3.24.4 \(\int (5-x) \sqrt {3+2 x} (2+5 x+3 x^2)^3 \, dx\)

Optimal. Leaf size=105 \[ -\frac {27 (2 x+3)^{17/2}}{2176}+\frac {189}{640} (2 x+3)^{15/2}-\frac {3519 (2 x+3)^{13/2}}{1664}+\frac {10475 (2 x+3)^{11/2}}{1408}-\frac {17201 (2 x+3)^{9/2}}{1152}+\frac {16005}{896} (2 x+3)^{7/2}-\frac {1585}{128} (2 x+3)^{5/2}+\frac {1625}{384} (2 x+3)^{3/2} \]

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Rubi [A]  time = 0.03, 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)^{17/2}}{2176}+\frac {189}{640} (2 x+3)^{15/2}-\frac {3519 (2 x+3)^{13/2}}{1664}+\frac {10475 (2 x+3)^{11/2}}{1408}-\frac {17201 (2 x+3)^{9/2}}{1152}+\frac {16005}{896} (2 x+3)^{7/2}-\frac {1585}{128} (2 x+3)^{5/2}+\frac {1625}{384} (2 x+3)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1625*(3 + 2*x)^(3/2))/384 - (1585*(3 + 2*x)^(5/2))/128 + (16005*(3 + 2*x)^(7/2))/896 - (17201*(3 + 2*x)^(9/2)
)/1152 + (10475*(3 + 2*x)^(11/2))/1408 - (3519*(3 + 2*x)^(13/2))/1664 + (189*(3 + 2*x)^(15/2))/640 - (27*(3 +
2*x)^(17/2))/2176

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) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^3 \, dx &=\int \left (\frac {1625}{128} \sqrt {3+2 x}-\frac {7925}{128} (3+2 x)^{3/2}+\frac {16005}{128} (3+2 x)^{5/2}-\frac {17201}{128} (3+2 x)^{7/2}+\frac {10475}{128} (3+2 x)^{9/2}-\frac {3519}{128} (3+2 x)^{11/2}+\frac {567}{128} (3+2 x)^{13/2}-\frac {27}{128} (3+2 x)^{15/2}\right ) \, dx\\ &=\frac {1625}{384} (3+2 x)^{3/2}-\frac {1585}{128} (3+2 x)^{5/2}+\frac {16005}{896} (3+2 x)^{7/2}-\frac {17201 (3+2 x)^{9/2}}{1152}+\frac {10475 (3+2 x)^{11/2}}{1408}-\frac {3519 (3+2 x)^{13/2}}{1664}+\frac {189}{640} (3+2 x)^{15/2}-\frac {27 (3+2 x)^{17/2}}{2176}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 48, normalized size = 0.46 \begin {gather*} -\frac {(2 x+3)^{3/2} \left (1216215 x^7-1702701 x^6-20968794 x^5-47286540 x^4-50880095 x^3-29756385 x^2-9013014 x-1197186\right )}{765765} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/765765*((3 + 2*x)^(3/2)*(-1197186 - 9013014*x - 29756385*x^2 - 50880095*x^3 - 47286540*x^4 - 20968794*x^5 -
1702701*x^6 + 1216215*x^7))

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IntegrateAlgebraic [A]  time = 0.06, size = 93, normalized size = 0.89 \begin {gather*} \frac {-1216215 (2 x+3)^{17/2}[phone] (2 x+3)^{15/2}-207286695 (2 x+3)^{13/2}[phone] (2 x+3)^{11/2}-1463547085 (2 x+3)^{9/2}[phone] (2 x+3)^{7/2}-1213737525 (2 x+3)^{5/2}[phone] (2 x+3)^{3/2}}{98017920} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(414789375*(3 + 2*x)^(3/2) - 1213737525*(3 + 2*x)^(5/2) + 1750866975*(3 + 2*x)^(7/2) - 1463547085*(3 + 2*x)^(9
/2) + 729217125*(3 + 2*x)^(11/2) - 207286695*(3 + 2*x)^(13/2) + 28945917*(3 + 2*x)^(15/2) - 1216215*(3 + 2*x)^
(17/2))/98017920

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fricas [A]  time = 0.38, size = 49, normalized size = 0.47 \begin {gather*} -\frac {1}{765765} \, {\left (2432430 \, x^{8} + 243243 \, x^{7} - 47045691 \, x^{6} - 157479462 \, x^{5} - 243619810 \, x^{4} - 212153055 \, x^{3} - 107295183 \, x^{2} - 29433414 \, x - 3591558\right )} \sqrt {2 \, x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^3*(3+2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/765765*(2432430*x^8 + 243243*x^7 - 47045691*x^6 - 157479462*x^5 - 243619810*x^4 - 212153055*x^3 - 107295183
*x^2 - 29433414*x - 3591558)*sqrt(2*x + 3)

