\frac{391}{128} (2 x+3)^{9/2}+\frac{10475}{896} (2 x+3)^{7/2}-\frac{17201}{640} (2 x+3)^{5/2}+\frac{5335}{128} (2 x+3)^{3/2}-\frac{7925}{128} \sqrt{2 x+3}-\frac{1625}{128 \sqrt{2 x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1625/(128*Sqrt[3 + 2*x]) - (7925*Sqrt[3 + 2*x])/128 + (5335*(3 + 2*x)^(3/2))/128 - (17201*(3 + 2*x)^(5/2))/64
0 + (10475*(3 + 2*x)^(7/2))/896 - (391*(3 + 2*x)^(9/2))/128 + (567*(3 + 2*x)^(11/2))/1408 - (27*(3 + 2*x)^(13/
2))/1664

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)^{3/2}} \, dx &=\int \left (\frac{1625}{128 (3+2 x)^{3/2}}-\frac{7925}{128 \sqrt{3+2 x}}+\frac{16005}{128} \sqrt{3+2 x}-\frac{17201}{128} (3+2 x)^{3/2}+\frac{10475}{128} (3+2 x)^{5/2}-\frac{3519}{128} (3+2 x)^{7/2}+\frac{567}{128} (3+2 x)^{9/2}-\frac{27}{128} (3+2 x)^{11/2}\right ) \, dx\\ &=-\frac{1625}{128 \sqrt{3+2 x}}-\frac{7925}{128} \sqrt{3+2 x}+\frac{5335}{128} (3+2 x)^{3/2}-\frac{17201}{640} (3+2 x)^{5/2}+\frac{10475}{896} (3+2 x)^{7/2}-\frac{391}{128} (3+2 x)^{9/2}+\frac{567 (3+2 x)^{11/2}}{1408}-\frac{27 (3+2 x)^{13/2}}{1664}\\ \end{align*}

Mathematica [A]  time = 0.0189482, size = 48, normalized size = 0.46 \[ -\frac{10395 x^7-19845 x^6-180530 x^5-392500 x^4-398339 x^3-256433 x^2+77138 x+431614}{5005 \sqrt{2 x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(431614 + 77138*x - 256433*x^2 - 398339*x^3 - 392500*x^4 - 180530*x^5 - 19845*x^6 + 10395*x^7)/(5005*Sqrt[3 +
2*x])

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Maple [A]  time = 0.004, size = 45, normalized size = 0.4 \begin{align*} -{\frac{10395\,{x}^{7}-19845\,{x}^{6}-180530\,{x}^{5}-392500\,{x}^{4}-398339\,{x}^{3}-256433\,{x}^{2}+77138\,x+431614}{5005}{\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)^3/(3+2*x)^(3/2),x)

[Out]

-1/5005*(10395*x^7-19845*x^6-180530*x^5-392500*x^4-398339*x^3-256433*x^2+77138*x+431614)/(3+2*x)^(1/2)

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Maxima [A]  time = 1.00495, size = 99, normalized size = 0.94 \begin{align*} -\frac{27}{1664} \,{\left (2 \, x + 3\right )}^{\frac{13}{2}} + \frac{567}{1408} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} - \frac{391}{128} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} + \frac{10475}{896} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - \frac{17201}{640} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{5335}{128} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - \frac{7925}{128} \, \sqrt{2 \, x + 3} - \frac{1625}{128 \, \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)^3/(3+2*x)^(3/2),x, algorithm="maxima")

[Out]

-27/1664*(2*x + 3)^(13/2) + 567/1408*(2*x + 3)^(11/2) - 391/128*(2*x + 3)^(9/2) + 10475/896*(2*x + 3)^(7/2) -
17201/640*(2*x + 3)^(5/2) + 5335/128*(2*x + 3)^(3/2) - 7925/128*sqrt(2*x + 3) - 1625/128/sqrt(2*x + 3)

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Fricas [A]  time = 1.78757, size = 159, normalized size = 1.51 \begin{align*} -\frac{10395 \, x^{7} - 19845 \, x^{6} - 180530 \, x^{5} - 392500 \, x^{4} - 398339 \, x^{3} - 256433 \, x^{2} + 77138 \, x + 431614}{5005 \, \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)^3/(3+2*x)^(3/2),x, algorithm="fricas")

[Out]

-1/5005*(10395*x^7 - 19845*x^6 - 180530*x^5 - 392500*x^4 - 398339*x^3 - 256433*x^2 + 77138*x + 431614)/sqrt(2*
x + 3)

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Sympy [A]  time = 37.9559, size = 94, normalized size = 0.9 \begin{align*} - \frac{27 \left (2 x + 3\right )^{\frac{13}{2}}}{1664} + \frac{567 \left (2 x + 3\right )^{\frac{11}{2}}}{1408} - \frac{391 \left (2 x + 3\right )^{\frac{9}{2}}}{128} + \frac{10475 \left (2 x + 3\right )^{\frac{7}{2}}}{896} - \frac{17201 \left (2 x + 3\right )^{\frac{5}{2}}}{640} + \frac{5335 \left (2 x + 3\right )^{\frac{3}{2}}}{128} - \frac{7925 \sqrt{2 x + 3}}{128} - \frac{1625}{128 \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)**3/(3+2*x)**(3/2),x)

[Out]

-27*(2*x + 3)**(13/2)/1664 + 567*(2*x + 3)**(11/2)/1408 - 391*(2*x + 3)**(9/2)/128 + 10475*(2*x + 3)**(7/2)/89
6 - 17201*(2*x + 3)**(5/2)/640 + 5335*(2*x + 3)**(3/2)/128 - 7925*sqrt(2*x + 3)/128 - 1625/(128*sqrt(2*x + 3))

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Giac [A]  time = 1.10308, size = 99, normalized size = 0.94 \begin{align*} -\frac{27}{1664} \,{\left (2 \, x + 3\right )}^{\frac{13}{2}} + \frac{567}{1408} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} - \frac{391}{128} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} + \frac{10475}{896} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - \frac{17201}{640} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{5335}{128} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - \frac{7925}{128} \, \sqrt{2 \, x + 3} - \frac{1625}{128 \, \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)^3/(3+2*x)^(3/2),x, algorithm="giac")

[Out]

-27/1664*(2*x + 3)^(13/2) + 567/1408*(2*x + 3)^(11/2) - 391/128*(2*x + 3)^(9/2) + 10475/896*(2*x + 3)^(7/2) -
17201/640*(2*x + 3)^(5/2) + 5335/128*(2*x + 3)^(3/2) - 7925/128*sqrt(2*x + 3) - 1625/128/sqrt(2*x + 3)