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

Optimal. Leaf size=53 \[ -\frac {1}{8} (2 x+3)^{3/2}+\frac {47}{8} \sqrt {2 x+3}+\frac {109}{8 \sqrt {2 x+3}}-\frac {65}{24 (2 x+3)^{3/2}} \]

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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 {1}{8} (2 x+3)^{3/2}+\frac {47}{8} \sqrt {2 x+3}+\frac {109}{8 \sqrt {2 x+3}}-\frac {65}{24 (2 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-65/(24*(3 + 2*x)^(3/2)) + 109/(8*Sqrt[3 + 2*x]) + (47*Sqrt[3 + 2*x])/8 - (3 + 2*x)^(3/2)/8

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

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Mathematica [A]  time = 0.01, size = 28, normalized size = 0.53 \begin {gather*} -\frac {3 x^3-57 x^2-273 x-263}{3 (2 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(-263 - 273*x - 57*x^2 + 3*x^3)/(3 + 2*x)^(3/2)

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IntegrateAlgebraic [A]  time = 0.05, size = 40, normalized size = 0.75 \begin {gather*} \frac {-3 (2 x+3)^3+141 (2 x+3)^2+327 (2 x+3)-65}{24 (2 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-65 + 327*(3 + 2*x) + 141*(3 + 2*x)^2 - 3*(3 + 2*x)^3)/(24*(3 + 2*x)^(3/2))

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fricas [A]  time = 0.38, size = 36, normalized size = 0.68 \begin {gather*} -\frac {{\left (3 \, x^{3} - 57 \, x^{2} - 273 \, x - 263\right )} \sqrt {2 \, x + 3}}{3 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*x^3 - 57*x^2 - 273*x - 263)*sqrt(2*x + 3)/(4*x^2 + 12*x + 9)

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giac [A]  time = 0.20, size = 33, normalized size = 0.62 \begin {gather*} -\frac {1}{8} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {47}{8} \, \sqrt {2 \, x + 3} + \frac {327 \, x + 458}{12 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(2*x + 3)^(3/2) + 47/8*sqrt(2*x + 3) + 1/12*(327*x + 458)/(2*x + 3)^(3/2)

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maple [A]  time = 0.00, size = 25, normalized size = 0.47 \begin {gather*} -\frac {3 x^{3}-57 x^{2}-273 x -263}{3 \left (2 x +3\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*x^3-57*x^2-273*x-263)/(2*x+3)^(3/2)

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maxima [A]  time = 0.58, size = 33, normalized size = 0.62 \begin {gather*} -\frac {1}{8} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {47}{8} \, \sqrt {2 \, x + 3} + \frac {327 \, x + 458}{12 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(2*x + 3)^(3/2) + 47/8*sqrt(2*x + 3) + 1/12*(327*x + 458)/(2*x + 3)^(3/2)

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mupad [B]  time = 0.05, size = 38, normalized size = 0.72 \begin {gather*} \frac {654\,x+141\,{\left (2\,x+3\right )}^2-3\,{\left (2\,x+3\right )}^3+916}{\sqrt {2\,x+3}\,\left (48\,x+72\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(654*x + 141*(2*x + 3)^2 - 3*(2*x + 3)^3 + 916)/((2*x + 3)^(1/2)*(48*x + 72))

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sympy [B]  time = 0.66, size = 102, normalized size = 1.92 \begin {gather*} - \frac {3 x^{3}}{6 x \sqrt {2 x + 3} + 9 \sqrt {2 x + 3}} + \frac {57 x^{2}}{6 x \sqrt {2 x + 3} + 9 \sqrt {2 x + 3}} + \frac {273 x}{6 x \sqrt {2 x + 3} + 9 \sqrt {2 x + 3}} + \frac {263}{6 x \sqrt {2 x + 3} + 9 \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)/(3+2*x)**(5/2),x)

[Out]

-3*x**3/(6*x*sqrt(2*x + 3) + 9*sqrt(2*x + 3)) + 57*x**2/(6*x*sqrt(2*x + 3) + 9*sqrt(2*x + 3)) + 273*x/(6*x*sqr
t(2*x + 3) + 9*sqrt(2*x + 3)) + 263/(6*x*sqrt(2*x + 3) + 9*sqrt(2*x + 3))

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