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giac [A]  time = 0.21, size = 73, normalized size = 0.70 \begin {gather*} -\frac {27}{2176} \, {\left (2 \, x + 3\right )}^{\frac {17}{2}} + \frac {189}{640} \, {\left (2 \, x + 3\right )}^{\frac {15}{2}} - \frac {3519}{1664} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} + \frac {10475}{1408} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} - \frac {17201}{1152} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + \frac {16005}{896} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - \frac {1585}{128} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {1625}{384} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^3*(3+2*x)^(1/2),x, algorithm="giac")

[Out]

-27/2176*(2*x + 3)^(17/2) + 189/640*(2*x + 3)^(15/2) - 3519/1664*(2*x + 3)^(13/2) + 10475/1408*(2*x + 3)^(11/2
) - 17201/1152*(2*x + 3)^(9/2) + 16005/896*(2*x + 3)^(7/2) - 1585/128*(2*x + 3)^(5/2) + 1625/384*(2*x + 3)^(3/
2)

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maple [A]  time = 0.00, size = 45, normalized size = 0.43 \begin {gather*} -\frac {\left (1216215 x^{7}-1702701 x^{6}-20968794 x^{5}-47286540 x^{4}-50880095 x^{3}-29756385 x^{2}-9013014 x -1197186\right ) \left (2 x +3\right )^{\frac {3}{2}}}{765765} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+5*x+2)^3*(2*x+3)^(1/2),x)

[Out]

-1/765765*(1216215*x^7-1702701*x^6-20968794*x^5-47286540*x^4-50880095*x^3-29756385*x^2-9013014*x-1197186)*(2*x
+3)^(3/2)

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maxima [A]  time = 0.67, size = 73, normalized size = 0.70 \begin {gather*} -\frac {27}{2176} \, {\left (2 \, x + 3\right )}^{\frac {17}{2}} + \frac {189}{640} \, {\left (2 \, x + 3\right )}^{\frac {15}{2}} - \frac {3519}{1664} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} + \frac {10475}{1408} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} - \frac {17201}{1152} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + \frac {16005}{896} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - \frac {1585}{128} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {1625}{384} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^3*(3+2*x)^(1/2),x, algorithm="maxima")

[Out]

-27/2176*(2*x + 3)^(17/2) + 189/640*(2*x + 3)^(15/2) - 3519/1664*(2*x + 3)^(13/2) + 10475/1408*(2*x + 3)^(11/2
) - 17201/1152*(2*x + 3)^(9/2) + 16005/896*(2*x + 3)^(7/2) - 1585/128*(2*x + 3)^(5/2) + 1625/384*(2*x + 3)^(3/
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 )}^{3/2}}{384}-\frac {1585\,{\left (2\,x+3\right )}^{5/2}}{128}+\frac {16005\,{\left (2\,x+3\right )}^{7/2}}{896}-\frac {17201\,{\left (2\,x+3\right )}^{9/2}}{1152}+\frac {10475\,{\left (2\,x+3\right )}^{11/2}}{1408}-\frac {3519\,{\left (2\,x+3\right )}^{13/2}}{1664}+\frac {189\,{\left (2\,x+3\right )}^{15/2}}{640}-\frac {27\,{\left (2\,x+3\right )}^{17/2}}{2176} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x + 3)^(1/2)*(x - 5)*(5*x + 3*x^2 + 2)^3,x)

[Out]

(1625*(2*x + 3)^(3/2))/384 - (1585*(2*x + 3)^(5/2))/128 + (16005*(2*x + 3)^(7/2))/896 - (17201*(2*x + 3)^(9/2)
)/1152 + (10475*(2*x + 3)^(11/2))/1408 - (3519*(2*x + 3)^(13/2))/1664 + (189*(2*x + 3)^(15/2))/640 - (27*(2*x
+ 3)^(17/2))/2176

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sympy [A]  time = 4.02, size = 94, normalized size = 0.90 \begin {gather*} - \frac {27 \left (2 x + 3\right )^{\frac {17}{2}}}{2176} + \frac {189 \left (2 x + 3\right )^{\frac {15}{2}}}{640} - \frac {3519 \left (2 x + 3\right )^{\frac {13}{2}}}{1664} + \frac {10475 \left (2 x + 3\right )^{\frac {11}{2}}}{1408} - \frac {17201 \left (2 x + 3\right )^{\frac {9}{2}}}{1152} + \frac {16005 \left (2 x + 3\right )^{\frac {7}{2}}}{896} - \frac {1585 \left (2 x + 3\right )^{\frac {5}{2}}}{128} + \frac {1625 \left (2 x + 3\right )^{\frac {3}{2}}}{384} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x**2+5*x+2)**3*(3+2*x)**(1/2),x)

[Out]

-27*(2*x + 3)**(17/2)/2176 + 189*(2*x + 3)**(15/2)/640 - 3519*(2*x + 3)**(13/2)/1664 + 10475*(2*x + 3)**(11/2)
/1408 - 17201*(2*x + 3)**(9/2)/1152 + 16005*(2*x + 3)**(7/2)/896 - 1585*(2*x + 3)**(5/2)/128 + 1625*(2*x + 3)*
*(3/2)/384

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