∫ (5 − x)(3 + 2x)7∕2(2 + 5x + 3x2)3dx

3.2542 \(\int (5-x) (3+2 x)^{7/2} (2+5 x+3 x^2)^3 \, dx\)

Optimal. Leaf size=105 \[ -\frac{27 (2 x+3)^{23/2}}{2944}+\frac{27}{128} (2 x+3)^{21/2}-\frac{3519 (2 x+3)^{19/2}}{2432}+\frac{10475 (2 x+3)^{17/2}}{2176}-\frac{17201 (2 x+3)^{15/2}}{1920}+\frac{16005 (2 x+3)^{13/2}}{1664}-\frac{7925 (2 x+3)^{11/2}}{1408}+\frac{1625 (2 x+3)^{9/2}}{1152} \]

[Out]

(1625*(3 + 2*x)^(9/2))/1152 - (7925*(3 + 2*x)^(11/2))/1408 + (16005*(3 + 2*x)^(13/2))/1664 - (17201*(3 + 2*x)^

(15/2))/1920 + (10475*(3 + 2*x)^(17/2))/2176 - (3519*(3 + 2*x)^(19/2))/2432 + (27*(3 + 2*x)^(21/2))/128 - (27*

(3 + 2*x)^(23/2))/2944

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Rubi [A] time = 0.0352645, 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} \[ -\frac{27 (2 x+3)^{23/2}}{2944}+\frac{27}{128} (2 x+3)^{21/2}-\frac{3519 (2 x+3)^{19/2}}{2432}+\frac{10475 (2 x+3)^{17/2}}{2176}-\frac{17201 (2 x+3)^{15/2}}{1920}+\frac{16005 (2 x+3)^{13/2}}{1664}-\frac{7925 (2 x+3)^{11/2}}{1408}+\frac{1625 (2 x+3)^{9/2}}{1152} \]

Antiderivative was successfully veriﬁed.

[In]

Int[(5 - x)*(3 + 2*x)^(7/2)*(2 + 5*x + 3*x^2)^3,x]

[Out]

(1625*(3 + 2*x)^(9/2))/1152 - (7925*(3 + 2*x)^(11/2))/1408 + (16005*(3 + 2*x)^(13/2))/1664 - (17201*(3 + 2*x)^

(15/2))/1920 + (10475*(3 + 2*x)^(17/2))/2176 - (3519*(3 + 2*x)^(19/2))/2432 + (27*(3 + 2*x)^(21/2))/128 - (27*

(3 + 2*x)^(23/2))/2944

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)^{7/2} \left (2+5 x+3 x^2\right )^3 \, dx &=\int \left (\frac{1625}{128} (3+2 x)^{7/2}-\frac{7925}{128} (3+2 x)^{9/2}+\frac{16005}{128} (3+2 x)^{11/2}-\frac{17201}{128} (3+2 x)^{13/2}+\frac{10475}{128} (3+2 x)^{15/2}-\frac{3519}{128} (3+2 x)^{17/2}+\frac{567}{128} (3+2 x)^{19/2}-\frac{27}{128} (3+2 x)^{21/2}\right ) \, dx\\ &=\frac{1625 (3+2 x)^{9/2}}{1152}-\frac{7925 (3+2 x)^{11/2}}{1408}+\frac{16005 (3+2 x)^{13/2}}{1664}-\frac{17201 (3+2 x)^{15/2}}{1920}+\frac{10475 (3+2 x)^{17/2}}{2176}-\frac{3519 (3+2 x)^{19/2}}{2432}+\frac{27}{128} (3+2 x)^{21/2}-\frac{27 (3+2 x)^{23/2}}{2944}\\ \end{align*}

Mathematica [A] time = 0.0214748, size = 48, normalized size = 0.46 \[ -\frac{(2 x+3)^{9/2} \left (56119635 x^7-56119635 x^6-943203690 x^5-2232945000 x^4-2481091899 x^3-1481619843 x^2-460865502 x-58847566\right )}{47805615} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[(5 - x)*(3 + 2*x)^(7/2)*(2 + 5*x + 3*x^2)^3,x]

[Out]

-((3 + 2*x)^(9/2)*(-58847566 - 460865502*x - 1481619843*x^2 - 2481091899*x^3 - 2232945000*x^4 - 943203690*x^5

- 56119635*x^6 + 56119635*x^7))/47805615

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Maple [A] time = 0.005, size = 45, normalized size = 0.4 \begin{align*} -{\frac{56119635\,{x}^{7}-56119635\,{x}^{6}-943203690\,{x}^{5}-2232945000\,{x}^{4}-2481091899\,{x}^{3}-1481619843\,{x}^{2}-460865502\,x-58847566}{47805615} \left ( 3+2\,x \right ) ^{{\frac{9}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3+2*x)^(7/2)*(3*x^2+5*x+2)^3,x)

[Out]

-1/47805615*(56119635*x^7-56119635*x^6-943203690*x^5-2232945000*x^4-2481091899*x^3-1481619843*x^2-460865502*x-

58847566)*(3+2*x)^(9/2)

