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

Optimal. Leaf size=79 \[ -\frac {9}{352} (2 x+3)^{11/2}+\frac {55}{96} (2 x+3)^{9/2}-\frac {359}{112} (2 x+3)^{7/2}+\frac {651}{80} (2 x+3)^{5/2}-\frac {355}{32} (2 x+3)^{3/2}+\frac {325}{32} \sqrt {2 x+3} \]

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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}{352} (2 x+3)^{11/2}+\frac {55}{96} (2 x+3)^{9/2}-\frac {359}{112} (2 x+3)^{7/2}+\frac {651}{80} (2 x+3)^{5/2}-\frac {355}{32} (2 x+3)^{3/2}+\frac {325}{32} \sqrt {2 x+3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(325*Sqrt[3 + 2*x])/32 - (355*(3 + 2*x)^(3/2))/32 + (651*(3 + 2*x)^(5/2))/80 - (359*(3 + 2*x)^(7/2))/112 + (55
*(3 + 2*x)^(9/2))/96 - (9*(3 + 2*x)^(11/2))/352

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 \frac {(5-x) \left (2+5 x+3 x^2\right )^2}{\sqrt {3+2 x}} \, dx &=\int \left (\frac {325}{32 \sqrt {3+2 x}}-\frac {1065}{32} \sqrt {3+2 x}+\frac {651}{16} (3+2 x)^{3/2}-\frac {359}{16} (3+2 x)^{5/2}+\frac {165}{32} (3+2 x)^{7/2}-\frac {9}{32} (3+2 x)^{9/2}\right ) \, dx\\ &=\frac {325}{32} \sqrt {3+2 x}-\frac {355}{32} (3+2 x)^{3/2}+\frac {651}{80} (3+2 x)^{5/2}-\frac {359}{112} (3+2 x)^{7/2}+\frac {55}{96} (3+2 x)^{9/2}-\frac {9}{352} (3+2 x)^{11/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 38, normalized size = 0.48 \begin {gather*} -\frac {\sqrt {2 x+3} \left (945 x^5-3500 x^4-12645 x^3-15354 x^2-6252 x-4344\right )}{1155} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/1155*(Sqrt[3 + 2*x]*(-4344 - 6252*x - 15354*x^2 - 12645*x^3 - 3500*x^4 + 945*x^5))

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IntegrateAlgebraic [A]  time = 0.04, size = 71, normalized size = 0.90 \begin {gather*} \frac {-945 (2 x+3)^{11/2}+21175 (2 x+3)^{9/2}-118470 (2 x+3)^{7/2}+300762 (2 x+3)^{5/2}-410025 (2 x+3)^{3/2}+375375 \sqrt {2 x+3}}{36960} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(375375*Sqrt[3 + 2*x] - 410025*(3 + 2*x)^(3/2) + 300762*(3 + 2*x)^(5/2) - 118470*(3 + 2*x)^(7/2) + 21175*(3 +
2*x)^(9/2) - 945*(3 + 2*x)^(11/2))/36960

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fricas [A]  time = 0.39, size = 34, normalized size = 0.43 \begin {gather*} -\frac {1}{1155} \, {\left (945 \, x^{5} - 3500 \, x^{4} - 12645 \, x^{3} - 15354 \, x^{2} - 6252 \, x - 4344\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]

-1/1155*(945*x^5 - 3500*x^4 - 12645*x^3 - 15354*x^2 - 6252*x - 4344)*sqrt(2*x + 3)

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giac [A]  time = 0.17, size = 55, normalized size = 0.70 \begin {gather*} -\frac {9}{352} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + \frac {55}{96} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} - \frac {359}{112} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + \frac {651}{80} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} - \frac {355}{32} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {325}{32} \, \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="giac")

[Out]

-9/352*(2*x + 3)^(11/2) + 55/96*(2*x + 3)^(9/2) - 359/112*(2*x + 3)^(7/2) + 651/80*(2*x + 3)^(5/2) - 355/32*(2
*x + 3)^(3/2) + 325/32*sqrt(2*x + 3)

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maple [A]  time = 0.00, size = 35, normalized size = 0.44 \begin {gather*} -\frac {\left (945 x^{5}-3500 x^{4}-12645 x^{3}-15354 x^{2}-6252 x -4344\right ) \sqrt {2 x +3}}{1155} \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]

-1/1155*(945*x^5-3500*x^4-12645*x^3-15354*x^2-6252*x-4344)*(2*x+3)^(1/2)

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maxima [A]  time = 0.52, size = 55, normalized size = 0.70 \begin {gather*} -\frac {9}{352} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + \frac {55}{96} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} - \frac {359}{112} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + \frac {651}{80} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} - \frac {355}{32} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {325}{32} \, \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="maxima")

[Out]

-9/352*(2*x + 3)^(11/2) + 55/96*(2*x + 3)^(9/2) - 359/112*(2*x + 3)^(7/2) + 651/80*(2*x + 3)^(5/2) - 355/32*(2
*x + 3)^(3/2) + 325/32*sqrt(2*x + 3)

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mupad [B]  time = 0.03, size = 55, normalized size = 0.70 \begin {gather*} \frac {325\,\sqrt {2\,x+3}}{32}-\frac {355\,{\left (2\,x+3\right )}^{3/2}}{32}+\frac {651\,{\left (2\,x+3\right )}^{5/2}}{80}-\frac {359\,{\left (2\,x+3\right )}^{7/2}}{112}+\frac {55\,{\left (2\,x+3\right )}^{9/2}}{96}-\frac {9\,{\left (2\,x+3\right )}^{11/2}}{352} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(325*(2*x + 3)^(1/2))/32 - (355*(2*x + 3)^(3/2))/32 + (651*(2*x + 3)^(5/2))/80 - (359*(2*x + 3)^(7/2))/112 + (
55*(2*x + 3)^(9/2))/96 - (9*(2*x + 3)^(11/2))/352

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sympy [A]  time = 85.81, size = 70, normalized size = 0.89 \begin {gather*} - \frac {9 \left (2 x + 3\right )^{\frac {11}{2}}}{352} + \frac {55 \left (2 x + 3\right )^{\frac {9}{2}}}{96} - \frac {359 \left (2 x + 3\right )^{\frac {7}{2}}}{112} + \frac {651 \left (2 x + 3\right )^{\frac {5}{2}}}{80} - \frac {355 \left (2 x + 3\right )^{\frac {3}{2}}}{32} + \frac {325 \sqrt {2 x + 3}}{32} \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)**(11/2)/352 + 55*(2*x + 3)**(9/2)/96 - 359*(2*x + 3)**(7/2)/112 + 651*(2*x + 3)**(5/2)/80 - 355*(
2*x + 3)**(3/2)/32 + 325*sqrt(2*x + 3)/32

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