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

Optimal. Leaf size=79 \[ -\frac {9}{416} (2 x+3)^{13/2}+\frac {15}{32} (2 x+3)^{11/2}-\frac {359}{144} (2 x+3)^{9/2}+\frac {93}{16} (2 x+3)^{7/2}-\frac {213}{32} (2 x+3)^{5/2}+\frac {325}{96} (2 x+3)^{3/2} \]

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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}{416} (2 x+3)^{13/2}+\frac {15}{32} (2 x+3)^{11/2}-\frac {359}{144} (2 x+3)^{9/2}+\frac {93}{16} (2 x+3)^{7/2}-\frac {213}{32} (2 x+3)^{5/2}+\frac {325}{96} (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)^2,x]

[Out]

(325*(3 + 2*x)^(3/2))/96 - (213*(3 + 2*x)^(5/2))/32 + (93*(3 + 2*x)^(7/2))/16 - (359*(3 + 2*x)^(9/2))/144 + (1
5*(3 + 2*x)^(11/2))/32 - (9*(3 + 2*x)^(13/2))/416

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 )^2 \, dx &=\int \left (\frac {325}{32} \sqrt {3+2 x}-\frac {1065}{32} (3+2 x)^{3/2}+\frac {651}{16} (3+2 x)^{5/2}-\frac {359}{16} (3+2 x)^{7/2}+\frac {165}{32} (3+2 x)^{9/2}-\frac {9}{32} (3+2 x)^{11/2}\right ) \, dx\\ &=\frac {325}{96} (3+2 x)^{3/2}-\frac {213}{32} (3+2 x)^{5/2}+\frac {93}{16} (3+2 x)^{7/2}-\frac {359}{144} (3+2 x)^{9/2}+\frac {15}{32} (3+2 x)^{11/2}-\frac {9}{416} (3+2 x)^{13/2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 38, normalized size = 0.48 \begin {gather*} -\frac {1}{117} (2 x+3)^{3/2} \left (81 x^5-270 x^4-1109 x^3-1332 x^2-648 x-132\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/117*((3 + 2*x)^(3/2)*(-132 - 648*x - 1332*x^2 - 1109*x^3 - 270*x^4 + 81*x^5))

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IntegrateAlgebraic [A]  time = 0.06, size = 58, normalized size = 0.73 \begin {gather*} -\frac {(2 x+3)^{3/2} \left (81 (2 x+3)^5-1755 (2 x+3)^4+9334 (2 x+3)^3-21762 (2 x+3)^2+24921 (2 x+3)-12675\right )}{3744} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3744*((3 + 2*x)^(3/2)*(-12675 + 24921*(3 + 2*x) - 21762*(3 + 2*x)^2 + 9334*(3 + 2*x)^3 - 1755*(3 + 2*x)^4 +
81*(3 + 2*x)^5))

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fricas [A]  time = 0.39, size = 39, normalized size = 0.49 \begin {gather*} -\frac {1}{117} \, {\left (162 \, x^{6} - 297 \, x^{5} - 3028 \, x^{4} - 5991 \, x^{3} - 5292 \, x^{2} - 2208 \, x - 396\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)^2*(3+2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/117*(162*x^6 - 297*x^5 - 3028*x^4 - 5991*x^3 - 5292*x^2 - 2208*x - 396)*sqrt(2*x + 3)

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giac [A]  time = 0.16, size = 55, normalized size = 0.70 \begin {gather*} -\frac {9}{416} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} + \frac {15}{32} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} - \frac {359}{144} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + \frac {93}{16} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - \frac {213}{32} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {325}{96} \, {\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)^2*(3+2*x)^(1/2),x, algorithm="giac")

[Out]

-9/416*(2*x + 3)^(13/2) + 15/32*(2*x + 3)^(11/2) - 359/144*(2*x + 3)^(9/2) + 93/16*(2*x + 3)^(7/2) - 213/32*(2
*x + 3)^(5/2) + 325/96*(2*x + 3)^(3/2)

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maple [A]  time = 0.00, size = 35, normalized size = 0.44 \begin {gather*} -\frac {\left (81 x^{5}-270 x^{4}-1109 x^{3}-1332 x^{2}-648 x -132\right ) \left (2 x +3\right )^{\frac {3}{2}}}{117} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/117*(81*x^5-270*x^4-1109*x^3-1332*x^2-648*x-132)*(2*x+3)^(3/2)

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maxima [A]  time = 0.64, size = 55, normalized size = 0.70 \begin {gather*} -\frac {9}{416} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} + \frac {15}{32} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} - \frac {359}{144} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + \frac {93}{16} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - \frac {213}{32} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {325}{96} \, {\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)^2*(3+2*x)^(1/2),x, algorithm="maxima")

[Out]

-9/416*(2*x + 3)^(13/2) + 15/32*(2*x + 3)^(11/2) - 359/144*(2*x + 3)^(9/2) + 93/16*(2*x + 3)^(7/2) - 213/32*(2
*x + 3)^(5/2) + 325/96*(2*x + 3)^(3/2)

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mupad [B]  time = 0.03, size = 55, normalized size = 0.70 \begin {gather*} \frac {325\,{\left (2\,x+3\right )}^{3/2}}{96}-\frac {213\,{\left (2\,x+3\right )}^{5/2}}{32}+\frac {93\,{\left (2\,x+3\right )}^{7/2}}{16}-\frac {359\,{\left (2\,x+3\right )}^{9/2}}{144}+\frac {15\,{\left (2\,x+3\right )}^{11/2}}{32}-\frac {9\,{\left (2\,x+3\right )}^{13/2}}{416} \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)^2,x)

[Out]

(325*(2*x + 3)^(3/2))/96 - (213*(2*x + 3)^(5/2))/32 + (93*(2*x + 3)^(7/2))/16 - (359*(2*x + 3)^(9/2))/144 + (1
5*(2*x + 3)^(11/2))/32 - (9*(2*x + 3)^(13/2))/416

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sympy [A]  time = 3.56, size = 70, normalized size = 0.89 \begin {gather*} - \frac {9 \left (2 x + 3\right )^{\frac {13}{2}}}{416} + \frac {15 \left (2 x + 3\right )^{\frac {11}{2}}}{32} - \frac {359 \left (2 x + 3\right )^{\frac {9}{2}}}{144} + \frac {93 \left (2 x + 3\right )^{\frac {7}{2}}}{16} - \frac {213 \left (2 x + 3\right )^{\frac {5}{2}}}{32} + \frac {325 \left (2 x + 3\right )^{\frac {3}{2}}}{96} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9*(2*x + 3)**(13/2)/416 + 15*(2*x + 3)**(11/2)/32 - 359*(2*x + 3)**(9/2)/144 + 93*(2*x + 3)**(7/2)/16 - 213*(
2*x + 3)**(5/2)/32 + 325*(2*x + 3)**(3/2)/96

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