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Maxima [A] time = 0.968793, size = 99, normalized size = 0.94 \begin{align*} -\frac{27}{2944} \,{\left (2 \, x + 3\right )}^{\frac{23}{2}} + \frac{27}{128} \,{\left (2 \, x + 3\right )}^{\frac{21}{2}} - \frac{3519}{2432} \,{\left (2 \, x + 3\right )}^{\frac{19}{2}} + \frac{10475}{2176} \,{\left (2 \, x + 3\right )}^{\frac{17}{2}} - \frac{17201}{1920} \,{\left (2 \, x + 3\right )}^{\frac{15}{2}} + \frac{16005}{1664} \,{\left (2 \, x + 3\right )}^{\frac{13}{2}} - \frac{7925}{1408} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} + \frac{1625}{1152} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^(7/2)*(3*x^2+5*x+2)^3,x, algorithm="maxima")

[Out]

-27/2944*(2*x + 3)^(23/2) + 27/128*(2*x + 3)^(21/2) - 3519/2432*(2*x + 3)^(19/2) + 10475/2176*(2*x + 3)^(17/2)

- 17201/1920*(2*x + 3)^(15/2) + 16005/1664*(2*x + 3)^(13/2) - 7925/1408*(2*x + 3)^(11/2) + 1625/1152*(2*x + 3

)^(9/2)

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Fricas [A] time = 1.83013, size = 328, normalized size = 3.12 \begin{align*} -\frac{1}{47805615} \,{\left (897914160 \, x^{11} + 4489570800 \, x^{10} - 8356902840 \, x^{9} - 126274674240 \, x^{8} - 465368338149 \, x^{7} - 952484547267 \, x^{6} - 1244240822034 \, x^{5} - 1081998930520 \, x^{4} - 626194644675 \, x^{3} - 232269229971 \, x^{2} - 50041179918 \, x - 4766652846\right )} \sqrt{2 \, x + 3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^(7/2)*(3*x^2+5*x+2)^3,x, algorithm="fricas")

[Out]

-1/47805615*(897914160*x^11 + 4489570800*x^10 - 8356902840*x^9 - 126274674240*x^8 - 465368338149*x^7 - 9524845

47267*x^6 - 1244240822034*x^5 - 1081998930520*x^4 - 626194644675*x^3 - 232269229971*x^2 - 50041179918*x - 4766

652846)*sqrt(2*x + 3)

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Sympy [A] time = 46.5302, size = 94, normalized size = 0.9 \begin{align*} - \frac{27 \left (2 x + 3\right )^{\frac{23}{2}}}{2944} + \frac{27 \left (2 x + 3\right )^{\frac{21}{2}}}{128} - \frac{3519 \left (2 x + 3\right )^{\frac{19}{2}}}{2432} + \frac{10475 \left (2 x + 3\right )^{\frac{17}{2}}}{2176} - \frac{17201 \left (2 x + 3\right )^{\frac{15}{2}}}{1920} + \frac{16005 \left (2 x + 3\right )^{\frac{13}{2}}}{1664} - \frac{7925 \left (2 x + 3\right )^{\frac{11}{2}}}{1408} + \frac{1625 \left (2 x + 3\right )^{\frac{9}{2}}}{1152} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)**(7/2)*(3*x**2+5*x+2)**3,x)

[Out]

-27*(2*x + 3)**(23/2)/2944 + 27*(2*x + 3)**(21/2)/128 - 3519*(2*x + 3)**(19/2)/2432 + 10475*(2*x + 3)**(17/2)/

2176 - 17201*(2*x + 3)**(15/2)/1920 + 16005*(2*x + 3)**(13/2)/1664 - 7925*(2*x + 3)**(11/2)/1408 + 1625*(2*x +

3)**(9/2)/1152

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Giac [A] time = 1.11385, size = 99, normalized size = 0.94 \begin{align*} -\frac{27}{2944} \,{\left (2 \, x + 3\right )}^{\frac{23}{2}} + \frac{27}{128} \,{\left (2 \, x + 3\right )}^{\frac{21}{2}} - \frac{3519}{2432} \,{\left (2 \, x + 3\right )}^{\frac{19}{2}} + \frac{10475}{2176} \,{\left (2 \, x + 3\right )}^{\frac{17}{2}} - \frac{17201}{1920} \,{\left (2 \, x + 3\right )}^{\frac{15}{2}} + \frac{16005}{1664} \,{\left (2 \, x + 3\right )}^{\frac{13}{2}} - \frac{7925}{1408} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} + \frac{1625}{1152} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^(7/2)*(3*x^2+5*x+2)^3,x, algorithm="giac")

[Out]

-27/2944*(2*x + 3)^(23/2) + 27/128*(2*x + 3)^(21/2) - 3519/2432*(2*x + 3)^(19/2) + 10475/2176*(2*x + 3)^(17/2)

- 17201/1920*(2*x + 3)^(15/2) + 16005/1664*(2*x + 3)^(13/2) - 7925/1408*(2*x + 3)^(11/2) + 1625/1152*(2*x + 3

)^(9/2)

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