3.2526 \(\int (5-x) (3+2 x)^{7/2} (2+5 x+3 x^2) \, dx\)

Optimal. Leaf size=53 \[ -\frac{1}{40} (2 x+3)^{15/2}+\frac{47}{104} (2 x+3)^{13/2}-\frac{109}{88} (2 x+3)^{11/2}+\frac{65}{72} (2 x+3)^{9/2} \]

[Out]

(65*(3 + 2*x)^(9/2))/72 - (109*(3 + 2*x)^(11/2))/88 + (47*(3 + 2*x)^(13/2))/104 - (3 + 2*x)^(15/2)/40

________________________________________________________________________________________

Rubi [A]  time = 0.015605, 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} \[ -\frac{1}{40} (2 x+3)^{15/2}+\frac{47}{104} (2 x+3)^{13/2}-\frac{109}{88} (2 x+3)^{11/2}+\frac{65}{72} (2 x+3)^{9/2} \]

Antiderivative was successfully verified.

[In]

Int[(5 - x)*(3 + 2*x)^(7/2)*(2 + 5*x + 3*x^2),x]

[Out]

(65*(3 + 2*x)^(9/2))/72 - (109*(3 + 2*x)^(11/2))/88 + (47*(3 + 2*x)^(13/2))/104 - (3 + 2*x)^(15/2)/40

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)^{7/2} \left (2+5 x+3 x^2\right ) \, dx &=\int \left (\frac{65}{8} (3+2 x)^{7/2}-\frac{109}{8} (3+2 x)^{9/2}+\frac{47}{8} (3+2 x)^{11/2}-\frac{3}{8} (3+2 x)^{13/2}\right ) \, dx\\ &=\frac{65}{72} (3+2 x)^{9/2}-\frac{109}{88} (3+2 x)^{11/2}+\frac{47}{104} (3+2 x)^{13/2}-\frac{1}{40} (3+2 x)^{15/2}\\ \end{align*}

Mathematica [A]  time = 0.01342, size = 28, normalized size = 0.53 \[ -\frac{(2 x+3)^{9/2} \left (1287 x^3-5841 x^2-10269 x-3727\right )}{6435} \]

Antiderivative was successfully verified.

[In]

Integrate[(5 - x)*(3 + 2*x)^(7/2)*(2 + 5*x + 3*x^2),x]

[Out]

-((3 + 2*x)^(9/2)*(-3727 - 10269*x - 5841*x^2 + 1287*x^3))/6435

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 25, normalized size = 0.5 \begin{align*} -{\frac{1287\,{x}^{3}-5841\,{x}^{2}-10269\,x-3727}{6435} \left ( 3+2\,x \right ) ^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3+2*x)^(7/2)*(3*x^2+5*x+2),x)

[Out]

-1/6435*(1287*x^3-5841*x^2-10269*x-3727)*(3+2*x)^(9/2)

________________________________________________________________________________________

Maxima [A]  time = 0.97649, size = 50, normalized size = 0.94 \begin{align*} -\frac{1}{40} \,{\left (2 \, x + 3\right )}^{\frac{15}{2}} + \frac{47}{104} \,{\left (2 \, x + 3\right )}^{\frac{13}{2}} - \frac{109}{88} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} + \frac{65}{72} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^(7/2)*(3*x^2+5*x+2),x, algorithm="maxima")

[Out]

-1/40*(2*x + 3)^(15/2) + 47/104*(2*x + 3)^(13/2) - 109/88*(2*x + 3)^(11/2) + 65/72*(2*x + 3)^(9/2)

________________________________________________________________________________________

Fricas [A]  time = 2.00177, size = 166, normalized size = 3.13 \begin{align*} -\frac{1}{6435} \,{\left (20592 \, x^{7} + 30096 \, x^{6} - 447048 \, x^{5} - 2029120 \, x^{4} - 3733305 \, x^{3} - 3496257 \, x^{2} - 1636821 \, x - 301887\right )} \sqrt{2 \, x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^(7/2)*(3*x^2+5*x+2),x, algorithm="fricas")

[Out]

-1/6435*(20592*x^7 + 30096*x^6 - 447048*x^5 - 2029120*x^4 - 3733305*x^3 - 3496257*x^2 - 1636821*x - 301887)*sq
rt(2*x + 3)

________________________________________________________________________________________

Sympy [B]  time = 5.02764, size = 116, normalized size = 2.19 \begin{align*} - \frac{16 x^{7} \sqrt{2 x + 3}}{5} - \frac{304 x^{6} \sqrt{2 x + 3}}{65} + \frac{49672 x^{5} \sqrt{2 x + 3}}{715} + \frac{405824 x^{4} \sqrt{2 x + 3}}{1287} + \frac{248887 x^{3} \sqrt{2 x + 3}}{429} + \frac{388473 x^{2} \sqrt{2 x + 3}}{715} + \frac{181869 x \sqrt{2 x + 3}}{715} + \frac{33543 \sqrt{2 x + 3}}{715} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)**(7/2)*(3*x**2+5*x+2),x)

[Out]

-16*x**7*sqrt(2*x + 3)/5 - 304*x**6*sqrt(2*x + 3)/65 + 49672*x**5*sqrt(2*x + 3)/715 + 405824*x**4*sqrt(2*x + 3
)/1287 + 248887*x**3*sqrt(2*x + 3)/429 + 388473*x**2*sqrt(2*x + 3)/715 + 181869*x*sqrt(2*x + 3)/715 + 33543*sq
rt(2*x + 3)/715

________________________________________________________________________________________

Giac [A]  time = 1.10495, size = 50, normalized size = 0.94 \begin{align*} -\frac{1}{40} \,{\left (2 \, x + 3\right )}^{\frac{15}{2}} + \frac{47}{104} \,{\left (2 \, x + 3\right )}^{\frac{13}{2}} - \frac{109}{88} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} + \frac{65}{72} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^(7/2)*(3*x^2+5*x+2),x, algorithm="giac")

[Out]

-1/40*(2*x + 3)^(15/2) + 47/104*(2*x + 3)^(13/2) - 109/88*(2*x + 3)^(11/2) + 65/72*(2*x + 3)^(9/2)