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

Optimal. Leaf size=79 \[ -\frac {9}{544} (2 x+3)^{17/2}+\frac {11}{32} (2 x+3)^{15/2}-\frac {359}{208} (2 x+3)^{13/2}+\frac {651}{176} (2 x+3)^{11/2}-\frac {355}{96} (2 x+3)^{9/2}+\frac {325}{224} (2 x+3)^{7/2} \]

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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}{544} (2 x+3)^{17/2}+\frac {11}{32} (2 x+3)^{15/2}-\frac {359}{208} (2 x+3)^{13/2}+\frac {651}{176} (2 x+3)^{11/2}-\frac {355}{96} (2 x+3)^{9/2}+\frac {325}{224} (2 x+3)^{7/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(325*(3 + 2*x)^(7/2))/224 - (355*(3 + 2*x)^(9/2))/96 + (651*(3 + 2*x)^(11/2))/176 - (359*(3 + 2*x)^(13/2))/208
+ (11*(3 + 2*x)^(15/2))/32 - (9*(3 + 2*x)^(17/2))/544

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)^{5/2} \left (2+5 x+3 x^2\right )^2 \, dx &=\int \left (\frac {325}{32} (3+2 x)^{5/2}-\frac {1065}{32} (3+2 x)^{7/2}+\frac {651}{16} (3+2 x)^{9/2}-\frac {359}{16} (3+2 x)^{11/2}+\frac {165}{32} (3+2 x)^{13/2}-\frac {9}{32} (3+2 x)^{15/2}\right ) \, dx\\ &=\frac {325}{224} (3+2 x)^{7/2}-\frac {355}{96} (3+2 x)^{9/2}+\frac {651}{176} (3+2 x)^{11/2}-\frac {359}{208} (3+2 x)^{13/2}+\frac {11}{32} (3+2 x)^{15/2}-\frac {9}{544} (3+2 x)^{17/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 38, normalized size = 0.48 \begin {gather*} -\frac {(2 x+3)^{7/2} \left (27027 x^5-78078 x^4-371679 x^3-461664 x^2-236768 x-44388\right )}{51051} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/51051*((3 + 2*x)^(7/2)*(-44388 - 236768*x - 461664*x^2 - 371679*x^3 - 78078*x^4 + 27027*x^5))

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IntegrateAlgebraic [A]  time = 0.05, size = 71, normalized size = 0.90 \begin {gather*} \frac {-27027 (2 x+3)^{17/2}+561561 (2 x+3)^{15/2}-2819586 (2 x+3)^{13/2}+6042582 (2 x+3)^{11/2}-6041035 (2 x+3)^{9/2}+2370225 (2 x+3)^{7/2}}{1633632} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2370225*(3 + 2*x)^(7/2) - 6041035*(3 + 2*x)^(9/2) + 6042582*(3 + 2*x)^(11/2) - 2819586*(3 + 2*x)^(13/2) + 561
561*(3 + 2*x)^(15/2) - 27027*(3 + 2*x)^(17/2))/1633632

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fricas [A]  time = 0.38, size = 49, normalized size = 0.62 \begin {gather*} -\frac {1}{51051} \, {\left (216216 \, x^{8} + 348348 \, x^{7} - 4324782 \, x^{6} - 20560239 \, x^{5} - 40692820 \, x^{4} - 43843941 \, x^{3} - 26848368 \, x^{2} - 8789688 \, x - 1198476\right )} \sqrt {2 \, x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/51051*(216216*x^8 + 348348*x^7 - 4324782*x^6 - 20560239*x^5 - 40692820*x^4 - 43843941*x^3 - 26848368*x^2 -
8789688*x - 1198476)*sqrt(2*x + 3)

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giac [A]  time = 0.16, size = 55, normalized size = 0.70 \begin {gather*} -\frac {9}{544} \, {\left (2 \, x + 3\right )}^{\frac {17}{2}} + \frac {11}{32} \, {\left (2 \, x + 3\right )}^{\frac {15}{2}} - \frac {359}{208} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} + \frac {651}{176} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} - \frac {355}{96} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + \frac {325}{224} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/544*(2*x + 3)^(17/2) + 11/32*(2*x + 3)^(15/2) - 359/208*(2*x + 3)^(13/2) + 651/176*(2*x + 3)^(11/2) - 355/9
6*(2*x + 3)^(9/2) + 325/224*(2*x + 3)^(7/2)

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maple [A]  time = 0.01, size = 35, normalized size = 0.44 \begin {gather*} -\frac {\left (27027 x^{5}-78078 x^{4}-371679 x^{3}-461664 x^{2}-236768 x -44388\right ) \left (2 x +3\right )^{\frac {7}{2}}}{51051} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/51051*(27027*x^5-78078*x^4-371679*x^3-461664*x^2-236768*x-44388)*(2*x+3)^(7/2)

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maxima [A]  time = 0.48, size = 55, normalized size = 0.70 \begin {gather*} -\frac {9}{544} \, {\left (2 \, x + 3\right )}^{\frac {17}{2}} + \frac {11}{32} \, {\left (2 \, x + 3\right )}^{\frac {15}{2}} - \frac {359}{208} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} + \frac {651}{176} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} - \frac {355}{96} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + \frac {325}{224} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/544*(2*x + 3)^(17/2) + 11/32*(2*x + 3)^(15/2) - 359/208*(2*x + 3)^(13/2) + 651/176*(2*x + 3)^(11/2) - 355/9
6*(2*x + 3)^(9/2) + 325/224*(2*x + 3)^(7/2)

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mupad [B]  time = 0.03, size = 55, normalized size = 0.70 \begin {gather*} \frac {325\,{\left (2\,x+3\right )}^{7/2}}{224}-\frac {355\,{\left (2\,x+3\right )}^{9/2}}{96}+\frac {651\,{\left (2\,x+3\right )}^{11/2}}{176}-\frac {359\,{\left (2\,x+3\right )}^{13/2}}{208}+\frac {11\,{\left (2\,x+3\right )}^{15/2}}{32}-\frac {9\,{\left (2\,x+3\right )}^{17/2}}{544} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(325*(2*x + 3)^(7/2))/224 - (355*(2*x + 3)^(9/2))/96 + (651*(2*x + 3)^(11/2))/176 - (359*(2*x + 3)^(13/2))/208
+ (11*(2*x + 3)^(15/2))/32 - (9*(2*x + 3)^(17/2))/544

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sympy [A]  time = 29.68, size = 70, normalized size = 0.89 \begin {gather*} - \frac {9 \left (2 x + 3\right )^{\frac {17}{2}}}{544} + \frac {11 \left (2 x + 3\right )^{\frac {15}{2}}}{32} - \frac {359 \left (2 x + 3\right )^{\frac {13}{2}}}{208} + \frac {651 \left (2 x + 3\right )^{\frac {11}{2}}}{176} - \frac {355 \left (2 x + 3\right )^{\frac {9}{2}}}{96} + \frac {325 \left (2 x + 3\right )^{\frac {7}{2}}}{224} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9*(2*x + 3)**(17/2)/544 + 11*(2*x + 3)**(15/2)/32 - 359*(2*x + 3)**(13/2)/208 + 651*(2*x + 3)**(11/2)/176 - 3
55*(2*x + 3)**(9/2)/96 + 325*(2*x + 3)**(7/2)/224

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