3.2544 \(\int (5-x) (3+2 x)^{3/2} (2+5 x+3 x^2)^3 \, dx\)

Optimal. Leaf size=105 \[ -\frac{27 (2 x+3)^{19/2}}{2432}+\frac{567 (2 x+3)^{17/2}}{2176}-\frac{1173}{640} (2 x+3)^{15/2}+\frac{10475 (2 x+3)^{13/2}}{1664}-\frac{17201 (2 x+3)^{11/2}}{1408}+\frac{5335}{384} (2 x+3)^{9/2}-\frac{7925}{896} (2 x+3)^{7/2}+\frac{325}{128} (2 x+3)^{5/2} \]

[Out]

(325*(3 + 2*x)^(5/2))/128 - (7925*(3 + 2*x)^(7/2))/896 + (5335*(3 + 2*x)^(9/2))/384 - (17201*(3 + 2*x)^(11/2))
/1408 + (10475*(3 + 2*x)^(13/2))/1664 - (1173*(3 + 2*x)^(15/2))/640 + (567*(3 + 2*x)^(17/2))/2176 - (27*(3 + 2
*x)^(19/2))/2432

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Rubi [A]  time = 0.0325764, 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} \[ -\frac{27 (2 x+3)^{19/2}}{2432}+\frac{567 (2 x+3)^{17/2}}{2176}-\frac{1173}{640} (2 x+3)^{15/2}+\frac{10475 (2 x+3)^{13/2}}{1664}-\frac{17201 (2 x+3)^{11/2}}{1408}+\frac{5335}{384} (2 x+3)^{9/2}-\frac{7925}{896} (2 x+3)^{7/2}+\frac{325}{128} (2 x+3)^{5/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(325*(3 + 2*x)^(5/2))/128 - (7925*(3 + 2*x)^(7/2))/896 + (5335*(3 + 2*x)^(9/2))/384 - (17201*(3 + 2*x)^(11/2))
/1408 + (10475*(3 + 2*x)^(13/2))/1664 - (1173*(3 + 2*x)^(15/2))/640 + (567*(3 + 2*x)^(17/2))/2176 - (27*(3 + 2
*x)^(19/2))/2432

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) (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^3 \, dx &=\int \left (\frac{1625}{128} (3+2 x)^{3/2}-\frac{7925}{128} (3+2 x)^{5/2}+\frac{16005}{128} (3+2 x)^{7/2}-\frac{17201}{128} (3+2 x)^{9/2}+\frac{10475}{128} (3+2 x)^{11/2}-\frac{3519}{128} (3+2 x)^{13/2}+\frac{567}{128} (3+2 x)^{15/2}-\frac{27}{128} (3+2 x)^{17/2}\right ) \, dx\\ &=\frac{325}{128} (3+2 x)^{5/2}-\frac{7925}{896} (3+2 x)^{7/2}+\frac{5335}{384} (3+2 x)^{9/2}-\frac{17201 (3+2 x)^{11/2}}{1408}+\frac{10475 (3+2 x)^{13/2}}{1664}-\frac{1173}{640} (3+2 x)^{15/2}+\frac{567 (3+2 x)^{17/2}}{2176}-\frac{27 (3+2 x)^{19/2}}{2432}\\ \end{align*}

Mathematica [A]  time = 0.0197936, size = 48, normalized size = 0.46 \[ -\frac{(2 x+3)^{5/2} \left (6891885 x^7-8513505 x^6-117819702 x^5-270695040 x^4-295054725 x^3-173763625 x^2-53367570 x-6778218\right )}{4849845} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((3 + 2*x)^(5/2)*(-6778218 - 53367570*x - 173763625*x^2 - 295054725*x^3 - 270695040*x^4 - 117819702*x^5 - 851
3505*x^6 + 6891885*x^7))/4849845

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Maple [A]  time = 0.004, size = 45, normalized size = 0.4 \begin{align*} -{\frac{6891885\,{x}^{7}-8513505\,{x}^{6}-117819702\,{x}^{5}-270695040\,{x}^{4}-295054725\,{x}^{3}-173763625\,{x}^{2}-53367570\,x-6778218}{4849845} \left ( 3+2\,x \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3+2*x)^(3/2)*(3*x^2+5*x+2)^3,x)

[Out]

-1/4849845*(6891885*x^7-8513505*x^6-117819702*x^5-270695040*x^4-295054725*x^3-173763625*x^2-53367570*x-6778218
)*(3+2*x)^(5/2)

