\(\int \frac {(5-x) (3+2 x)^2}{(2+5 x+3 x^2)^{5/2}} \, dx\) [2518]

Optimal result

Integrand size = 27, antiderivative size = 54 \[ \int \frac {(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2 (3+2 x)^2 (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {376 (7+8 x)}{3 \sqrt {2+5 x+3 x^2}} \]

[Out]

-2/3*(3+2*x)^2*(29+35*x)/(3*x^2+5*x+2)^(3/2)+376/3*(7+8*x)/(3*x^2+5*x+2)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {818, 650} \[ \int \frac {(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {376 (8 x+7)}{3 \sqrt {3 x^2+5 x+2}}-\frac {2 (2 x+3)^2 (35 x+29)}{3 \left (3 x^2+5 x+2\right )^{3/2}} \]

[In]

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

[Out]

(-2*(3 + 2*x)^2*(29 + 35*x))/(3*(2 + 5*x + 3*x^2)^(3/2)) + (376*(7 + 8*x))/(3*Sqrt[2 + 5*x + 3*x^2])

Rule 650

Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[-2*((b*d - 2*a*e + (2*c*
d - b*e)*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] &&
NeQ[b^2 - 4*a*c, 0]

Rule 818

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*((b*f - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c))), x] - Dist[m*
((b*(e*f + d*g) - 2*(c*d*f + a*e*g))/((p + 1)*(b^2 - 4*a*c))), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)
, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0] && LtQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (3+2 x)^2 (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {188}{3} \int \frac {3+2 x}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx \\ & = -\frac {2 (3+2 x)^2 (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {376 (7+8 x)}{3 \sqrt {2+5 x+3 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.83 \[ \int \frac {(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {2+5 x+3 x^2} \left (2371+8925 x+10932 x^2+4372 x^3\right )}{3 (1+x)^2 (2+3 x)^2} \]

[In]

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

[Out]

(2*Sqrt[2 + 5*x + 3*x^2]*(2371 + 8925*x + 10932*x^2 + 4372*x^3))/(3*(1 + x)^2*(2 + 3*x)^2)

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.56

method	result	size
trager	\(\frac {\frac {8744}{3} x^{3}+7288 x^{2}+5950 x +\frac {4742}{3}}{\left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}\)	\(30\)
risch	\(\frac {\frac {8744}{3} x^{3}+7288 x^{2}+5950 x +\frac {4742}{3}}{\left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}\)	\(30\)
gosper	\(\frac {2 \left (4372 x^{3}+10932 x^{2}+8925 x +2371\right ) \left (1+x \right ) \left (2+3 x \right )}{3 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}\)	\(38\)
default	\(-\frac {1093 \left (5+6 x \right )}{162 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {\frac {21860}{27}+\frac {8744 x}{9}}{\sqrt {3 x^{2}+5 x +2}}-\frac {787}{162 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {4 x^{2}}{3 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {7 x}{9 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}\)	\(86\)

[In]

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

[Out]

2/3*(4372*x^3+10932*x^2+8925*x+2371)/(3*x^2+5*x+2)^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.94 \[ \int \frac {(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (4372 \, x^{3} + 10932 \, x^{2} + 8925 \, x + 2371\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{3 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]

[In]

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

[Out]

2/3*(4372*x^3 + 10932*x^2 + 8925*x + 2371)*sqrt(3*x^2 + 5*x + 2)/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)

Sympy [F]

\[ \int \frac {(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=- \int \left (- \frac {51 x}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {8 x^{2}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {4 x^{3}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {45}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]

[In]

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

[Out]

-Integral(-51*x/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x +
2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-8*x**2/(9*x**4*sqrt(3*x**2 + 5*x
+ 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(
3*x**2 + 5*x + 2)), x) - Integral(4*x**3/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 37*
x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-45/(9*x*
*4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2
+ 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.41 \[ \int \frac {(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {8744 \, x}{9 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} + \frac {4 \, x^{2}}{3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} + \frac {21860}{27 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {1114 \, x}{27 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {1042}{27 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \]

[In]

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

[Out]

8744/9*x/sqrt(3*x^2 + 5*x + 2) + 4/3*x^2/(3*x^2 + 5*x + 2)^(3/2) + 21860/27/sqrt(3*x^2 + 5*x + 2) - 1114/27*x/
(3*x^2 + 5*x + 2)^(3/2) - 1042/27/(3*x^2 + 5*x + 2)^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.52 \[ \int \frac {(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left (4 \, {\left (1093 \, x + 2733\right )} x + 8925\right )} x + 2371\right )}}{3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \]

[In]

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

[Out]

2/3*((4*(1093*x + 2733)*x + 8925)*x + 2371)/(3*x^2 + 5*x + 2)^(3/2)

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.89 \[ \int \frac {(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {36062\,x+8744\,x\,\left (3\,x^2+5\,x+2\right )+21872\,x^2+14226}{\sqrt {3\,x^2+5\,x+2}\,\left (27\,x^2+45\,x+18\right )} \]

[In]

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

[Out]

(36062*x + 8744*x*(5*x + 3*x^2 + 2) + 21872*x^2 + 14226)/((5*x + 3*x^2 + 2)^(1/2)*(45*x + 27*x^2 + 18))