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

Optimal. Leaf size=79 \[ -\frac {9}{608} (2 x+3)^{19/2}+\frac {165}{544} (2 x+3)^{17/2}-\frac {359}{240} (2 x+3)^{15/2}+\frac {651}{208} (2 x+3)^{13/2}-\frac {1065}{352} (2 x+3)^{11/2}+\frac {325}{288} (2 x+3)^{9/2} \]

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Rubi [A]  time = 0.02, antiderivative size = 79, 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 {9}{608} (2 x+3)^{19/2}+\frac {165}{544} (2 x+3)^{17/2}-\frac {359}{240} (2 x+3)^{15/2}+\frac {651}{208} (2 x+3)^{13/2}-\frac {1065}{352} (2 x+3)^{11/2}+\frac {325}{288} (2 x+3)^{9/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(325*(3 + 2*x)^(9/2))/288 - (1065*(3 + 2*x)^(11/2))/352 + (651*(3 + 2*x)^(13/2))/208 - (359*(3 + 2*x)^(15/2))/
240 + (165*(3 + 2*x)^(17/2))/544 - (9*(3 + 2*x)^(19/2))/608

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 )^2 \, dx &=\int \left (\frac {325}{32} (3+2 x)^{7/2}-\frac {1065}{32} (3+2 x)^{9/2}+\frac {651}{16} (3+2 x)^{11/2}-\frac {359}{16} (3+2 x)^{13/2}+\frac {165}{32} (3+2 x)^{15/2}-\frac {9}{32} (3+2 x)^{17/2}\right ) \, dx\\ &=\frac {325}{288} (3+2 x)^{9/2}-\frac {1065}{352} (3+2 x)^{11/2}+\frac {651}{208} (3+2 x)^{13/2}-\frac {359}{240} (3+2 x)^{15/2}+\frac {165}{544} (3+2 x)^{17/2}-\frac {9}{608} (3+2 x)^{19/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 38, normalized size = 0.48 \begin {gather*} -\frac {(2 x+3)^{9/2} \left (984555 x^5-2702700 x^4-13495911 x^3-17037702 x^2-8846388 x-1670104\right )}{2078505} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2078505*((3 + 2*x)^(9/2)*(-1670104 - 8846388*x - 17037702*x^2 - 13495911*x^3 - 2702700*x^4 + 984555*x^5))

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IntegrateAlgebraic [A]  time = 0.04, size = 71, normalized size = 0.90 \begin {gather*} \frac {-984555 (2 x+3)^{19/2}[phone] (2 x+3)^{17/2}-99491106 (2 x+3)^{15/2}[phone] (2 x+3)^{13/2}-201237075 (2 x+3)^{11/2}[phone] (2 x+3)^{9/2}}{66512160} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(75057125*(3 + 2*x)^(9/2) - 201237075*(3 + 2*x)^(11/2) + 208170270*(3 + 2*x)^(13/2) - 99491106*(3 + 2*x)^(15/2
) + 20173725*(3 + 2*x)^(17/2) - 984555*(3 + 2*x)^(19/2))/66512160

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fricas [A]  time = 0.40, size = 54, normalized size = 0.68 \begin {gather*} -\frac {1}{2078505} \, {\left (15752880 \, x^{9} + 51274080 \, x^{8} - 262729896 \, x^{7} - 1939330008 \, x^{6} - 5196312621 \, x^{5} - 7690154020 \, x^{4} - 6844462215 \, x^{3} - 3651616134 \, x^{2} - 1077299892 \, x - 135278424\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)^2,x, algorithm="fricas")

[Out]

-1/2078505*(15752880*x^9 + 51274080*x^8 - 262729896*x^7 - 1939330008*x^6 - 5196312621*x^5 - 7690154020*x^4 - 6
844462215*x^3 - 3651616134*x^2 - 1077299892*x - 135278424)*sqrt(2*x + 3)

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giac [A]  time = 0.17, size = 55, normalized size = 0.70 \begin {gather*} -\frac {9}{608} \, {\left (2 \, x + 3\right )}^{\frac {19}{2}} + \frac {165}{544} \, {\left (2 \, x + 3\right )}^{\frac {17}{2}} - \frac {359}{240} \, {\left (2 \, x + 3\right )}^{\frac {15}{2}} + \frac {651}{208} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} - \frac {1065}{352} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + \frac {325}{288} \, {\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)^2,x, algorithm="giac")

