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

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

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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)^{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} \end {gather*}

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*}

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

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.05, size = 93, normalized size = 0.89 \begin {gather*} \frac {-6891885 (2 x+3)^{19/2}[phone] (2 x+3)^{17/2}-1137773637 (2 x+3)^{15/2}[phone] (2 x+3)^{13/2}-7583834895 (2 x+3)^{11/2}[phone] (2 x+3)^{9/2}-5490717375 (2 x+3)^{7/2}[phone] (2 x+3)^{5/2}}{620780160} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1576199625*(3 + 2*x)^(5/2) - 5490717375*(3 + 2*x)^(7/2) + 8624641025*(3 + 2*x)^(9/2) - 7583834895*(3 + 2*x)^(
11/2) + 3907855875*(3 + 2*x)^(13/2) - 1137773637*(3 + 2*x)^(15/2) + 161756595*(3 + 2*x)^(17/2) - 6891885*(3 +
2*x)^(19/2))/620780160

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fricas [A]  time = 0.38, size = 54, normalized size = 0.51 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.17, size = 73, normalized size = 0.70 \begin {gather*} -\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 {gather*}

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
)

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maple [A]  time = 0.00, size = 45, normalized size = 0.43 \begin {gather*} -\frac {\left (6891885 x^{7}-8513505 x^{6}-117819702 x^{5}-270695040 x^{4}-295054725 x^{3}-173763625 x^{2}-53367570 x -6778218\right ) \left (2 x +3\right )^{\frac {5}{2}}}{4849845} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maxima [A]  time = 0.53, size = 73, normalized size = 0.70 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 0.03, size = 73, normalized size = 0.70 \begin {gather*} \frac {325\,{\left (2\,x+3\right )}^{5/2}}{128}-\frac {7925\,{\left (2\,x+3\right )}^{7/2}}{896}+\frac {5335\,{\left (2\,x+3\right )}^{9/2}}{384}-\frac {17201\,{\left (2\,x+3\right )}^{11/2}}{1408}+\frac {10475\,{\left (2\,x+3\right )}^{13/2}}{1664}-\frac {1173\,{\left (2\,x+3\right )}^{15/2}}{640}+\frac {567\,{\left (2\,x+3\right )}^{17/2}}{2176}-\frac {27\,{\left (2\,x+3\right )}^{19/2}}{2432} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 35.46, size = 94, normalized size = 0.90 \begin {gather*} - \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 {gather*}

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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