3.2529 \(\int (5-x) \sqrt{3+2 x} (2+5 x+3 x^2) \, dx\)

Optimal. Leaf size=53 \[ -\frac{1}{24} (2 x+3)^{9/2}+\frac{47}{56} (2 x+3)^{7/2}-\frac{109}{40} (2 x+3)^{5/2}+\frac{65}{24} (2 x+3)^{3/2} \]

[Out]

(65*(3 + 2*x)^(3/2))/24 - (109*(3 + 2*x)^(5/2))/40 + (47*(3 + 2*x)^(7/2))/56 - (3 + 2*x)^(9/2)/24

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Rubi [A]  time = 0.015023, 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} \[ -\frac{1}{24} (2 x+3)^{9/2}+\frac{47}{56} (2 x+3)^{7/2}-\frac{109}{40} (2 x+3)^{5/2}+\frac{65}{24} (2 x+3)^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(65*(3 + 2*x)^(3/2))/24 - (109*(3 + 2*x)^(5/2))/40 + (47*(3 + 2*x)^(7/2))/56 - (3 + 2*x)^(9/2)/24

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 ) \, dx &=\int \left (\frac{65}{8} \sqrt{3+2 x}-\frac{109}{8} (3+2 x)^{3/2}+\frac{47}{8} (3+2 x)^{5/2}-\frac{3}{8} (3+2 x)^{7/2}\right ) \, dx\\ &=\frac{65}{24} (3+2 x)^{3/2}-\frac{109}{40} (3+2 x)^{5/2}+\frac{47}{56} (3+2 x)^{7/2}-\frac{1}{24} (3+2 x)^{9/2}\\ \end{align*}

Mathematica [A]  time = 0.0112848, size = 28, normalized size = 0.53 \[ -\frac{1}{105} (2 x+3)^{3/2} \left (35 x^3-195 x^2-249 x-101\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((3 + 2*x)^(3/2)*(-101 - 249*x - 195*x^2 + 35*x^3))/105

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Maple [A]  time = 0.005, size = 25, normalized size = 0.5 \begin{align*} -{\frac{35\,{x}^{3}-195\,{x}^{2}-249\,x-101}{105} \left ( 3+2\,x \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(35*x^3-195*x^2-249*x-101)*(3+2*x)^(3/2)

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Maxima [A]  time = 0.979925, size = 50, normalized size = 0.94 \begin{align*} -\frac{1}{24} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} + \frac{47}{56} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - \frac{109}{40} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{65}{24} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)*(3+2*x)^(1/2),x, algorithm="maxima")

[Out]

-1/24*(2*x + 3)^(9/2) + 47/56*(2*x + 3)^(7/2) - 109/40*(2*x + 3)^(5/2) + 65/24*(2*x + 3)^(3/2)

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Fricas [A]  time = 1.67587, size = 89, normalized size = 1.68 \begin{align*} -\frac{1}{105} \,{\left (70 \, x^{4} - 285 \, x^{3} - 1083 \, x^{2} - 949 \, x - 303\right )} \sqrt{2 \, x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)*(3+2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/105*(70*x^4 - 285*x^3 - 1083*x^2 - 949*x - 303)*sqrt(2*x + 3)

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Sympy [A]  time = 2.13686, size = 44, normalized size = 0.83 \begin{align*} - \frac{\left (2 x + 3\right )^{\frac{9}{2}}}{24} + \frac{47 \left (2 x + 3\right )^{\frac{7}{2}}}{56} - \frac{109 \left (2 x + 3\right )^{\frac{5}{2}}}{40} + \frac{65 \left (2 x + 3\right )^{\frac{3}{2}}}{24} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*x + 3)**(9/2)/24 + 47*(2*x + 3)**(7/2)/56 - 109*(2*x + 3)**(5/2)/40 + 65*(2*x + 3)**(3/2)/24

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Giac [A]  time = 1.08475, size = 50, normalized size = 0.94 \begin{align*} -\frac{1}{24} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} + \frac{47}{56} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - \frac{109}{40} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{65}{24} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)*(3+2*x)^(1/2),x, algorithm="giac")

[Out]

-1/24*(2*x + 3)^(9/2) + 47/56*(2*x + 3)^(7/2) - 109/40*(2*x + 3)^(5/2) + 65/24*(2*x + 3)^(3/2)