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

Optimal. Leaf size=53 \[ -\frac {3}{104} (2 x+3)^{13/2}+\frac {47}{88} (2 x+3)^{11/2}-\frac {109}{72} (2 x+3)^{9/2}+\frac {65}{56} (2 x+3)^{7/2} \]

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Rubi [A]  time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {771} \begin {gather*} -\frac {3}{104} (2 x+3)^{13/2}+\frac {47}{88} (2 x+3)^{11/2}-\frac {109}{72} (2 x+3)^{9/2}+\frac {65}{56} (2 x+3)^{7/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(65*(3 + 2*x)^(7/2))/56 - (109*(3 + 2*x)^(9/2))/72 + (47*(3 + 2*x)^(11/2))/88 - (3*(3 + 2*x)^(13/2))/104

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)^{5/2} \left (2+5 x+3 x^2\right ) \, dx &=\int \left (\frac {65}{8} (3+2 x)^{5/2}-\frac {109}{8} (3+2 x)^{7/2}+\frac {47}{8} (3+2 x)^{9/2}-\frac {3}{8} (3+2 x)^{11/2}\right ) \, dx\\ &=\frac {65}{56} (3+2 x)^{7/2}-\frac {109}{72} (3+2 x)^{9/2}+\frac {47}{88} (3+2 x)^{11/2}-\frac {3}{104} (3+2 x)^{13/2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 28, normalized size = 0.53 \begin {gather*} -\frac {(2 x+3)^{7/2} \left (2079 x^3-9891 x^2-16429 x-5829\right )}{9009} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/9009*((3 + 2*x)^(7/2)*(-5829 - 16429*x - 9891*x^2 + 2079*x^3))

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IntegrateAlgebraic [A]  time = 0.05, size = 49, normalized size = 0.92 \begin {gather*} \frac {-2079 (2 x+3)^{13/2}+38493 (2 x+3)^{11/2}-109109 (2 x+3)^{9/2}+83655 (2 x+3)^{7/2}}{72072} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(83655*(3 + 2*x)^(7/2) - 109109*(3 + 2*x)^(9/2) + 38493*(3 + 2*x)^(11/2) - 2079*(3 + 2*x)^(13/2))/72072

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fricas [A]  time = 0.40, size = 39, normalized size = 0.74 \begin {gather*} -\frac {1}{9009} \, {\left (16632 \, x^{6} - 4284 \, x^{5} - 375242 \, x^{4} - 1116057 \, x^{3} - 1364067 \, x^{2} - 758349 \, x - 157383\right )} \sqrt {2 \, x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9009*(16632*x^6 - 4284*x^5 - 375242*x^4 - 1116057*x^3 - 1364067*x^2 - 758349*x - 157383)*sqrt(2*x + 3)

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giac [A]  time = 0.19, size = 37, normalized size = 0.70 \begin {gather*} -\frac {3}{104} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} + \frac {47}{88} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} - \frac {109}{72} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + \frac {65}{56} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/104*(2*x + 3)^(13/2) + 47/88*(2*x + 3)^(11/2) - 109/72*(2*x + 3)^(9/2) + 65/56*(2*x + 3)^(7/2)

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maple [A]  time = 0.00, size = 25, normalized size = 0.47 \begin {gather*} -\frac {\left (2079 x^{3}-9891 x^{2}-16429 x -5829\right ) \left (2 x +3\right )^{\frac {7}{2}}}{9009} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9009*(2079*x^3-9891*x^2-16429*x-5829)*(2*x+3)^(7/2)

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maxima [A]  time = 0.50, size = 37, normalized size = 0.70 \begin {gather*} -\frac {3}{104} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} + \frac {47}{88} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} - \frac {109}{72} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + \frac {65}{56} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/104*(2*x + 3)^(13/2) + 47/88*(2*x + 3)^(11/2) - 109/72*(2*x + 3)^(9/2) + 65/56*(2*x + 3)^(7/2)

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mupad [B]  time = 0.04, size = 37, normalized size = 0.70 \begin {gather*} \frac {65\,{\left (2\,x+3\right )}^{7/2}}{56}-\frac {109\,{\left (2\,x+3\right )}^{9/2}}{72}+\frac {47\,{\left (2\,x+3\right )}^{11/2}}{88}-\frac {3\,{\left (2\,x+3\right )}^{13/2}}{104} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(65*(2*x + 3)^(7/2))/56 - (109*(2*x + 3)^(9/2))/72 + (47*(2*x + 3)^(11/2))/88 - (3*(2*x + 3)^(13/2))/104

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sympy [B]  time = 1.85, size = 100, normalized size = 1.89 \begin {gather*} - \frac {24 x^{6} \sqrt {2 x + 3}}{13} + \frac {68 x^{5} \sqrt {2 x + 3}}{143} + \frac {53606 x^{4} \sqrt {2 x + 3}}{1287} + \frac {372019 x^{3} \sqrt {2 x + 3}}{3003} + \frac {151563 x^{2} \sqrt {2 x + 3}}{1001} + \frac {84261 x \sqrt {2 x + 3}}{1001} + \frac {17487 \sqrt {2 x + 3}}{1001} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-24*x**6*sqrt(2*x + 3)/13 + 68*x**5*sqrt(2*x + 3)/143 + 53606*x**4*sqrt(2*x + 3)/1287 + 372019*x**3*sqrt(2*x +
3)/3003 + 151563*x**2*sqrt(2*x + 3)/1001 + 84261*x*sqrt(2*x + 3)/1001 + 17487*sqrt(2*x + 3)/1001

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