3.2536 \(\int (5-x) (3+2 x)^{3/2} (2+5 x+3 x^2)^2 \, dx\)

Optimal. Leaf size=79 \[ -\frac{3}{160} (2 x+3)^{15/2}+\frac{165}{416} (2 x+3)^{13/2}-\frac{359}{176} (2 x+3)^{11/2}+\frac{217}{48} (2 x+3)^{9/2}-\frac{1065}{224} (2 x+3)^{7/2}+\frac{65}{32} (2 x+3)^{5/2} \]

[Out]

(65*(3 + 2*x)^(5/2))/32 - (1065*(3 + 2*x)^(7/2))/224 + (217*(3 + 2*x)^(9/2))/48 - (359*(3 + 2*x)^(11/2))/176 +
(165*(3 + 2*x)^(13/2))/416 - (3*(3 + 2*x)^(15/2))/160

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Rubi [A]  time = 0.023153, antiderivative size = 79, normalized size of antiderivative = 1., 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} \[ -\frac{3}{160} (2 x+3)^{15/2}+\frac{165}{416} (2 x+3)^{13/2}-\frac{359}{176} (2 x+3)^{11/2}+\frac{217}{48} (2 x+3)^{9/2}-\frac{1065}{224} (2 x+3)^{7/2}+\frac{65}{32} (2 x+3)^{5/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(65*(3 + 2*x)^(5/2))/32 - (1065*(3 + 2*x)^(7/2))/224 + (217*(3 + 2*x)^(9/2))/48 - (359*(3 + 2*x)^(11/2))/176 +
(165*(3 + 2*x)^(13/2))/416 - (3*(3 + 2*x)^(15/2))/160

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 )^2 \, dx &=\int \left (\frac{325}{32} (3+2 x)^{3/2}-\frac{1065}{32} (3+2 x)^{5/2}+\frac{651}{16} (3+2 x)^{7/2}-\frac{359}{16} (3+2 x)^{9/2}+\frac{165}{32} (3+2 x)^{11/2}-\frac{9}{32} (3+2 x)^{13/2}\right ) \, dx\\ &=\frac{65}{32} (3+2 x)^{5/2}-\frac{1065}{224} (3+2 x)^{7/2}+\frac{217}{48} (3+2 x)^{9/2}-\frac{359}{176} (3+2 x)^{11/2}+\frac{165}{416} (3+2 x)^{13/2}-\frac{3}{160} (3+2 x)^{15/2}\\ \end{align*}

Mathematica [A]  time = 0.0158143, size = 38, normalized size = 0.48 \[ -\frac{(2 x+3)^{5/2} \left (9009 x^5-27720 x^4-124005 x^3-151270 x^2-76260 x-14304\right )}{15015} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((3 + 2*x)^(5/2)*(-14304 - 76260*x - 151270*x^2 - 124005*x^3 - 27720*x^4 + 9009*x^5))/15015

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Maple [A]  time = 0.004, size = 35, normalized size = 0.4 \begin{align*} -{\frac{9009\,{x}^{5}-27720\,{x}^{4}-124005\,{x}^{3}-151270\,{x}^{2}-76260\,x-14304}{15015} \left ( 3+2\,x \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15015*(9009*x^5-27720*x^4-124005*x^3-151270*x^2-76260*x-14304)*(3+2*x)^(5/2)

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Maxima [A]  time = 0.955985, size = 74, normalized size = 0.94 \begin{align*} -\frac{3}{160} \,{\left (2 \, x + 3\right )}^{\frac{15}{2}} + \frac{165}{416} \,{\left (2 \, x + 3\right )}^{\frac{13}{2}} - \frac{359}{176} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} + \frac{217}{48} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} - \frac{1065}{224} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} + \frac{65}{32} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} \end{align*}

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)^2,x, algorithm="maxima")

[Out]

-3/160*(2*x + 3)^(15/2) + 165/416*(2*x + 3)^(13/2) - 359/176*(2*x + 3)^(11/2) + 217/48*(2*x + 3)^(9/2) - 1065/
224*(2*x + 3)^(7/2) + 65/32*(2*x + 3)^(5/2)

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Fricas [A]  time = 1.74544, size = 165, normalized size = 2.09 \begin{align*} -\frac{1}{15015} \,{\left (36036 \, x^{7} - 2772 \, x^{6} - 747579 \, x^{5} - 2342620 \, x^{4} - 3236325 \, x^{3} - 2333766 \, x^{2} - 857988 \, x - 128736\right )} \sqrt{2 \, x + 3} \end{align*}

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)^2,x, algorithm="fricas")

[Out]

-1/15015*(36036*x^7 - 2772*x^6 - 747579*x^5 - 2342620*x^4 - 3236325*x^3 - 2333766*x^2 - 857988*x - 128736)*sqr
t(2*x + 3)

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Sympy [A]  time = 20.5515, size = 70, normalized size = 0.89 \begin{align*} - \frac{3 \left (2 x + 3\right )^{\frac{15}{2}}}{160} + \frac{165 \left (2 x + 3\right )^{\frac{13}{2}}}{416} - \frac{359 \left (2 x + 3\right )^{\frac{11}{2}}}{176} + \frac{217 \left (2 x + 3\right )^{\frac{9}{2}}}{48} - \frac{1065 \left (2 x + 3\right )^{\frac{7}{2}}}{224} + \frac{65 \left (2 x + 3\right )^{\frac{5}{2}}}{32} \end{align*}

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)**2,x)

[Out]

-3*(2*x + 3)**(15/2)/160 + 165*(2*x + 3)**(13/2)/416 - 359*(2*x + 3)**(11/2)/176 + 217*(2*x + 3)**(9/2)/48 - 1
065*(2*x + 3)**(7/2)/224 + 65*(2*x + 3)**(5/2)/32

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Giac [A]  time = 1.12704, size = 74, normalized size = 0.94 \begin{align*} -\frac{3}{160} \,{\left (2 \, x + 3\right )}^{\frac{15}{2}} + \frac{165}{416} \,{\left (2 \, x + 3\right )}^{\frac{13}{2}} - \frac{359}{176} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} + \frac{217}{48} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} - \frac{1065}{224} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} + \frac{65}{32} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} \end{align*}

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)^2,x, algorithm="giac")

[Out]

-3/160*(2*x + 3)^(15/2) + 165/416*(2*x + 3)^(13/2) - 359/176*(2*x + 3)^(11/2) + 217/48*(2*x + 3)^(9/2) - 1065/
224*(2*x + 3)^(7/2) + 65/32*(2*x + 3)^(5/2)