Integrand size = 27, antiderivative size = 52 \[ \int \frac {(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2 (3+2 x) (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (1869+2186 x)}{9 \sqrt {2+5 x+3 x^2}} \] Output:
-2/9*(3+2*x)*(121+139*x)/(3*x^2+5*x+2)^(3/2)+4/9*(1869+2186*x)/(3*x^2+5*x+ 2)^(1/2)
Time = 0.30 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.87 \[ \int \frac {(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {2+5 x+3 x^2} \left (2371+8925 x+10932 x^2+4372 x^3\right )}{3 (1+x)^2 (2+3 x)^2} \] Input:
Integrate[((5 - x)*(3 + 2*x)^2)/(2 + 5*x + 3*x^2)^(5/2),x]
Output:
(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)
Time = 0.30 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1227, 1158}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(5-x) (2 x+3)^2}{\left (3 x^2+5 x+2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 1227 |
\(\displaystyle -\frac {188}{3} \int \frac {2 x+3}{\left (3 x^2+5 x+2\right )^{3/2}}dx-\frac {2 (35 x+29) (2 x+3)^2}{3 \left (3 x^2+5 x+2\right )^{3/2}}\) |
\(\Big \downarrow \) 1158 |
\(\displaystyle \frac {376 (8 x+7)}{3 \sqrt {3 x^2+5 x+2}}-\frac {2 (2 x+3)^2 (35 x+29)}{3 \left (3 x^2+5 x+2\right )^{3/2}}\) |
Input:
Int[((5 - x)*(3 + 2*x)^2)/(2 + 5*x + 3*x^2)^(5/2),x]
Output:
(-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])
Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbo l] :> 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]
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] - Simp[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] && EqQ[Simplify[m + 2*p + 3], 0] && LtQ[p, -1]
Time = 1.56 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.58
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 | \(\frac {2 \left (4372 x^{3}+10932 x^{2}+8925 x +2371\right ) \left (x +1\right ) \left (3 x +2\right )}{3 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}\) | \(38\) |
orering | \(-\frac {2 \left (4372 x^{3}+10932 x^{2}+8925 x +2371\right ) \left (x +1\right ) \left (3 x +2\right ) \left (5-x \right )}{3 \left (-5+x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}\) | \(48\) |
default | \(-\frac {1093 \left (6 x +5\right )}{162 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {\frac {8744 x}{9}+\frac {21860}{27}}{\sqrt {3 x^{2}+5 x +2}}-\frac {787}{162 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {7 x}{9 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {4 x^{2}}{3 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}\) | \(86\) |
Input:
int((5-x)*(2*x+3)^2/(3*x^2+5*x+2)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/3*(4372*x^3+10932*x^2+8925*x+2371)/(3*x^2+5*x+2)^(3/2)
Time = 0.11 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.98 \[ \int \frac {(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (4372 \, x^{3} + 10932 \, x^{2} + 8925 \, x + 2371\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{3 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \] Input:
integrate((5-x)*(3+2*x)^2/(3*x^2+5*x+2)^(5/2),x, algorithm="fricas")
Output:
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)
\[ \int \frac {(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=- \int \left (- \frac {51 x}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {8 x^{2}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {4 x^{3}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {45}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \] Input:
integrate((5-x)*(3+2*x)**2/(3*x**2+5*x+2)**(5/2),x)
Output:
-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*s qrt(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)
Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.46 \[ \int \frac {(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {8744 \, x}{9 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} + \frac {4 \, x^{2}}{3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} + \frac {21860}{27 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {1114 \, x}{27 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {1042}{27 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \] Input:
integrate((5-x)*(3+2*x)^2/(3*x^2+5*x+2)^(5/2),x, algorithm="maxima")
Output:
8744/9*x/sqrt(3*x^2 + 5*x + 2) + 4/3*x^2/(3*x^2 + 5*x + 2)^(3/2) + 21860/2 7/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)
Time = 0.38 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.54 \[ \int \frac {(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left (4 \, {\left (1093 \, x + 2733\right )} x + 8925\right )} x + 2371\right )}}{3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \] Input:
integrate((5-x)*(3+2*x)^2/(3*x^2+5*x+2)^(5/2),x, algorithm="giac")
Output:
2/3*((4*(1093*x + 2733)*x + 8925)*x + 2371)/(3*x^2 + 5*x + 2)^(3/2)
Time = 0.12 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.92 \[ \int \frac {(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {36062\,x+8744\,x\,\left (3\,x^2+5\,x+2\right )+21872\,x^2+14226}{\sqrt {3\,x^2+5\,x+2}\,\left (27\,x^2+45\,x+18\right )} \] Input:
int(-((2*x + 3)^2*(x - 5))/(5*x + 3*x^2 + 2)^(5/2),x)
Output:
(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))
Time = 0.22 (sec) , antiderivative size = 114, normalized size of antiderivative = 2.19 \[ \int \frac {(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {78696 \sqrt {3 x^{2}+5 x +2}\, x^{3}+196776 \sqrt {3 x^{2}+5 x +2}\, x^{2}+160650 \sqrt {3 x^{2}+5 x +2}\, x +42678 \sqrt {3 x^{2}+5 x +2}-82584 \sqrt {3}\, x^{4}-275280 \sqrt {3}\, x^{3}-339512 \sqrt {3}\, x^{2}-183520 \sqrt {3}\, x -36704 \sqrt {3}}{243 x^{4}+810 x^{3}+999 x^{2}+540 x +108} \] Input:
int((5-x)*(3+2*x)^2/(3*x^2+5*x+2)^(5/2),x)
Output:
(2*(39348*sqrt(3*x**2 + 5*x + 2)*x**3 + 98388*sqrt(3*x**2 + 5*x + 2)*x**2 + 80325*sqrt(3*x**2 + 5*x + 2)*x + 21339*sqrt(3*x**2 + 5*x + 2) - 41292*sq rt(3)*x**4 - 137640*sqrt(3)*x**3 - 169756*sqrt(3)*x**2 - 91760*sqrt(3)*x - 18352*sqrt(3)))/(27*(9*x**4 + 30*x**3 + 37*x**2 + 20*x + 4))