[Out]

-9/608*(2*x + 3)^(19/2) + 165/544*(2*x + 3)^(17/2) - 359/240*(2*x + 3)^(15/2) + 651/208*(2*x + 3)^(13/2) - 106
5/352*(2*x + 3)^(11/2) + 325/288*(2*x + 3)^(9/2)

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maple [A]  time = 0.01, size = 35, normalized size = 0.44 \begin {gather*} -\frac {\left (984555 x^{5}-2702700 x^{4}-13495911 x^{3}-17037702 x^{2}-8846388 x -1670104\right ) \left (2 x +3\right )^{\frac {9}{2}}}{2078505} \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)^2,x)

[Out]

-1/2078505*(984555*x^5-2702700*x^4-13495911*x^3-17037702*x^2-8846388*x-1670104)*(2*x+3)^(9/2)

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maxima [A]  time = 0.53, size = 55, normalized size = 0.70 \begin {gather*} -\frac {9}{608} \, {\left (2 \, x + 3\right )}^{\frac {19}{2}} + \frac {165}{544} \, {\left (2 \, x + 3\right )}^{\frac {17}{2}} - \frac {359}{240} \, {\left (2 \, x + 3\right )}^{\frac {15}{2}} + \frac {651}{208} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} - \frac {1065}{352} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + \frac {325}{288} \, {\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)^2,x, algorithm="maxima")

[Out]

-9/608*(2*x + 3)^(19/2) + 165/544*(2*x + 3)^(17/2) - 359/240*(2*x + 3)^(15/2) + 651/208*(2*x + 3)^(13/2) - 106
5/352*(2*x + 3)^(11/2) + 325/288*(2*x + 3)^(9/2)

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mupad [B]  time = 2.39, size = 55, normalized size = 0.70 \begin {gather*} \frac {325\,{\left (2\,x+3\right )}^{9/2}}{288}-\frac {1065\,{\left (2\,x+3\right )}^{11/2}}{352}+\frac {651\,{\left (2\,x+3\right )}^{13/2}}{208}-\frac {359\,{\left (2\,x+3\right )}^{15/2}}{240}+\frac {165\,{\left (2\,x+3\right )}^{17/2}}{544}-\frac {9\,{\left (2\,x+3\right )}^{19/2}}{608} \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)^2,x)

[Out]

(325*(2*x + 3)^(9/2))/288 - (1065*(2*x + 3)^(11/2))/352 + (651*(2*x + 3)^(13/2))/208 - (359*(2*x + 3)^(15/2))/
240 + (165*(2*x + 3)^(17/2))/544 - (9*(2*x + 3)^(19/2))/608

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sympy [B]  time = 5.88, size = 146, normalized size = 1.85 \begin {gather*} - \frac {144 x^{9} \sqrt {2 x + 3}}{19} - \frac {7968 x^{8} \sqrt {2 x + 3}}{323} + \frac {612424 x^{7} \sqrt {2 x + 3}}{4845} + \frac {19589192 x^{6} \sqrt {2 x + 3}}{20995} + \frac {577368069 x^{5} \sqrt {2 x + 3}}{230945} + \frac {1538030804 x^{4} \sqrt {2 x + 3}}{415701} + \frac {456297481 x^{3} \sqrt {2 x + 3}}{138567} + \frac {405735126 x^{2} \sqrt {2 x + 3}}{230945} + \frac {119699988 x \sqrt {2 x + 3}}{230945} + \frac {15030936 \sqrt {2 x + 3}}{230945} \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)**2,x)

[Out]

-144*x**9*sqrt(2*x + 3)/19 - 7968*x**8*sqrt(2*x + 3)/323 + 612424*x**7*sqrt(2*x + 3)/4845 + 19589192*x**6*sqrt
(2*x + 3)/20995 + 577368069*x**5*sqrt(2*x + 3)/230945 + 1538030804*x**4*sqrt(2*x + 3)/415701 + 456297481*x**3*
sqrt(2*x + 3)/138567 + 405735126*x**2*sqrt(2*x + 3)/230945 + 119699988*x*sqrt(2*x + 3)/230945 + 15030936*sqrt(
2*x + 3)/230945

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