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Maxima [A]  time = 0.98429, size = 99, normalized size = 0.94 \begin{align*} -\frac{27}{2432} \,{\left (2 \, x + 3\right )}^{\frac{19}{2}} + \frac{567}{2176} \,{\left (2 \, x + 3\right )}^{\frac{17}{2}} - \frac{1173}{640} \,{\left (2 \, x + 3\right )}^{\frac{15}{2}} + \frac{10475}{1664} \,{\left (2 \, x + 3\right )}^{\frac{13}{2}} - \frac{17201}{1408} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} + \frac{5335}{384} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} - \frac{7925}{896} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} + \frac{325}{128} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^(3/2)*(3*x^2+5*x+2)^3,x, algorithm="maxima")

[Out]

-27/2432*(2*x + 3)^(19/2) + 567/2176*(2*x + 3)^(17/2) - 1173/640*(2*x + 3)^(15/2) + 10475/1664*(2*x + 3)^(13/2
) - 17201/1408*(2*x + 3)^(11/2) + 5335/384*(2*x + 3)^(9/2) - 7925/896*(2*x + 3)^(7/2) + 325/128*(2*x + 3)^(5/2
)

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Fricas [A]  time = 1.76531, size = 246, normalized size = 2.34 \begin{align*} -\frac{1}{4849845} \,{\left (27567540 \, x^{9} + 48648600 \, x^{8} - 511413903 \, x^{7} - 2573238129 \, x^{6} - 5488936698 \, x^{5} - 6671966560 \, x^{4} - 4954126305 \, x^{3} - 2231396337 \, x^{2} - 561646746 \, x - 61003962\right )} \sqrt{2 \, x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^(3/2)*(3*x^2+5*x+2)^3,x, algorithm="fricas")

[Out]

-1/4849845*(27567540*x^9 + 48648600*x^8 - 511413903*x^7 - 2573238129*x^6 - 5488936698*x^5 - 6671966560*x^4 - 4
954126305*x^3 - 2231396337*x^2 - 561646746*x - 61003962)*sqrt(2*x + 3)

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Sympy [A]  time = 31.9655, size = 94, normalized size = 0.9 \begin{align*} - \frac{27 \left (2 x + 3\right )^{\frac{19}{2}}}{2432} + \frac{567 \left (2 x + 3\right )^{\frac{17}{2}}}{2176} - \frac{1173 \left (2 x + 3\right )^{\frac{15}{2}}}{640} + \frac{10475 \left (2 x + 3\right )^{\frac{13}{2}}}{1664} - \frac{17201 \left (2 x + 3\right )^{\frac{11}{2}}}{1408} + \frac{5335 \left (2 x + 3\right )^{\frac{9}{2}}}{384} - \frac{7925 \left (2 x + 3\right )^{\frac{7}{2}}}{896} + \frac{325 \left (2 x + 3\right )^{\frac{5}{2}}}{128} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)**(3/2)*(3*x**2+5*x+2)**3,x)

[Out]

-27*(2*x + 3)**(19/2)/2432 + 567*(2*x + 3)**(17/2)/2176 - 1173*(2*x + 3)**(15/2)/640 + 10475*(2*x + 3)**(13/2)
/1664 - 17201*(2*x + 3)**(11/2)/1408 + 5335*(2*x + 3)**(9/2)/384 - 7925*(2*x + 3)**(7/2)/896 + 325*(2*x + 3)**
(5/2)/128

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Giac [A]  time = 1.10778, size = 99, normalized size = 0.94 \begin{align*} -\frac{27}{2432} \,{\left (2 \, x + 3\right )}^{\frac{19}{2}} + \frac{567}{2176} \,{\left (2 \, x + 3\right )}^{\frac{17}{2}} - \frac{1173}{640} \,{\left (2 \, x + 3\right )}^{\frac{15}{2}} + \frac{10475}{1664} \,{\left (2 \, x + 3\right )}^{\frac{13}{2}} - \frac{17201}{1408} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} + \frac{5335}{384} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} - \frac{7925}{896} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} + \frac{325}{128} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^(3/2)*(3*x^2+5*x+2)^3,x, algorithm="giac")

[Out]

-27/2432*(2*x + 3)^(19/2) + 567/2176*(2*x + 3)^(17/2) - 1173/640*(2*x + 3)^(15/2) + 10475/1664*(2*x + 3)^(13/2
) - 17201/1408*(2*x + 3)^(11/2) + 5335/384*(2*x + 3)^(9/2) - 7925/896*(2*x + 3)^(7/2) + 325/128*(2*x + 3)^(5/2
)