3.2528 \(\int (5-x) (3+2 x)^{3/2} (2+5 x+3 x^2) \, dx\)

Optimal. Leaf size=53 \[ -\frac{3}{88} (2 x+3)^{11/2}+\frac{47}{72} (2 x+3)^{9/2}-\frac{109}{56} (2 x+3)^{7/2}+\frac{13}{8} (2 x+3)^{5/2} \]

[Out]

(13*(3 + 2*x)^(5/2))/8 - (109*(3 + 2*x)^(7/2))/56 + (47*(3 + 2*x)^(9/2))/72 - (3*(3 + 2*x)^(11/2))/88

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Rubi [A]  time = 0.0197465, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {771} \[ -\frac{3}{88} (2 x+3)^{11/2}+\frac{47}{72} (2 x+3)^{9/2}-\frac{109}{56} (2 x+3)^{7/2}+\frac{13}{8} (2 x+3)^{5/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(13*(3 + 2*x)^(5/2))/8 - (109*(3 + 2*x)^(7/2))/56 + (47*(3 + 2*x)^(9/2))/72 - (3*(3 + 2*x)^(11/2))/88

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 ) \, dx &=\int \left (\frac{65}{8} (3+2 x)^{3/2}-\frac{109}{8} (3+2 x)^{5/2}+\frac{47}{8} (3+2 x)^{7/2}-\frac{3}{8} (3+2 x)^{9/2}\right ) \, dx\\ &=\frac{13}{8} (3+2 x)^{5/2}-\frac{109}{56} (3+2 x)^{7/2}+\frac{47}{72} (3+2 x)^{9/2}-\frac{3}{88} (3+2 x)^{11/2}\\ \end{align*}

Mathematica [A]  time = 0.011594, size = 28, normalized size = 0.53 \[ -\frac{1}{693} (2 x+3)^{5/2} \left (189 x^3-959 x^2-1455 x-513\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((3 + 2*x)^(5/2)*(-513 - 1455*x - 959*x^2 + 189*x^3))/693

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Maple [A]  time = 0.003, size = 25, normalized size = 0.5 \begin{align*} -{\frac{189\,{x}^{3}-959\,{x}^{2}-1455\,x-513}{693} \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),x)

[Out]

-1/693*(189*x^3-959*x^2-1455*x-513)*(3+2*x)^(5/2)

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Maxima [A]  time = 1.05848, size = 50, normalized size = 0.94 \begin{align*} -\frac{3}{88} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} + \frac{47}{72} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} - \frac{109}{56} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} + \frac{13}{8} \,{\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),x, algorithm="maxima")

[Out]

-3/88*(2*x + 3)^(11/2) + 47/72*(2*x + 3)^(9/2) - 109/56*(2*x + 3)^(7/2) + 13/8*(2*x + 3)^(5/2)

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Fricas [A]  time = 1.90197, size = 113, normalized size = 2.13 \begin{align*} -\frac{1}{693} \,{\left (756 \, x^{5} - 1568 \, x^{4} - 15627 \, x^{3} - 28143 \, x^{2} - 19251 \, x - 4617\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),x, algorithm="fricas")

[Out]

-1/693*(756*x^5 - 1568*x^4 - 15627*x^3 - 28143*x^2 - 19251*x - 4617)*sqrt(2*x + 3)

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Sympy [A]  time = 11.973, size = 46, normalized size = 0.87 \begin{align*} - \frac{3 \left (2 x + 3\right )^{\frac{11}{2}}}{88} + \frac{47 \left (2 x + 3\right )^{\frac{9}{2}}}{72} - \frac{109 \left (2 x + 3\right )^{\frac{7}{2}}}{56} + \frac{13 \left (2 x + 3\right )^{\frac{5}{2}}}{8} \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),x)

[Out]

-3*(2*x + 3)**(11/2)/88 + 47*(2*x + 3)**(9/2)/72 - 109*(2*x + 3)**(7/2)/56 + 13*(2*x + 3)**(5/2)/8

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Giac [A]  time = 1.09702, size = 50, normalized size = 0.94 \begin{align*} -\frac{3}{88} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} + \frac{47}{72} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} - \frac{109}{56} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} + \frac{13}{8} \,{\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),x, algorithm="giac")

[Out]

-3/88*(2*x + 3)^(11/2) + 47/72*(2*x + 3)^(9/2) - 109/56*(2*x + 3)^(7/2) + 13/8*(2*x + 3)^(5/2)