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

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

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Rubi [A]  time = 0.04, antiderivative size = 105, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.02, size = 48, normalized size = 0.46 \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/47805615*((3 + 2*x)^(9/2)*(-58847566 - 460865502*x - 1481619843*x^2 - 2481091899*x^3 - 2232945000*x^4 - 943
203690*x^5 - 56119635*x^6 + 56119635*x^7))

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IntegrateAlgebraic [A]  time = 0.05, size = 93, normalized size = 0.89 \begin {gather*} \frac {-56119635 (2 x+3)^{23/2}+1290751605 (2 x+3)^{21/2}-8854103115 (2 x+3)^{19/2}+29456695125 (2 x+3)^{17/2}-54820292241 (2 x+3)^{15/2}+58856066775 (2 x+3)^{13/2}-34441772625 (2 x+3)^{11/2}+8631569375 (2 x+3)^{9/2}}{6119118720} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(8631569375*(3 + 2*x)^(9/2) - 34441772625*(3 + 2*x)^(11/2) + 58856066775*(3 + 2*x)^(13/2) - 54820292241*(3 + 2
*x)^(15/2) + 29456695125*(3 + 2*x)^(17/2) - 8854103115*(3 + 2*x)^(19/2) + 1290751605*(3 + 2*x)^(21/2) - 561196
35*(3 + 2*x)^(23/2))/6119118720

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fricas [A]  time = 0.40, size = 64, normalized size = 0.61 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.18, size = 73, normalized size = 0.70 \begin {gather*} -\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 {gather*}

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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maple [A]  time = 0.00, size = 45, normalized size = 0.43 \begin {gather*} -\frac {\left (56119635 x^{7}-56119635 x^{6}-943203690 x^{5}-2232945000 x^{4}-2481091899 x^{3}-1481619843 x^{2}-460865502 x -58847566\right ) \left (2 x +3\right )^{\frac {9}{2}}}{47805615} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(2*x+3)^(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)*(2*x+3)^(9/2)

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maxima [A]  time = 0.71, size = 73, normalized size = 0.70 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 0.03, size = 73, normalized size = 0.70 \begin {gather*} \frac {1625\,{\left (2\,x+3\right )}^{9/2}}{1152}-\frac {7925\,{\left (2\,x+3\right )}^{11/2}}{1408}+\frac {16005\,{\left (2\,x+3\right )}^{13/2}}{1664}-\frac {17201\,{\left (2\,x+3\right )}^{15/2}}{1920}+\frac {10475\,{\left (2\,x+3\right )}^{17/2}}{2176}-\frac {3519\,{\left (2\,x+3\right )}^{19/2}}{2432}+\frac {27\,{\left (2\,x+3\right )}^{21/2}}{128}-\frac {27\,{\left (2\,x+3\right )}^{23/2}}{2944} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1625*(2*x + 3)^(9/2))/1152 - (7925*(2*x + 3)^(11/2))/1408 + (16005*(2*x + 3)^(13/2))/1664 - (17201*(2*x + 3)^
(15/2))/1920 + (10475*(2*x + 3)^(17/2))/2176 - (3519*(2*x + 3)^(19/2))/2432 + (27*(2*x + 3)^(21/2))/128 - (27*
(2*x + 3)^(23/2))/2944

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sympy [A]  time = 49.99, size = 94, normalized size = 0.90 \begin {gather*} - \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 {gather*}

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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